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Confidence intervals construction for difference of two means with incomplete correlated data

BMC Medical Research MethodologyBMC series – open, inclusive and trusted201616:31

https://doi.org/10.1186/s12874-016-0125-3

Received: 23 July 2015

Accepted: 10 February 2016

Published: 11 March 2016

Abstract

Background

Incomplete data often arise in various clinical trials such as crossover trials, equivalence trials, and pre and post-test comparative studies. Various methods have been developed to construct confidence interval (CI) of risk difference or risk ratio for incomplete paired binary data. But, there is little works done on incomplete continuous correlated data. To this end, this manuscript aims to develop several approaches to construct CI of the difference of two means for incomplete continuous correlated data.

Methods

Large sample method, hybrid method, simple Bootstrap-resampling method based on the maximum likelihood estimates (B 1) and Ekbohm’s unbiased estimator (B 2), and percentile Bootstrap-resampling method based on the maximum likelihood estimates (B 3) and Ekbohm’s unbiased estimator (B 4) are presented to construct CI of the difference of two means for incomplete continuous correlated data. Simulation studies are conducted to evaluate the performance of the proposed CIs in terms of empirical coverage probability, expected interval width, and mesial and distal non-coverage probabilities.

Results

Empirical results show that the Bootstrap-resampling-based CIs B 1, B 2, B 4 behave satisfactorily for small to moderate sample sizes in the sense that their coverage probabilities could be well controlled around the pre-specified nominal confidence level and the ratio of their mesial non-coverage probabilities to the non-coverage probabilities could be well controlled in the interval [0.4, 0.6].

Conclusions

If one would like a CI with the shortest interval width, the Bootstrap-resampling-based CIs B 1 is the optimal choice.

Keywords

BootstrapConfidence intervalCorrelated dataIncomplete data

Background

Incomplete data often arise in various research fields such as crossover trials, equivalence trials, and pre and post-test comparative studies. For instance, ([1] pp. 212) designed a crossover clinical trial to measure the onset of action of two doses of formoterol solution aerosol: 12 ug and 24 ug. In this study, twenty-four patients were randomly allocated in equal numbers to one of the six possible sequences of two treatments at a time. Each patient was received two aerosols at each of visits 2 and 4. After four weeks, researchers measured the forced expiratory volume of a second (FEV1) indicators for twenty-four patients. Due to the fact that researches did not consider all possible combinations of three treatments (e.g., placebo, 12 ug and 24 ug aerosols), which indicates that the missing data mechanism is missing completely at random (MCAR) thus FEV1 was only observed for 7 patients under both treatments (e.g., 12 ug and 24 ug aerosols), 9 patients only for 12 ug aerosol, and 8 patients only for 24 ug aerosol. The resultant data are shown in Table 1, which consist of two parts: the complete observations and the incomplete observations.
Table 1

FEV1 indicators of patients for 12 ug and 24 ug formoterol solution aerosol

12 ug(x 1)

24 ug(x 2)

2.250

2.700

0.925

0.900

1.010

1.270

2.100

2.150

2.500

2.450

1.750

1.725

1.370

1.120

3.400

 

2.250

 

1.460

 

1.480

 

2.050

 

3.500

 

2.650

 

2.190

 

0.840

 
 

1.750

 

2.525

 

1.080

 

3.120

 

3.100

 

2.700

 

1.870

 

0.940

For the above crossover clinical trial, our main interest is to test the equivalence between 12 ug and 24 ug formoterol solution aerosols with respect to the FEV1 value. To this end, we can construct a (1−α)100 % confidence interval for the difference of two FEV1 values. If the resultant confidence interval (CI) lies entirely in the interval (−δ 0,δ 0) with δ 0(>0) being some pre-specified clinical acceptable threshold, we thus could conclude the equivalence between two doses of formoterol solution aerosol at the α significance level. As a result, reliable CIs for the difference in the presence of incomplete data are necessary.

The problem of testing the equality and constructing CI for the difference of two correlated proportions in the presence of incomplete paired binary data has received considerable attention in past years. For example, ones can refer to [26] for the large sample method, and [7] for the corrected profile likelihood method. When sample size is small, [8] proposed the exact unconditional test procedure for testing equality of two correlated proportions with incomplete correlated data. Tang, Ling and Tian [9] developed the exact unconditional and approximate unconditional CIs for proportion difference in the presence of incomplete paired binary data. Lin et al. [10] presented a Bayesian method to test equality of two correlated proportions with incomplete correlated data. Li et al. [11] discussed the confidence interval construction for rate ratio in matched-pair studies with incomplete data. However, all the aforementioned methods were developed for incomplete paired binary data.

Statistical inference on the difference of two means with incomplete correlated data has received a limited attention. For example, [12] discussed the problem of testing the equality of two means with missing data on one response and recommended [13] statistic when the variances were not too different. Lin and Stivers [14] also gave a similar comparison. Lin and Stivers [15] and [12] suggested some test statistics for testing the equality of two means with incomplete data on both response. However, to our knowledge, little work has been done on CI construction for the difference of two means with incomplete correlated data under the MCAR assumption.

Inspired by [1619], we develop several CIs for the difference of two means with incomplete correlated data under the MCAR assumption based on the large sample method, hybrid method and Bootstrap-resampling method. The presented Bootstrap-resampling CIs have not been considered in the literature related to missing observations.

The rest of this article is organized as follows. Several methods are presented to construct CIs for the difference of the two means with incomplete correlated data in Section “Methods”. Simulation studies and an example are conducted to evaluate the finite performance of the proposed CIs in terms of coverage probability, expected interval width, and mesial and distal non-coverage probabilities in Section “Results”. A brief discussion is given in Section “Discussion”. Some concluding remarks are given in Section “Conclusion”.

Methods

Suppose that x=(x 1,x 2) is a 2×1 vector of random variables, and follows a distribution with mean μ and covariance matrix Σ given by
$$\boldsymbol{\mu}=\left(\begin{array}{l} \mu_{1} \\ \mu_{2} \end{array} \right) \,\text{and}\,\, \vspace*{-4pt} \boldsymbol{\Sigma}=\left(\begin{array}{ll} {\sigma_{1}^{2}}&\, \rho\sigma_{1}\sigma_{2}\\ \rho\sigma_{1}\sigma_{2}& \,{\sigma_{2}^{2}} \end{array} \right), $$
respectively. Let {(x 1m ,x 2m ):m=1,,n} be n paired observations on x 1 and x 2, \(\left \{x_{1,n+1}, \cdots, x_{1,n+n_{1}}\right \}\) be n 1 additional observations on x 1, \(\left \{x_{2,n+1}, \cdots, x_{2,n+n_{2}}\right \}\) be n 2 additional observations on x 2. Thus, there are n 1 missing observations on x 2, and n 2 missing observations on x 1. Without loss of generality, the data may be presented as follows:
$$ \begin{array}{lll} x_{11}, \cdots, x_{1n}, & x_{1,n+1}, \cdots, x_{1,n+n_{1}},& \\ x_{21}, \cdots, x_{2n}, & &x_{2,n+1}, \cdots, x_{2,n+n_{2}}, \end{array} $$

where (x 1m ,x 2m ) is referred to as a paired observation, while x 1,n+j and x 2,n+k are referred to as incomplete or unpaired observations. Similar to [20, 21], throughout this article, it is assumed that the missing data mechanism is MCAR (i.e., independent of treatment and outcome). Based on these observations, we here want to construct reliable explicit CIs for the difference of two means δ=μ 1μ 2 under MCAR assumption.

Confidence interval based on the large sample method

To make a comparison with the following proposed methods, we assume that x follows a bivariate normal distribution in this subsection. In this case, if only variable x 1 or x 2 is subject to missingness (i.e., n 1=0 or n 2=0), one can obtain the closed forms of the maximum likelihood estimates (MLEs) of μ and Σ [22]. However, there are no closed forms of the MLEs for μ and Σ when variables x 1 and x 2 are simultaneously subject to missingness (i.e., n 1≠0 and n 2≠0), though one can find the MLEs of μ and Σ using an iterative algorithm [23]. To get the closed forms of MLEs for μ and Σ, [15] proposed the modified MLEs using a non-iterative procedure and provided several test statistics based on the obtained estimators of μ and Σ.

(i) Confidence interval based on Lin and Stivers’s test statistics

Let \(\hat {\delta }=\hat {\mu }_{1}-\hat {\mu }_{2}\) be the MLE of δ under the bivariate normal assumption of x. When Σ is known, it follows from [15] that the MLE of δ is
$$ \hat{\delta}=a\overline{x}_{1}^{(n)}+\left(1-a\right)\overline{x}_{1}^{(n_{1})}-b\overline{x}_{2}^{(n)}-(1-b)\overline{x}_{2}^{(n_{2})}, $$
and the asymptotic variance of \(\hat {\delta }\) can be expressed as
$$ \begin{aligned}\text{Var}(\hat{\delta})=~ &h\left\{\left[n+n_{2}\left(1-\rho\right)^{2}\right]\right\}{\sigma_{1}^{2}}-2n\rho\sigma_{1}\sigma_{2}\\ &\left.+\left[n+n_{1}\left(1-\rho^{2}\right)\right]{\sigma_{2}^{2}}\right\},\end{aligned} $$

respectively, where \(\overline {x}_{1}^{(n)}=\frac {1}{n}\sum _{j=1}^{n}x_{1j}\), \(\overline {x}_{2}^{(n)}=\frac {1}{n}\sum _{j=1}^{n}x_{2j}\), \(\overline {x}_{1}^{(n_{1})}=\frac {1}{n_{1}}\sum _{j=1}^{n_{1}}x_{1,n+j}\), \(\overline {x}_{2}^{(n_{2})}=\frac {1}{n_{2}}\sum _{k=1}^{n_{2}}x_{2,n+k}\), a=nh(n+n 2+n 1 β 21), b=nh(n+n 1+n 2 β 12), β 21=ρσ 2/σ 1, β 12=ρσ 1/σ 2, h=1/{(n+n 1)(n+n 2)−n 1 n 2 ρ 2}. An approximate 100(1−α) % CI of δ is given by \(\left (\hat {\delta }-\textit {z}_{\alpha /2}\sqrt {\text {Var}(\hat {\delta })}, \hat {\delta }+\textit {z}_{\alpha /2}\sqrt {\text {Var}(\hat {\delta })}\right)\), which is denoted as T w1-CI.

Following [15], when Σ is unknown, the statistic for testing H 0:δ=δ 0 versus H 1:δδ 0 is given by
$$ {}T_{1}\,=\,\frac{A\!\left(\overline{x}_{1}^{(n)}\!-\overline{x}_{1}^{(n_{1})}\right)\,-\,B\left(\overline{x}_{2}^{(n)}\!-\overline{x}_{2}^{(n_{2})}\right)\!+\overline{x}_{1}^{(n_{1})}\!-\overline{x}_{2}^{(n_{2})}\!-\delta_{0}}{\sqrt{V_{1}}}, $$

which is asymptotically distributed as t-distribution with n degrees of freedom under H 0, where V 1=[{A 2/n+(1−A)2/n 1 }m 1+{ B 2/n+(1−B)2/n 2 } m 2−2ABm 12/n]/(n−1), A={n(n+n 2+n 1 m 12/m 1}/{ (n+n 1)(n+n 2)−n 1 n 2 r 2}−1, B={n(n+n 1+n 2 m 12/ m 2} /{ (n + n 1)(n + n 2)−n 1 n 2 r 2}−1, \(m_{1}=\sum _{j=1}^{n} \left (x_{1j}-\overline {x}_{1}^{(n)}\right)^{2}\), \(m_{2}=\sum _{j=1}^{n}\left (x_{2j}-\overline {x}_{2}^{(n)}\right)^{2}\), \(m_{12}=\sum _{j=1}^{n}\left (x_{1j}\,-\,\overline {x}_{1}^{(n)}\right)\left (x_{2j}-\overline {x}_{2}^{(n)}\right)\), \(r=m_{12}/\sqrt {m_{1}m_{2}}\). Therefore, the approximate 100(1−α) % CI on the basis of T 1 is given by (L, U), where \(L=A\left (\overline {x}_{1}^{(n)}-\overline {x}_{1}^{(n_{1})}\right)-B\left (\overline {x}_{2}^{(n)}-\overline {x}_{2}^{(n_{2})}\right)+\overline {x}_{1}^{(n_{1})}-\overline {x}_{2}^{(n_{2})}-t_{\alpha /2}(n)\sqrt {V_{1}}\), and \(U=A\left (\overline {x}_{1}^{(n)}-\overline {x}_{1}^{(n_{1})}\right)-B\left (\overline {x}_{2}^{(n)}-\overline {x}_{2}^{(n_{2})}\right)+\overline {x}_{1}^{(n_{1})}-\overline {x}_{2}^{(n_{2})}+t_{\alpha /2}(n)\sqrt {V_{1}}\), which is denoted as T 1-CI.

Another test statistic defined by [15] for testing H 0:δ=δ 0 versus H 1:δδ 0, which is a generalization of [24] test statistic for two independent samples, is given by
$$ T_{2}=\frac{\bar{x}_{1}^{\left(n+n_{1}\right)}-\bar{x}_{2}^{\left(n+n_{2}\right)}-\delta_{0}}{\sqrt{h_{1}+h_{2}+h_{3}}}, $$

which is asymptotically distributed as t distribution with degrees ν of freedom, where \(\bar {x}_{1}^{(n+n_{1})}=(n+n_{1})^{-1}\sum _{j=1}^{n+n_{1}}x_{1j}\), \(\bar {x}_{2}^{(n+n_{2})}=(n+n_{2})^{-1}\sum _{j=1}^{n+n_{2}}x_{2j}\), h 1=n{(n+n 2)m 1/(n+n 1)+(n+n 1)m 2/(n+n 2)−2m 12}/{(n−1)(n+n 1)(n+n 2)}, h 2=n 1 b 1/{(n 1−1)(n+n 1)2}, h 3=n 2 b 2/{(n 2−1)(n+n 2)2}, \(b_{1}=\sum _{j=n+1}^{n+n_{1}}\left (x_{1j}-\overline {x}_{1}^{(n_{1})}\right)^{2}\), \(b_{2}=\sum _{j=n+1}^{n+n_{2}}\left (x_{2j}-\overline {x}_{2}^{(n_{1})}\right)^{2}\), and \(\nu =\left (h_{1}+h_{2}+h_{3}\right)^{2}/\{{h_{1}^{2}}/(n-1)+{h_{2}^{2}}/(n_{1}-1)+{h_{3}^{2}}/(n_{2}-1)\}\). Therefore, the approximate 100(1−α) % CI of δ for statistic T 2 is denoted as T 2-CI.

When σ 1=σ 2, it follows from [15] that the statistic for testing H 0:δ=δ 0 versus H 1:δδ 0 can be expressed as
$$ \begin{aligned} T_{3}=\left\{\bar{x}_{1}^{(n+n_{1})}-\bar{x}_{2}^{(n+n_{2})}-\delta_{0}\right\} \sqrt{\frac{(n+n_{1}+n_{2}-2)(n+n_{1})(n+n_{2})}{(b_{1}+c_{2})(2n-2nr+n_{1}+n_{2})}}, \end{aligned} $$

which is asymptotically distribution as t-distribution with degrees n+n 1+n 2−4 of freedom. Note that when n 2>n 1, b 1+c 2 should be replaced by b 2+c 1. Thus, the approximate 100(1−α) % CI of δ for T 3 is denoted as T 3-CI, where \(c_{1}=\sum _{j=1}^{n+n_{1}}\left (x_{1j}-{n+n_{1}}\sum _{j=1}^{n+n_{1}}x_{1j}\right)^{2}\), and \(c_{2}=\sum _{j=1}^{n+n_{2}}\left (x_{2j}-\frac {1}{n+n_{2}}\sum _{j=1}^{n+n_{2}}x_{2j}\right)^{2}\).

Also, [12] presented the similar but simpler test statistics for testing the mean difference δ=μ 1μ 2, which are adopted to construct CIs of δ as follows.

(ii) Confidence interval based on Ekbohm’s test statistics

Following [12], an unbiased estimator of δ is given by \(\hat {\delta }=\bar {x}_{1}^{(n+n_{1})}-\bar {x}_{2}^{(n+n_{2})}\), and its variance is given by \(\text {Var}(\hat {\delta })= \text {Var}(\hat {\mu })=\left \{(n+n_{2}){\sigma _{1}^{2}}+(n+n_{2}){\sigma _{2}^{2}}-2n\rho \sigma _{1}\sigma _{2}\right \}/\left \{(n+n_{1})(n+n_{2})\right \}\). An approximate 100(1 − α) % CI of δ can be obtained by \(\left (\hat {\delta }-\textit {z}_{\alpha /2}\sqrt {\text {Var}(\hat {\delta })}, \hat {\delta }+\right.\) \(\left.\textit {z}_{\alpha /2}\sqrt {\text {Var}(\hat {\delta })}\right),\) which is denoted as T w2-CI.

When σ 1=σ 2, Ekbothm (1976) proposed the following statistic for testing H 0: \(T_{4}=(\tilde {\delta }-\delta _{0})\sqrt {\!(n\,+\,\!n_{1})(n\,+\,\!n_{2})\,-\,n_{1}n_{2}\lambda ^{2}}/\) \( \left \{\hat \sigma \!\sqrt {2n(1-\lambda)+(n_{1}+n_{2})(1-\lambda ^{2})}\right \}\), where \(\tilde {\delta }=\left [n\left (n+n_{2}+n_{1}\lambda \right)\overline {x}_{1}^{(n)}\!-n\left (n+n_{1}+n_{2}\lambda \right)\overline {x}_{2}^{(n)}+n_{1}\!\left \{n\,+\,n_{2}\!\left (1\,-\,\lambda ^{2}\right)-\!n\lambda \right \}\overline {x}_{1}^{(n_{1})}-n_{2}\left \{n\,+\,n_{1}\!\left (1\!\,-\,\!\lambda \right)^{2}\!\,-\,n\lambda \right \}\overline {x}_{2}^{(n_{2})}\right ]\!\big /\!\!\left \{(n\,+\,n_{1})(n+n_{2})\,-\,n_{1}n_{2}\lambda ^{2}\right \}\), \(\hat {\sigma }^{2}\,=\,\left \{m_{1}\,+\,m_{2}\,+\,(1+\lambda ^{2})(b_{1}\,+\,b_{2})\right \}/\left \{2(n-\!1)+\!(1\,+\,\lambda ^{2})(n_{1}\,+\,n_{2}\,-\,2)\right \}\), and λ=2m 12/(m 1+m 2). Under H 0, T 4 is asymptotically distributed as t-distribution with degrees n of freedom. Therefore, the approximate 100(1−α) % CI is denoted as T 4-CI.

Following [12], when σ 1=σ 2, another statistic for testing H 0 can be expressed as \(T_{5}= \left (\bar {x}_{1}^{(n+n_{1})}\!-\bar {x}_{2}^{(n+n_{2})}\,-\,\delta _{0}\!\right)\!\sqrt {(n\,+\,n_{1})(n\,+\,n_{2})/(R_{1}\!\,+\,R_{2})}\), which is asymptotically distributed as t distribution with degrees ν σ of freedom under H 0, where R 1 = n(m 1 + m 2 − 2m 12) /(n − 1), R 2 =(n 1 + n 2)(b 1+b 2)/(n 1+n 2−2), and \(\nu _{\sigma }=\left (R_{1}+R_{2}\right)^{2}\!\!\!~/\left \{{R_{1}^{2}}/(n+1)+{R_{2}^{2}}/(n_{1}+n_{2})\right \}-2\). Thus, an approximate 100(1−α) % CI of δ for T 5 is denoted as T 5-CI.

Confidence interval based on the generalized estimating equations(GEEs)

To relax the bivariate normality assumption of x, the method of the generalized estimating equations (GEEs) with exchangeable working correlation structure (e.g., [25]) can be adopted to make statistical inference on δ in the incomplete correlated data because the GEE approach have become one of the most widely used methods in dealing with correlated response data [26, 27]. Following [28], the GEEs with exchangeable working correlation structure can be used to estimate parameter vector μ; the so-called sandwich variance estimator can be used to consistently estimate the covariance matrix of μ; and the ML method under a bivariate normal assumption via available paired observations is used to estimate the correlation parameter. Thus, an approximate 100(1−α) % CI of δ based on GEE method is denoted as T g -CI.

Confidence interval based on the hybrid method

When the distribution function of x is unknown, a hybrid method is developed to construct CI of δ in this subsection. We first introduce the general concept of hybrid method. Let θ 1 and θ 2 be two parameters of interest. Now our main interest is to construct a 100(1−α) % two-sided CI (L,U) of θ 1θ 2 via hybrid method. Let \(\hat {\theta }_{1}\) and \(\hat {\theta }_{2}\) be two estimates of θ 1 and θ 2, respectively; and let (l 1,u 1) and (l 2,u 2) denote two approximate 100(1−α) % CIs for θ 1 and θ 2, respectively. Under the dependent assumption on \(\hat \theta _{1}\) and \(\hat \theta _{2}\), it follows from the central limit theorem that the approximate two-sided 100(1−α) % CI of θ 1θ 2 is given by (L,U), where
$${\kern100pt}L=\hat{\theta}_{1}-\hat{\theta}_{2}-z_{\alpha/2}\sqrt{\text{Var}(\hat{\theta}_{1})+\text{Var}(\hat{\theta}_{2})-2\text{Cov}(\hat\theta_{1}, \hat\theta_{2})}, $$
$${\kern100pt}U=\hat{\theta}_{1}-\hat{\theta}_{2}+z_{\alpha/2}\sqrt{\text{Var}(\hat{\theta}_{1})+\text{Var}(\hat{\theta}_{2})-2\text{Cov}(\hat\theta_{1}, \hat\theta_{2})}. $$
Because \(\text {Cov}(\hat \theta _{1}, \hat \theta _{2})=\text {corr}(\hat \theta _{1}, \hat \theta _{2})\left \{\text {Var}(\hat {\theta }_{1})\text {Var}(\hat {\theta }_{2})\right \}^{1/2}\), the lower limit L and the upper limit U can be rewritten as
$$\begin{aligned} &L=\hat{\theta}_{1}-\hat{\theta}_{2}-z_{\alpha/2}\sqrt{\text{Var}(\hat{\theta}_{1})+\text{Var}(\hat{\theta}_{2})-2\text{corr}(\hat\theta_{1}, \hat\theta_{2})\left\{\text{Var}(\hat{\theta}_{1})\text{Var}(\hat{\theta}_{2})\right\}^{1/2}}\\ &U=\hat{\theta}_{1}-\hat{\theta}_{2}+z_{\alpha/2}\sqrt{\text{Var}(\hat{\theta}_{1})+\text{Var}(\hat{\theta}_{2})-2\text{corr}(\hat\theta_{1}, \hat\theta_{2})\left\{\text{Var}(\hat{\theta}_{1})\text{Var}(\hat{\theta}_{2})\right\}^{1/2}}, \end{aligned} $$
respectively. Note that (l 1,u 1) contains the plausible parameter values of θ 1, and (l 2,u 2) contains the plausible parameter values for θ 2. Among these plausible values for θ 1 and θ 2, the values closest to the minimum L and maximum U are respectively l 1u 2 and u 1l 2 in spirit of the score-type CI [29]. From the central limit theorem, the variance estimates can now be recovered from θ 1=l 1 as \(\widehat {\text {Var}(\hat \theta _{1})}=(\hat \theta _{1}-l_{1})^{2}/z_{\alpha /2}^{2}\) and from θ 2=u 2 as \(\widehat {\text {Var}(\hat \theta _{2})}=\left (u_{2}-\hat \theta _{2}\right)^{2}\!\!\!\big /z_{\alpha /2}^{2}\) for setting L. As a result, the lower limit L for θ 1θ 2 is
$$\begin{array}{@{}rcl@{}} \begin{aligned}L\,=\,\hat\theta_{1}\!-\hat{\theta}_{2}-\!\sqrt{\left(\hat{\theta}_{1}\!-l_{1}\right)^{2}\!\,+\,\left(u_{2}-\hat{\theta}_{2}\right)^{2}\!\,-\,2\widehat{\text{corr}}\!\left(\hat{\theta}_{1}, \hat{\theta}_{2}\right)\!\!\left(\hat{\theta}_{1}\,-\,l_{1}\right)\!\!\left(u_{2}\,-\,\hat{\theta}_{2}\right)}\end{aligned} \end{array} $$
(1)
Similarly, we can obtain
$$\begin{array}{@{}rcl@{}} \begin{aligned}U\,=\,\hat{\theta}_{1}-\hat{\theta}_{2}\,+\,\sqrt{\!\left(u_{1}\!-\hat{\theta}_{1}\right)^{2}\!\,+\,\left(\hat{\theta}_{2}\,-\,l_{2}\right)^{2}\,-\,2\widehat{\text{corr}}\!\left(\hat{\theta}_{1}, \hat{\theta}_{2}\right)\!\!\left(u_{1}\!-\hat{\theta}_{1}\right)\!\!\left(\hat{\theta}_{2}\,-\,l_{2}\right)}\end{aligned} \end{array} $$
(2)

To obtain the above presented approximate 100(1−α) % hybrid CI for μ 1μ 2, one requires evaluating the (1−α) 100 % CIs of θ 1 = μ 1 (denoted as (l 1, u 1)) and θ 2=μ 2 (denoted as (l 2, u 2)), and estimating the correlation coefficient \(\widehat {\text {corr}}(\hat {\theta }_{1}, \hat \theta _{2})\). For the former, following [19], we consider the following two methods for getting the confidence limits (l 1, u 1) and (l 2, u 2) of θ 1 and θ 2.

(i) The Wilson score method
$$ l_{i}=\tilde{\theta}_{i}-\frac{z_{\alpha/2}}{N_{i}+z_{\alpha/2}^{2}}\sqrt{\frac{n}{n-1}{\sum}_{j=1}^{n}\left(x_{ij}-\hat{\theta}_{i}\right)^{2}+\frac{z_{\alpha/2}^{2}}{4}}, $$
$$ u_{i}=\tilde{\theta}_{i}+\frac{z_{\alpha/2}}{N_{i}+z_{\alpha/2}^{2}}\sqrt{\frac{n}{n-1}{\sum}_{j=1}^{n}\left(x_{ij}-\hat{\theta}_{i}\right)^{2}+\frac{z_{\alpha/2}^{2}}{4}}, $$

where N i =n+n i and \(\hat {\theta }_{i}=\frac {1}{N_{i}}\sum _{j=1}^{N_{i}}x_{ij}\) for i=1,2.

(ii)The Agresti-coull method
$$ {\small{\begin{aligned} {\kern20pt}l_{i}=\tilde{\theta}_{i}-z_{\alpha/2}\sqrt{\frac{\sum_{j=1}^{n}(x_{ij}-\hat{\theta}_{i})^{2}}{\left(N_{i}+z_{\alpha/2}^{2}\right)(n-1)}}, \end{aligned}}} $$
$$\kern20pt {\small{\begin{aligned} u_{i}=\tilde{\theta}_{i}+z_{\alpha/2}\sqrt{\frac{\sum_{j=1}^{n}(x_{ij}-\hat{\theta}_{i})^{2}}{\left(N_{i}+z_{\alpha/2}^{2}\right)(n-1)}}, \end{aligned}}} $$

where N i =n+n i and \(\tilde {\theta }_{i}=\left (\sum _{j=1}^{N_{i}}x_{ij}+0.5z_{\alpha /2}^{2}\right)/\left (N_{i}+z_{\alpha /2}^{2}\right)\) for i=1,2.

To construct CI for δ=μ 1μ 2 via the above described hybrid method, we can simply set θ 1=μ 1 and θ 2=μ 2. If Σ is known, the estimated correlation coefficient \(\widehat {\text {corr}}(\hat {\mu }_{1}, \hat {\mu }_{2})\) of \(\hat {\mu }_{1}\) and \(\hat {\mu }_{2}\) is given by \(\widehat {\text {corr}}(\hat {\mu }_{1}, \hat {\mu }_{2})=2n\rho /\sqrt {(n+n_{1})(n+n_{2})}\). If Σ is unknown, \(\widehat {\text {corr}}(\hat {\mu }_{1}, \hat {\mu }_{2})\) is given by \(\widehat {\text {corr}}(\hat {\mu }_{1}, \hat {\mu }_{2})=nr/\left \{(n+n_{1})(n+n_{2})-n_{1}n_{2}r^{2}\right \}\), where \(r=m_{12}/\sqrt {m_{1}m_{2}}\), \(m_{1}=\sum _{j=1}^{n}\left (x_{1j}-\overline {x}_{1}^{(n)}\right)^{2}\) and \(m_{2}=\sum _{j=1}^{n}\left (x_{2j}-\overline {x}_{2}^{(n)}\right)^{2}.\) Thus, using Eqs. (1) and (2) yields CIs of δ=μ 1μ 2. When l i and u i are estimated by the Wilson score method, we denote the corresponding CI as W s -CI; when l i and u i are estimated by the Agresti-coull method, the corresponding CI is denoted as W a -CI.

Bootstrap-resampling-based confidence intervals

When the distribution of x is known, one can obtain the approximate CIs of δ based on the asymptotic distributions of the constructed test statistics under the null hypotheses H 0:δ=δ 0. However, when the distribution of x is unknown, the asymptotic distributions of the constructed test statistics may not be reliable, especially with small sample size. On the other hand, estimators of some nuisance parameters have not the closed-form solutions even if the approximate distribution is reliable, and they must be obtained by using some iterative algorithms, which are computationally intensive. In this case, the Bootstrap method is often adopted to construct CIs of parameter of interest. The Bootstrap CIs can be constructed via the following steps.

Step 1. Given the paired observations and incomplete observations
$${} D=\left(\!\! \begin{array}{lll} x_{11}, \cdots, x_{1n}, & x_{1,n+\!1}, \cdots, x_{1,n+n_{1}},& \\ x_{21}, \cdots, x_{2n}, & &\!x_{2,n+\!1}, \cdots, x_{2,n+n_{2}} \end{array} \!\!\right)$$
we draw n paired observations \(\left \{(x_{1m}^{*},x_{2m}^{*}):\! m=1, \cdots,n\right \}\) with replacement from n paired observations {(x 11,x 21),,(x 1n ,x 2n )}, generate n 1 observations \(\{x_{1,n+j}^{*}: j=1, \cdots, n_{1}\}\) with replacement from \(\left \{x_{1,n+1}, \cdots, x_{1,n+n_{1}}\right \}\), and sample n 2 observations \(\left \{x_{2,n+k}^{*}: k=1, \cdots, n_{2}\right \}\) with replacement from \(\left \{x_{2,n+1}, \cdots, x_{2,n+n_{1}}\right \}\). Thus, we obtain the following Bootstrap resampling sample
$$D_{b}^{*}\,=\,\left(\! \begin{array}{lll} x_{11}^{*}, \cdots, x_{1n}^{*}, & x_{1,n+\!1}^{*}, \cdots, x_{1,n+n_{1}}^{*},& \\ x_{21}^{*}, \cdots, x_{2n}^{*}, & &,x_{2,n+\!1}^{*}, \cdots, x_{2,n+n_{2}}^{*} \end{array} \!\right). $$

Step 2. For the above generated Bootstrap resampling sample \(D_{b}^{*}\), we first compute \(\hat {\mu }_{1}^{*}=(n+n_{1})^{-1}\sum _{j=1}^{n+n_{1}}x_{1j}^{*}\) and \(\hat {\mu }_{2}^{*}=(n+n_{2})^{-1}\sum _{j=1}^{n+n_{2}}x_{2j}^{*}\), and then calculate the estimated value \(\hat {\delta }^{*}\) of δ via \(\hat {\delta }^{*}=\hat {\mu }_{1}^{*}-\hat {\mu }_{2}^{*}\).

Step 3. Repeating the above steps 1 and 2 for a total of G times yields G Bootstrap estimates \(\left \{\hat {\delta }_{g}^{*}: g=1,2,\cdots,G\right \}\) of δ. Let \(\hat \delta _{(1)}^{*}<\hat \delta _{(2)}<\cdots <\hat \delta _{(G)}^{*}\) be the ordered values of \(\left \{\hat \delta _{g}^{*}: g=1,2,\cdots,G\right \}\).

Step 4. Based on the bootstrap estimates \(\left \{\hat {\delta }_{g}^{*}, g=1,2,\ldots,G\vphantom {\left \{\hat {\delta }_{g}^{*}, g=\right.}\right \}\), Bootstrap-resampling-based CIs for δ can be constructed as follows.

Generally, the standard error se \((\hat {\delta })\) of \(\hat {\delta }\) can be estimated by the sample standard deviation of the G replications, i.e., \(\hat {\text {se}}(\hat {\delta })=\sqrt {(G-1)^{-1}\sum _{g=1}^{G}\left (\hat {\delta }_{g}^{*}-\bar {\delta }_{B}^{*}\right)^{2}}\), where \(\bar \delta _{B}^{*}=\left (\hat {\delta }_{1}^{*}+\cdots +\hat {\delta }_{G}^{*}\right)/G\). If \(\left \{\hat {\delta }_{g}^{*}: g=1,\cdots, G\right \}\) is approximately normally distributed, an approximate 100(1−α) % Bootstrap CI for δ is given by \(\left (\hat {\delta }-z_{\alpha /2}\hat {\text {se}}(\hat {\delta }), \hat {\delta }+z_{\alpha /2}\hat {\text {se}}(\hat \delta)\right)\), where z α/2 is the upper α/2-percentile of the standard normal distribution, which is referred as the simple Bootstrap confidence interval. When \(\hat {\delta }=a\overline {x}_{1}^{(n)}+(1-a)\overline {x}_{1}^{(n_{1})}-b\overline {x}_{2}^{(n)}-(1-b)\overline {x}_{2}^{(n_{2})}\), the corresponding simple Bootstrap CI is denoted as B 1. When \(\hat {\delta }=\bar {x}_{1}^{(n+n_{1})}-\bar {x}_{2}^{(n+n_{2})}\), the corresponding simple Bootstrap CI is denoted as B 2.

Alternatively, if \(\left \{\hat {\delta }_{g}^{*}: g=1,\cdots, G\right \}\) is not normally distributed, it follows from ([16] p.132) that the approximate 100(1−α) % Bootstrap-resampling-based percentile CI for δ is \(\left (\hat \delta _{\left ([G\alpha /2]\right)}^{*},\hat {\delta }_{([G(1-\alpha /2)])}^{*}\right)\), where [ a] represents the integer part of a, which is referred as the percentile Bootstrap CI. When \(\hat \delta =a\overline {x}_{1}^{(n)}+(1-a)\overline {x}_{1}^{(n_{1})}-b\overline {x}_{2}^{(n)}-\left (1-b\right)\overline {x}_{2}^{(n_{2})}\), the corresponding percentile Bootstrap CI is denoted as B 3. When \(\hat {\delta }=\bar {x}_{1}^{(n+n_{1})}-\bar {x}_{2}^{(n+n_{2})}\), the corresponding percentile Bootstrap CI is denoted as B 4.

Results

Simulation studies

In this subsection, we investigate the finite performance of various CIs in terms of empirical coverage probability (ECP), empirical confidence widths (ECW), and distal and mesial non-coverage probabilities (DNP and MNP) in various parameter settings via Monte Carlo simulation studies. A summary of abbreviation for various confidence intervals is presented in Table 2.
Table 2

Summary of various abbreviations

Abbreviation

Definition

T 1

CI based on T 1 statistic

T 2

CI based on T 2 statistic

T 3

CI based on T 3 statistic

T 4

CI based on T 4 statistic

T 5

CI based on T 5 statistic

T g

CI based on GEE method

W s

CI based on Wilson score method

W a

CI based on Agresti-coull method

B 1

Simple Bootstrap CI based on

 

\(\hat \delta =a\overline {x}_{1}^{(n)}+\left (1-a\right)\overline {x}_{1}^{(n_{1})}-b\overline {x}_{2}^{(n)}-(1-b)\overline {x}_{2}^{(n_{2})}\)

B 2

Simple Bootstrap CI based on \(\hat {\delta }=\bar {x}_{1}^{(n+n_{1})}-\bar {x}_{2}^{(n+n_{2})}\)

B 3

Percentile Bootstrap CI based on

 

\(\hat \delta =a\overline {x}_{1}^{(n)}+(1-a)\overline {x}_{1}^{(n_{1})}-b\overline {x}_{2}^{(n)}-(1-b)\overline {x}_{2}^{(n_{2})}\)

B 4

Percentile Bootstrap CI based on \(\hat {\delta }=\bar {x}_{1}^{(n+n_{1})}-\bar {x}_{2}^{(n+n_{2})}\)

ECPs

Empirical coverage probabilities, is defined by Eq. (3)

ECW

Empirical confidence widths, is defined by Eq. (3)

RNCP

The ratio of the mesial non-coverage probabilities to the

 

non-coverage probabilities, is defined by Eqs. (4) and (5)

In the first simulation study, we consider the following case that (n,n 1,n 2) is set to be (5,2,2); μ 1=0,1,2; μ 2=0.25,1,1.5; ρ=−0.9,−0.5,−0.1,0,0.1,0.5,0.9; δ=μ 1μ 2=−0.25,0,0.5; \({\sigma _{2}^{2}}=4\); \({\sigma _{1}^{2}}=1,8\) and α=0.05. For a given combination (n,n 1,n 2,μ 1,μ 2,ρ,σ 1,σ 2), we generate n+n 1+n 2 random samples of (x 1,x 2) from a bivariate normal distribution with μ=(μ 1,μ 2) and
$$\Sigma=\left(\begin{array}{ll} {\sigma_{1}^{2}}& \rho\sigma_{1}\sigma_{2}\\ \rho\sigma_{1}\sigma_{2}& {\sigma_{2}^{2}} \end{array} \right). $$
Then, for the generated n+n 1+n 2 random samples, the n 1 observations on x 2 are deleted randomly. For the remaining paired n+n 2 random samples, the n 2 observations on x 1 are deleted randomly. Thus, (x 1m ,x 2m )(m=1,,n) are n pairs observations on (x 1,x 2); x 1,n+j (j=1,,n 1) are n 1 additional observations on x 1; x 2,n+k (k=1,,n 2) are n 2 additional observations on x 2. Based on the observation {(x 1j ,x 2j ):m=1,,n}, {x 1,n+j :j=1,,n 1}, {x 2,n+k :k=1,,n 2}, we can draw 5000 bootstrap resampling samples. Independently repeating the above process M=10000 times, we can compute their corresponding ECP, ECW, MNP and DNP values. The ECP, ECW, MNP and DNP are defined by
$$ \begin{aligned} \text{ECP}&=\frac{1}{M}\sum\limits_{m=1}^{M}I\left\{\delta\in \left[L\left(\boldsymbol{x}^{(m)}\right),U\left(\boldsymbol{x}^{(m)}\right)\right]\right\},\\ \text{ECW}&=\frac{1}{M}\sum\limits_{m=1}^{M}\left[U\left(\boldsymbol{x}^{(m)})\right)-L\left(\boldsymbol{x}^{(m)}\right)\right], \end{aligned} $$
(3)
$$ \begin{aligned} \text{MNP}&=\frac{1}{M}\sum\limits_{m=1}^{M}I\left\{\delta\in \left[-\infty, L\left(\boldsymbol{x}^{(m)}\right)\right]\right\},\\ \text{DNP}&=\frac{1}{M}\sum\limits_{m=1}^{M}I\left\{\delta\in \left[U\left(\boldsymbol{x}^{(m)}\right), +\infty\right]\right\}, \end{aligned} $$
(4)
respectively, where \(I\{\delta \in \mathcal {A}\}\) is an indicator function, which is 1 if \(\delta \in \mathcal {A}\) and 0 otherwise. The ratio of the MNP to the non-coverage probability (NCP) is defined as
$$ \text{RNCP}=\frac{\text{MNP}}{\text{NCP}}=\frac{\text{MNP}}{1.0-\text{ECP}}. $$
(5)
Results are presented in Tables 3, 4 and 5. Also, to investigate the performance of the proposed CIs under the assumption \({\sigma _{1}^{2}}={\sigma _{2}^{2}}=\sigma ^{2}\), we calculate the corresponding results for T 3, T 4, T 5, hybrid CIs, Bootstrap-resampling-based CIs when σ 2=4 and (n,n 1,n 2)=(5,5,2), which are given in Tables 9, 10 and 11.
Table 3

ECPs of various confidence intervals under bivariate normal distribution with different ρ and δ, μ 1, \(\mu _{2} {\sigma _{1}^{2}}\) and (n,n 1,n 2)=(5,2,2) and \({\sigma _{2}^{2}}=4\)

ρ

\({\sigma _{1}^{2}}\)

δ

μ 1

μ 2

T 1

T 2

T g

W s

W a

B 1

B 2

B 3

B 4

-0.9

1

-0.25

0

0.25

0.9390

0.9590

0.9370

0.9350

0.8800

0.9520

0.9560

0.9370

0.9570

  

0

1

1

0.9440

0.9580

0.9470

0.9220

0.8760

0.9470

0.9490

0.9300

0.9490

  

0.5

2

1.5

0.9430

0.9670

0.9530

0.9400

0.8860

0.9470

0.9480

0.9310

0.9480

 

8

-0.25

0

0.25

0.9410

0.9630

0.9450

0.9180

0.8600

0.9430

0.9490

0.9340

0.9490

  

0

1

1

0.9370

0.9580

0.9350

0.9240

0.8640

0.9510

0.9500

0.9390

0.9500

  

0.5

2

1.5

0.9380

0.9570

0.9410

0.9240

0.8750

0.9570

0.9560

0.9470

0.9560

-0.5

1

-0.25

0

0.25

0.9440

0.9610

0.9530

0.9200

0.8570

0.9490

0.9430

0.9330

0.9420

  

0

1

1

0.9420

0.9660

0.9230

0.9270

0.8660

0.9570

0.9560

0.9460

0.9550

  

0.5

2

1.5

0.9460

0.9660

0.9380

0.9250

0.8640

0.9480

0.9560

0.9430

0.9540

 

8

-0.25

0

0.25

0.9290

0.9590

0.9480

0.9230

0.8730

0.9470

0.9450

0.9390

0.9440

  

0

1

1

0.9290

0.9560

0.9420

0.9210

0.8790

0.9460

0.9430

0.9380

0.9440

  

0.5

2

1.5

0.9350

0.9690

0.9410

0.9330

0.8880

0.9520

0.9540

0.9470

0.9520

-0.1

1

-0.25

0

0.25

0.9300

0.9570

0.9500

0.9170

0.8630

0.9550

0.9500

0.9450

0.9470

  

0

1

1

0.9380

0.9590

0.9450

0.9170

0.8600

0.9540

0.9500

0.9450

0.9520

  

0.5

2

1.5

0.9400

0.9620

0.9440

0.9140

0.8560

0.9510

0.9460

0.9420

0.9460

 

8

-0.25

0

0.25

0.9460

0.9600

0.9310

0.9050

0.8490

0.9460

0.9470

0.9440

0.9470

  

0

1

1

0.9450

0.9670

0.9440

0.9150

0.8590

0.9560

0.9500

0.9480

0.9510

  

0.5

2

1.5

0.9350

0.9610

0.9360

0.9150

0.8570

0.9500

0.9520

0.9440

0.9490

0

1

-0.25

0

0.25

0.9380

0.9610

0.9400

0.9330

0.8860

0.9550

0.9550

0.9530

0.9530

  

0

1

1

0.9290

0.9610

0.9280

0.9200

0.8680

0.9470

0.9480

0.9470

0.9470

  

0.5

2

1.5

0.9300

0.9580

0.9420

0.9230

0.8800

0.9520

0.9510

0.9500

0.9510

 

8

-0.25

0

0.25

0.9210

0.9590

0.9390

0.9090

0.8400

0.9430

0.9450

0.9450

0.9450

  

0

1

1

0.9240

0.9570

0.9400

0.9050

0.8520

0.9430

0.9440

0.9430

0.9430

  

0.5

2

1.5

0.9360

0.9680

0.9380

0.9140

0.8540

0.9530

0.9530

0.9530

0.9520

0.1

1

-0.25

0

0.25

0.9310

0.9690

0.9480

0.9150

0.8530

0.9510

0.9510

0.9490

0.9490

  

0

1

1

0.9330

0.9670

0.9440

0.9150

0.8550

0.9500

0.9500

0.9490

0.9510

  

0.5

2

1.5

0.9310

0.9570

0.9490

0.9150

0.8630

0.9520

0.9520

0.9510

0.9520

 

8

-0.25

0

0.25

0.9220

0.9520

0.9420

0.9190

0.8700

0.9510

0.9510

0.9520

0.9520

  

0

1

1

0.9290

0.9540

0.9360

0.9210

0.8690

0.9490

0.9490

0.9470

0.9470

  

0.5

2

1.5

0.9180

0.9530

0.9350

0.9340

0.8860

0.9520

0.9520

0.9500

0.9500

0.5

1

-0.25

0

0.25

0.9230

0.9530

0.9470

0.8980

0.8470

0.9540

0.9540

0.9530

0.9530

  

0

1

1

0.9330

0.9620

0.9390

0.9050

0.8510

0.9440

0.9440

0.9440

0.9440

  

0.5

2

1.5

0.9280

0.9640

0.9330

0.9140

0.8640

0.9520

0.9520

0.9500

0.9500

 

8

-0.25

0

0.25

0.9360

0.9660

0.9420

0.9030

0.8450

0.9470

0.9470

0.9460

0.9460

  

0

1

1

0.9220

0.9600

0.9350

0.9060

0.8410

0.9500

0.9500

0.9480

0.9480

  

0.5

2

1.5

0.9300

0.9650

0.9500

0.9140

0.8570

0.9580

0.9580

0.9570

0.9570

0.9

1

-0.25

0

0.25

0.9190

0.9540

0.9400

0.9300

0.8710

0.9450

0.9450

0.9440

0.9430

  

0

1

1

0.9390

0.9640

0.9460

0.9360

0.8870

0.9590

0.9580

0.9570

0.9580

  

0.5

2

1.5

0.9240

0.9610

0.9310

0.9220

0.8760

0.9470

0.9460

0.9470

0.9470

 

8

-0.25

0

0.25

0.9200

0.9590

0.9440

0.9050

0.8440

0.9440

0.9430

0.9430

0.9450

  

0

1

1

0.9310

0.9620

0.9430

0.9040

0.8390

0.9450

0.9450

0.9460

0.9460

  

0.5

2

1.5

0.9310

0.9620

0.9400

0.9190

0.8610

0.9530

0.9520

0.9520

0.9530

Table 4

ECW of various confidence intervals under bivariate normal distribution with different ρ and δ, μ 1, μ 2, \({\sigma _{1}^{2}}\) and (n,n 1,n 2)=(5,2,2) and \({\sigma _{2}^{2}}=4\)

ρ

\({\sigma _{1}^{2}}\)

δ

μ 1

μ 2

T 1

T 2

T g

W s

W a

B 1

B 2

B 3

B 4

-0.9

1

-0.25

0

0.25

8.0510

9.8480

7.6040

4.9790

4.0830

6.5400

6.9700

6.5380

6.9700

  

0

1

1

8.0980

9.8440

7.6290

4.9880

4.0930

6.5410

6.9710

6.5410

6.9710

  

0.5

2

1.5

8.1690

9.7210

7.6410

5.0880

4.2070

6.5420

6.9700

6.5410

6.9680

 

8

-0.25

0

0.25

10.8170

12.0750

9.6020

6.5090

5.2840

8.8020

9.1950

8.8010

9.1960

  

0

1

1

10.8350

12.1090

9.5830

6.5080

5.2840

8.8030

9.1950

8.8050

9.1940

  

0.5

2

1.5

10.8310

12.0670

9.5720

6.5610

5.3560

8.8080

9.2000

8.8070

9.1960

-0.5

1

-0.25

0

0.25

12.7390

14.0040

11.0300

7.6080

6.1620

10.2980

10.7370

10.2990

10.7370

  

0

1

1

12.7510

14.0800

11.0500

7.6310

6.1810

10.3020

10.7410

10.3000

10.7380

  

0.5

2

1.5

12.7460

14.0150

11.0120

7.6540

6.2220

10.3070

10.7470

10.3080

10.7450

 

8

-0.25

0

0.25

7.9520

9.4420

7.3030

4.7520

3.8910

6.4600

6.5990

6.4620

6.6000

  

0

1

1

7.9990

9.4880

7.3300

4.7760

3.9140

6.4630

6.6000

6.4630

6.6030

  

0.5

2

1.5

7.9410

9.4190

7.3300

4.8830

4.0460

6.4650

6.6040

6.4630

6.6010

-0.1

1

-0.25

0

0.25

10.1230

11.1210

8.9290

6.0060

4.8870

8.2480

8.3910

8.2510

8.3940

  

0

1

1

10.1150

11.2600

9.9060

6.0040

4.8850

8.2490

8.3920

8.2490

8.3930

  

0.5

2

1.5

10.0550

11.1650

9.8830

6.0750

4.9860

8.2460

8.3880

8.2480

8.3890

 

8

-0.25

0

0.25

11.8990

12.9260

10.2600

7.0330

5.7080

9.6020

9.7660

9.6030

9.7670

  

0

1

1

11.9170

12.9540

10.2910

7.0500

5.7240

9.6030

9.7670

9.6020

9.7670

  

0.5

2

1.5

11.9290

13.0050

10.2490

7.1130

5.8070

9.5990

9.7620

9.5980

9.7610

0

1

-0.25

0

0.25

7.4380

8.7970

6.9270

4.4570

3.6460

6.2020

6.2080

6.1980

6.2060

  

0

1

1

7.4070

9.0290

6.9140

4.4570

3.6480

6.2100

6.2160

6.2100

6.2160

  

0.5

2

1.5

7.4750

9.0040

6.9620

4.6380

3.8580

6.2020

6.2080

6.2000

6.2060

 

8

-0.25

0

0.25

9.0700

10.2520

8.2140

5.4680

4.4610

7.4900

7.4970

7.4910

7.4960

  

0

1

1

9.0480

10.0050

8.1310

5.4190

4.4290

7.4910

7.4980

7.4880

7.4960

  

0.5

2

1.5

9.1370

10.2100

8.2110

5.5930

4.6170

7.4920

7.5000

7.4910

7.4970

0.1

1

-0.25

0

0.25

10.5430

11.8910

9.3750

6.3650

5.1880

8.6680

8.6760

8.6700

8.6770

  

0

1

1

10.5330

11.7900

9.3610

6.3410

5.1710

8.6680

8.6760

8.6660

8.6740

  

0.5

2

1.5

10.6010

11.7180

9.3710

6.4860

5.3310

8.6700

8.6780

8.6680

8.6770

 

8

-0.25

0

0.25

7.3190

8.8790

6.8430

4.3920

3.5910

6.1080

6.1080

6.1070

6.1070

  

0

1

1

7.2750

8.7620

6.8270

4.3840

3.5900

6.1090

6.1090

6.1090

6.1090

  

0.5

2

1.5

7.3480

8.7970

6.8640

4.5800

3.8160

6.1070

6.1070

6.1040

6.1040

0.5

1

-0.25

0

0.25

8.7070

9.8380

7.9460

5.2650

4.3050

7.2590

7.2590

7.2570

7.2570

  

0

1

1

8.7510

9.9250

7.9940

5.3100

4.3450

7.2570

7.2570

7.2540

7.2540

  

0.5

2

1.5

8.8320

10.0890

8.0490

5.4970

4.5480

7.2590

7.2590

7.2590

7.2590

 

8

-0.25

0

0.25

10.2360

11.4530

9.1100

6.1750

5.0390

8.3820

8.3820

8.3810

8.3810

  

0

1

1

10.1380

11.2610

9.0610

6.1540

5.0260

8.3810

8.3810

8.3850

8.3850

  

0.5

2

1.5

10.1020

11.3160

9.0800

6.2300

5.1320

8.3830

8.3830

8.3830

8.3830

0.9

1

-0.25

0

0.25

7.2300

8.9110

6.8140

4.3740

3.5750

6.0000

6.0070

6.0020

6.0090

  

0

1

1

7.3030

8.6810

6.7940

4.3620

3.5700

5.9960

6.0020

5.9950

6.0020

  

0.5

2

1.5

7.2340

8.8310

6.7930

4.5270

3.7720

5.9990

6.0060

5.9990

6.0050

 

8

-0.25

0

0.25

8.4830

9.7340

7.8050

5.1900

4.2510

7.0030

7.0110

6.9980

7.0050

  

0

1

1

8.4410

9.6700

7.7630

5.1400

4.2100

7.0010

7.0080

6.9970

7.0050

  

0.5

2

1.5

8.4160

9.8250

7.8290

5.3240

4.4150

7.0000

7.0080

7.0020

7.0100

Table 5

RNCP of various confidence intervals under bivariate normal distribution with different ρ and δ, μ 1, μ 2, \({\sigma _{1}^{2}}\) and (n,n 1,n 2)=(5,2,2) and \({\sigma _{2}^{2}}=4\)

ρ

\({\sigma _{1}^{2}}\)

δ

μ 1

μ 2

T 1

T 2

T g

W s

W a

B 1

B 2

B 3

B 4

-0.9

1

-0.25

0

0.25

0.4754

0.4805

0.4731

0.4769

0.4660

0.5000

0.4091

0.4921

0.4186

  

0

1

1

0.4286

0.5286

0.4563

0.3846

0.4892

0.4528

0.4314

0.4286

0.4706

  

0.5

2

1.5

0.4737

0.5909

0.4839

0.4667

0.4590

0.4906

0.4231

0.4638

0.4038

 

8

-0.25

0

0.25

0.4237

0.5108

0.5048

0.4268

0.4574

0.5088

0.5686

0.5303

0.5686

  

0

1

1

0.4603

0.5143

0.4857

0.4474

0.5000

0.5102

0.5000

0.5082

0.5000

  

0.5

2

1.5

0.4677

0.5744

0.4545

0.5395

0.4983

0.4186

0.4545

0.4717

0.4773

-0.5

1

-0.25

0

0.25

0.5536

0.5436

0.5234

0.5375

0.5289

0.5686

0.5789

0.5821

0.5862

  

0

1

1

0.5000

0.5235

0.4948

0.4795

0.5389

0.4651

0.4773

0.4815

0.4667

  

0.5

2

1.5

0.5741

0.5176

0.5294

0.6533

0.5266

0.5577

0.6591

0.6140

0.6304

 

8

-0.25

0

0.25

0.5070

0.5829

0.5098

0.5195

0.5481

0.5472

0.5273

0.5410

0.5357

  

0

1

1

0.5352

0.5364

0.5306

0.4684

0.5585

0.5370

0.5263

0.5645

0.5536

  

0.5

2

1.5

0.4769

0.5355

0.4719

0.3731

0.5256

0.5208

0.4348

0.4717

0.4375

-0.1

1

-0.25

0

0.25

0.5000

0.5744

0.5300

0.4699

0.6086

0.5333

0.5000

0.4727

0.4717

  

0

1

1

0.4839

0.5585

0.4842

0.4458

0.5714

0.5000

0.5400

0.5091

0.5417

  

0.5

2

1.5

0.5333

0.5632

0.5000

0.5116

0.5000

0.5102

0.5185

0.5000

0.5000

 

8

-0.25

0

0.25

0.4630

0.5750

0.4848

0.4526

0.5176

0.4444

0.4151

0.4464

0.4528

  

0

1

1

0.5091

0.5879

0.5104

0.5059

0.5119

0.5455

0.4800

0.5000

0.4898

  

0.5

2

1.5

0.5385

0.5179

0.5288

0.5529

0.5248

0.5200

0.5208

0.5179

0.4902

0

1

-0.25

0

0.25

0.5484

0.5641

0.5667

0.6119

0.4800

0.4889

0.5333

0.5319

0.5319

  

0

1

1

0.4789

0.5923

0.5000

0.4000

0.4996

0.4906

0.4808

0.4906

0.4906

  

0.5

2

1.5

0.4286

0.5714

0.5000

0.2857

0.5097

0.5000

0.5102

0.5200

0.5306

 

8

-0.25

0

0.25

0.4684

0.5829

0.5149

0.4835

0.5397

0.4912

0.5091

0.5091

0.5091

  

0

1

1

0.5789

0.5977

0.4700

0.4737

0.5028

0.4561

0.4464

0.4912

0.4561

  

0.5

2

1.5

0.5313

0.5500

0.5000

0.5233

0.5100

0.4894

0.5106

0.5106

0.5000

0.1

1

-0.25

0

0.25

0.5217

0.5065

0.5488

0.5176

0.5566

0.5102

0.5102

0.4902

0.4902

  

0

1

1

0.5224

0.5788

0.4651

0.5176

0.5212

0.4200

0.4200

0.4314

0.4286

  

0.5

2

1.5

0.5362

0.5116

0.5824

0.6235

0.5852

0.5417

0.5417

0.5714

0.5417

 

8

-0.25

0

0.25

0.4359

0.5417

0.4490

0.5309

0.5833

0.4490

0.4490

0.4583

0.4583

  

0

1

1

0.4789

0.5304

0.4904

0.3544

0.4914

0.4118

0.4118

0.4528

0.4528

  

0.5

2

1.5

0.4878

0.5170

0.5053

0.2879

0.5314

0.4167

0.4167

0.4200

0.4200

0.5

1

-0.25

0

0.25

0.4935

0.5106

0.4563

0.4510

0.5125

0.5000

0.5000

0.5106

0.5106

  

0

1

1

0.5522

0.5947

0.4505

0.4211

0.5085

0.3929

0.3929

0.4107

0.4107

  

0.5

2

1.5

0.4861

0.5944

0.4943

0.5000

0.4692

0.5417

0.5417

0.5000

0.5000

 

8

-0.25

0

0.25

0.4688

0.5647

0.4592

0.4227

0.5081

0.5472

0.5472

0.5185

0.5185

  

0

1

1

0.5256

0.5750

0.5474

0.5426

0.5008

0.5200

0.5200

0.5577

0.5577

  

0.5

2

1.5

0.5286

0.5000

0.4875

0.5233

0.5093

0.5238

0.5238

0.5349

0.5349

0.9

1

-0.25

0

0.25

0.5062

0.5652

0.5000

0.5714

0.4861

0.5273

0.5273

0.5357

0.5263

  

0

1

1

0.5246

0.5111

0.5238

0.3281

0.5100

0.5122

0.5000

0.4884

0.5000

  

0.5

2

1.5

0.4605

0.5692

0.4141

0.2179

0.2217

0.4528

0.4444

0.4340

0.4340

 

8

-0.25

0

0.25

0.5250

0.5341

0.5104

0.5053

0.5045

0.5179

0.5088

0.5439

0.5455

  

0

1

1

0.5362

0.5579

0.5155

0.4688

0.6133

0.5273

0.5273

0.5370

0.5370

  

0.5

2

1.5

0.5217

0.5579

0.4778

0.4938

0.4672

0.4681

0.4583

0.5000

0.4681

Following [17, 30], an interval can be regarded as satisfactory if (i) its ECP is close to the pre-specified 95 % confidence level, (ii) it possesses shorter interval width, and (iii) its RNCP lies in the interval [0.4,0.6]; too mesially located if its RNCP is less than 0.4; and too distally if its RNCP is greater than 0.6.

In the second Monte Carlo simulation study, we assume that the random samples of bivariate variables x 1 and x 2 are generated from a bivariate t-distribution with five degrees of freedom, and mean μ and scale parameter Σ specified in the first simulation study. The corresponding results with (n,n 1,n 2)=(5,5,5) are given in Tables 6, 7 and 8. Similarly, we calculate the corresponding results for T 3, T 4, T 5, hybrid CIs, Bootstrap-resampling-based CIs when σ 2=4 and (n,n 1,n 2)=(5,5,2), which are given in Tables 9, 10 and 11.
Table 6

ECPs of various confidence intervals under bivariate t-distribution with different ρ and δ, μ 1, μ 2, \({\sigma _{1}^{2}}\) and (n,n 1,n 2)=(5,5,5) and \({\sigma _{2}^{2}}=4\)

ρ

\({\sigma _{1}^{2}}\)

δ

μ 1

μ 2

T 1

T 2

T g

W s

W a

B 1

B 2

B 3

B 4

-0.9

1

-0.25

0

0.25

0.9260

0.9750

0.9460

0.9510

0.9020

0.9470

0.9470

0.9500

0.9500

  

0

1

1

0.9060

0.9590

0.9490

0.9340

0.8820

0.9450

0.9450

0.9510

0.9510

  

0.5

2

1.5

0.9160

0.9710

0.9370

0.9480

0.8930

0.9490

0.9490

0.9530

0.9530

 

8

-0.25

0

0.25

0.8950

0.9630

0.9380

0.9460

0.8920

0.9490

0.9380

0.9410

0.9410

  

0

1

1

0.9030

0.9580

0.9430

0.9450

0.9020

0.9400

0.9410

0.9410

0.9410

  

0.5

2

1.5

0.9080

0.9640

0.9370

0.9490

0.9070

0.9500

0.9480

0.9520

0.9520

-0.5

1

-0.25

0

0.25

0.9160

0.9700

0.9460

0.9380

0.8810

0.9440

0.9410

0.9430

0.9420

  

0

1

1

0.9150

0.9670

0.9510

0.9380

0.8970

0.9470

0.9480

0.9480

0.9480

  

0.5

2

1.5

0.9190

0.9650

0.9440

0.9440

0.8940

0.9480

0.9520

0.9540

0.9540

 

8

-0.25

0

0.25

0.9160

0.9680

0.9490

0.9580

0.9160

0.9530

0.9480

0.9440

0.9510

  

0

1

1

0.9080

0.9690

0.9510

0.9590

0.9200

0.9460

0.9450

0.9400

0.9480

  

0.5

2

1.5

0.9130

0.9750

0.9400

0.9630

0.9200

0.9410

0.9410

0.9230

0.9460

-0.1

1

-0.25

0

0.25

0.9230

0.9660

0.9480

0.9500

0.9020

0.9530

0.9470

0.9410

0.9490

  

0

1

1

0.9060

0.9600

0.9380

0.9370

0.8920

0.9430

0.9450

0.9390

0.9500

  

0.5

2

1.5

0.9020

0.9660

0.9410

0.9400

0.8910

0.9530

0.9460

0.9350

0.9460

 

8

-0.25

0

0.25

0.9110

0.9670

0.9450

0.9650

0.9290

0.9440

0.9420

0.8800

0.9470

  

0

1

1

0.9190

0.9720

0.9360

0.9650

0.9270

0.9510

0.9450

0.8810

0.9470

  

0.5

2

1.5

0.9140

0.9700

0.9390

0.9630

0.9270

0.9480

0.9440

0.8890

0.9470

0

1

-0.25

0

0.25

0.9180

0.9580

0.9430

0.9500

0.8980

0.9470

0.9390

0.7900

0.9420

  

0

1

1

0.9150

0.9710

0.9550

0.9550

0.9130

0.9490

0.9500

0.8030

0.9500

  

0.5

2

1.5

0.9180

0.9670

0.9500

0.9590

0.9200

0.9450

0.9510

0.7940

0.9540

 

8

-0.25

0

0.25

0.9380

0.9660

0.9380

0.9560

0.9280

0.9510

0.9510

0.9380

0.9530

  

0

1

1

0.9360

0.9650

0.9340

0.9530

0.9220

0.9560

0.9520

0.9370

0.9540

  

0.5

2

1.5

0.9310

0.9540

0.9340

0.9510

0.9230

0.9450

0.9530

0.9400

0.9540

0.1

1

-0.25

0

0.25

0.9360

0.9640

0.9420

0.9530

0.9210

0.9480

0.9510

0.9430

0.9550

  

0

1

1

0.9350

0.9620

0.9340

0.9520

0.9190

0.9560

0.9520

0.9400

0.9520

  

0.5

2

1.5

0.9290

0.9600

0.9340

0.9440

0.9160

0.9440

0.9470

0.9340

0.9480

 

8

-0.25

0

0.25

0.9300

0.9530

0.9330

0.9470

0.9190

0.9400

0.9380

0.9350

0.9400

  

0

1

1

0.9340

0.9590

0.9310

0.9520

0.9160

0.9410

0.9410

0.9360

0.9420

  

0.5

2

1.5

0.9390

0.9660

0.9330

0.9520

0.9210

0.9530

0.9500

0.9490

0.9530

0.5

1

-0.25

0

0.25

0.9370

0.9640

0.9370

0.9490

0.9120

0.9450

0.9440

0.9430

0.9470

  

0

1

1

0.9450

0.9590

0.9360

0.9450

0.9080

0.9460

0.9420

0.9380

0.9440

  

0.5

2

1.5

0.9430

0.9680

0.9400

0.9520

0.9200

0.9540

0.9480

0.9490

0.9540

 

8

-0.25

0

0.25

0.9340

0.9580

0.9460

0.9520

0.9190

0.9420

0.9450

0.9470

0.9480

  

0

1

1

0.9400

0.9630

0.9470

0.9530

0.9210

0.9550

0.9560

0.9580

0.9580

  

0.5

2

1.5

0.9270

0.9610

0.9330

0.9470

0.9230

0.9420

0.9420

0.9470

0.9460

0.9

1

-0.25

0

0.25

0.9430

0.9660

0.9410

0.9500

0.9140

0.9470

0.9470

0.9480

0.9480

  

0

1

1

0.9410

0.9530

0.9440

0.9400

0.9040

0.9470

0.9460

0.9510

0.9500

  

0.5

2

1.5

0.9430

0.9660

0.9480

0.9490

0.9160

0.9540

0.9560

0.9550

0.9560

 

8

-0.25

0

0.25

0.9320

0.9540

0.9520

0.9450

0.9200

0.9460

0.9460

0.9490

0.9490

  

0

1

1

0.9460

0.9660

0.9470

0.9590

0.9300

0.9470

0.9470

0.9490

0.9490

  

0.5

2

1.5

0.9410

0.9580

0.9460

0.9510

0.9200

0.9550

0.9550

0.9580

0.9580

Table 7

ECW of various confidence intervals under bivariate t-distribution with different ρ and δ, μ 1, μ 2, \({\sigma _{1}^{2}}\) and (n,n 1,n 2)=(5,5,5) and \({\sigma _{2}^{2}}=4\)

ρ

\({\sigma _{1}^{2}}\)

δ

μ 1

μ 2

T 1

T 2

T g

W s

W a

B 1

B 2

B 3

B 4

-0.9

1

-0.25

0

0.25

39.9860

40.8450

35.0080

27.6870

23.6020

35.9410

35.9410

36.4110

36.4110

  

0

1

1

39.5890

40.3210

34.6710

27.5260

23.4670

35.9280

35.9280

36.4080

36.4080

  

0.5

2

1.5

39.2290

40.6160

34.8600

27.6570

23.5830

35.9050

35.9050

36.3930

36.3930

 

8

-0.25

0

0.25

32.6680

34.3430

29.0000

22.6520

19.2790

29.8250

29.8650

30.3280

30.3510

  

0

1

1

32.7540

34.2080

28.8030

22.5610

19.2030

29.8370

29.8760

30.3360

30.3630

  

0.5

2

1.5

32.3510

34.3530

28.9420

22.6560

19.2890

29.8380

29.8770

30.3350

30.3580

-0.5

1

-0.25

0

0.25

38.5200

39.5450

34.4200

27.2240

23.2190

35.0120

35.0610

35.4930

35.5290

  

0

1

1

37.8690

39.0530

33.9970

26.9350

22.9710

35.0060

35.0530

35.4800

35.5150

  

0.5

2

1.5

38.7140

39.7290

34.1600

27.1930

23.1930

35.0190

35.0680

35.4960

35.5300

 

8

-0.25

0

0.25

29.3790

32.8700

27.8810

21.7710

18.5360

27.1550

28.3790

27.6280

28.8530

  

0

1

1

28.9240

32.3540

27.5350

21.5080

18.3100

27.1670

28.4000

27.6550

28.8590

  

0.5

2

1.5

30.1060

33.6320

28.5080

22.3350

19.0220

27.1880

28.4140

27.6570

28.8780

-0.1

1

-0.25

0

0.25

31.3890

36.1820

31.7890

25.3610

21.6710

29.6610

31.4040

30.1280

31.7960

  

0

1

1

30.9880

35.1900

30.9350

24.6920

21.0960

29.6870

31.4300

30.1620

31.8160

  

0.5

2

1.5

31.1860

35.2420

31.2250

24.8410

21.2380

29.6730

31.4220

30.1440

31.8080

 

8

-0.25

0

0.25

23.8340

31.2500

26.6610

20.8190

17.7300

20.9570

26.8340

21.2360

27.2880

  

0

1

1

23.4990

31.1470

26.6370

20.8520

17.7590

20.9660

26.8530

21.2550

27.3120

  

0.5

2

1.5

23.1750

30.4390

26.1330

20.3920

17.3770

20.9520

26.8280

21.2470

27.2670

0

1

-0.25

0

0.25

16.7960

30.7840

27.3290

21.9590

18.8280

16.6250

27.2590

16.9850

27.6450

  

0

1

1

17.1650

30.5510

27.3190

21.8760

18.7550

16.6250

27.2600

16.9750

27.6580

  

0.5

2

1.5

16.9980

30.6500

27.2430

22.0410

18.9120

16.6160

27.2610

16.9700

27.6440

 

8

-0.25

0

0.25

27.2420

29.7100

27.2280

22.7000

20.2600

26.0380

27.9560

26.2890

28.2960

  

0

1

1

27.6030

29.9040

27.3850

22.8460

20.3900

26.0420

27.9600

26.2820

28.2960

  

0.5

2

1.5

27.4440

29.6420

27.2230

22.6840

20.2480

26.0420

27.9660

26.2770

28.2950

0.1

1

-0.25

0

0.25

36.7630

38.5960

35.2540

29.7190

26.5230

35.0020

36.7010

35.3140

37.1130

  

0

1

1

36.9580

38.9090

35.4500

29.9490

26.7290

34.9960

36.6930

35.3230

37.1390

  

0.5

2

1.5

36.6820

38.7640

35.2490

29.7940

26.5910

34.9890

36.6840

35.3050

37.1090

 

8

-0.25

0

0.25

26.8170

28.2480

25.9750

21.6000

19.2790

25.9390

26.5530

26.1980

26.8650

  

0

1

1

27.0150

28.2910

26.0250

21.6540

19.3270

25.9380

26.5470

26.1960

26.8510

  

0.5

2

1.5

27.1980

28.6990

26.3160

21.9060

19.5540

25.9400

26.5480

26.1920

26.8500

0.5

1

-0.25

0

0.25

35.0610

35.8660

32.9600

27.7210

24.7450

33.0080

33.6310

33.3300

34.0030

  

0

1

1

35.2510

35.8450

32.9690

27.7590

24.7790

32.9910

33.6170

33.3000

33.9910

  

0.5

2

1.5

34.6160

35.6320

32.7980

27.5930

24.6330

32.9950

33.6200

33.3110

33.9920

 

8

-0.25

0

0.25

26.0830

26.9080

24.8210

20.5850

18.3740

25.0540

25.0810

25.3290

25.3590

  

0

1

1

25.6840

26.7000

24.6450

20.4400

18.2450

25.0390

25.0650

25.3160

25.3480

  

0.5

2

1.5

25.9800

26.9400

24.8400

20.5900

18.3810

25.0410

25.0680

25.3280

25.3580

0.9

1

-0.25

0

0.25

31.7980

32.2880

29.9420

25.1170

22.4300

30.2210

30.2530

30.5230

30.5690

  

0

1

1

31.8710

32.0900

29.8060

24.9940

22.3200

30.1980

30.2290

30.5050

30.5500

  

0.5

2

1.5

31.3990

32.0560

29.7450

24.9700

22.3010

30.2140

30.2440

30.5180

30.5600

 

8

-0.25

0

0.25

25.4700

26.4860

24.4770

20.2660

18.0900

24.6850

24.6850

24.9600

24.9600

  

0

1

1

25.5190

26.3770

24.3630

20.1840

18.0160

24.6740

24.6740

24.9450

24.9450

  

0.5

2

1.5

25.4630

26.4500

24.4990

20.2850

18.1100

24.6930

24.6930

24.9760

24.9760

Table 8

RNCP of various confidence intervals under bivariate t-distribution with different ρ and δ, μ 1, μ 2, \({\sigma _{1}^{2}}\) and (n,n 1,n 2)=(5,5,5) and \({\sigma _{2}^{2}}=4\)

ρ

\({\sigma _{1}^{2}}\)

δ

μ 1

μ 2

T 1

T 2

T g

W s

W a

B 1

B 2

B 3

B 4

-0.9

1

-0.25

0

0.25

0.4324

0.5200

0.5000

0.5918

0.5102

0.4717

0.4717

0.4800

0.4800

  

0

1

1

0.4574

0.4634

0.5062

0.4848

0.5000

0.4727

0.4727

0.5102

0.5102

  

0.5

2

1.5

0.4524

0.5862

0.5238

0.5385

0.5047

0.4118

0.4118

0.4255

0.4255

 

8

-0.25

0

0.25

0.4762

0.4865

0.4878

0.4815

0.4815

0.4754

0.4677

0.4746

0.4746

  

0

1

1

0.5361

0.5238

0.4675

0.5091

0.5000

0.4833

0.4746

0.4746

0.4746

  

0.5

2

1.5

0.4783

0.5278

0.4795

0.5098

0.5484

0.5400

0.5000

0.5208

0.5208

-0.5

1

-0.25

0

0.25

0.4524

0.4000

0.4595

0.4839

0.4538

0.4464

0.4237

0.4211

0.4138

  

0

1

1

0.6000

0.6061

0.5797

0.5645

0.5534

0.5283

0.5385

0.5577

0.5385

  

0.5

2

1.5

0.5062

0.5429

0.5455

0.5357

0.5660

0.5000

0.5000

0.5435

0.5435

 

8

-0.25

0

0.25

0.4762

0.5000

0.5070

0.5952

0.5119

0.5106

0.5385

0.5179

0.5306

  

0

1

1

0.5217

0.5806

0.5085

0.5854

0.5500

0.4815

0.5273

0.5167

0.5000

  

0.5

2

1.5

0.4943

0.4000

0.4000

0.4595

0.5125

0.5316

0.5217

0.5195

0.5313

-0.1

1

-0.25

0

0.25

0.5584

0.4706

0.5323

0.5800

0.4796

0.5319

0.5660

0.5932

0.5686

  

0

1

1

0.5532

0.5500

0.5278

0.5714

0.5463

0.4737

0.4727

0.4754

0.5000

  

0.5

2

1.5

0.4490

0.4706

0.4348

0.4333

0.4679

0.4255

0.4815

0.4769

0.4630

 

8

-0.25

0

0.25

0.4831

0.4545

0.5231

0.4000

0.4648

0.5714

0.5000

0.4917

0.5283

  

0

1

1

0.5062

0.5000

0.4844

0.4571

0.5068

0.4898

0.4727

0.4958

0.4717

  

0.5

2

1.5

0.4651

0.5000

0.4590

0.5676

0.5479

0.4423

0.4464

0.4595

0.4717

0

1

-0.25

0

0.25

0.5244

0.5714

0.6140

0.5800

0.5196

0.5094

0.5246

0.4857

0.5000

  

0

1

1

0.5059

0.4483

0.4444

0.4444

0.4828

0.5490

0.5600

0.5228

0.5600

  

0.5

2

1.5

0.5366

0.3939

0.4800

0.4146

0.4875

0.4545

0.5306

0.5097

0.5217

 

8

-0.25

0

0.25

0.5161

0.6176

0.5968

0.6136

0.5972

0.5714

0.6122

0.5968

0.5957

  

0

1

1

0.4844

0.4571

0.4697

0.4468

0.4744

0.5682

0.5208

0.5079

0.5000

  

0.5

2

1.5

0.4928

0.4130

0.4545

0.4286

0.4545

0.4545

0.4255

0.4833

0.4348

0.1

1

-0.25

0

0.25

0.5469

0.5833

0.5862

0.6170

0.5696

0.5385

0.5102

0.5088

0.5333

  

0

1

1

0.5692

0.4737

0.5303

0.5208

0.5309

0.5227

0.5208

0.5000

0.5000

  

0.5

2

1.5

0.4507

0.4500

0.4394

0.4107

0.4286

0.4464

0.4717

0.4545

0.4808

 

8

-0.25

0

0.25

0.5000

0.5319

0.5373

0.5472

0.5062

0.5000

0.5161

0.5077

0.5000

  

0

1

1

0.5303

0.5366

0.5072

0.5417

0.4762

0.4746

0.4915

0.4844

0.5000

  

0.5

2

1.5

0.5246

0.5294

0.5373

0.5417

0.5443

0.5532

0.5400

0.5294

0.5319

0.5

1

-0.25

0

0.25

0.6190

0.5833

0.5397

0.6078

0.5341

0.5091

0.5714

0.5614

0.5660

  

0

1

1

0.4545

0.4878

0.4844

0.5091

0.4565

0.4444

0.4828

0.4677

0.4643

  

0.5

2

1.5

0.5088

0.5625

0.5000

0.5208

0.5125

0.5000

0.4615

0.4510

0.4565

 

8

-0.25

0

0.25

0.5303

0.4762

0.5000

0.4583

0.4815

0.5172

0.5273

0.5283

0.5385

  

0

1

1

0.5500

0.5676

0.5714

0.5532

0.6076

0.5333

0.5455

0.5238

0.5238

  

0.5

2

1.5

0.5479

0.5385

0.5224

0.5660

0.5584

0.4655

0.4828

0.4717

0.4630

0.9

1

-0.25

0

0.25

0.5088

0.5000

0.4746

0.4800

0.4884

0.4906

0.4717

0.4808

0.4808

  

0

1

1

0.4915

0.5106

0.4848

0.5000

0.4479

0.4151

0.4259

0.4286

0.4200

  

0.5

2

1.5

0.5789

0.5294

0.5161

0.5098

0.5595

0.4783

0.4773

0.5333

0.5455

 

8

-0.25

0

0.25

0.4559

0.5435

0.5147

0.4909

0.4875

0.5000

0.5000

0.4902

0.4902

  

0

1

1

0.4815

0.5294

0.5283

0.5366

0.5429

0.6038

0.6038

0.5686

0.5686

  

0.5

2

1.5

0.4407

0.4524

0.5000

0.5102

0.5250

0.4222

0.4222

0.4048

0.4048

To investigate powers for the proposed CIs, we calculated the power in both the first and second simulation study. The results are shown in Tables 12 and 13. There is very little power in both the first and second simulation study to exclude a difference of zero.

Results of simulation studies

From Tables 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13, we have the following findings. First, when Σ is unknown, the CIs based on the the Bootstrap-resampling-based methods except for B 3 behave satisfactorily in the sense that their ECPs are close to the pre-specified confidence level 95 % (e.g., see Tables 3 and 6); the CI based on the Bootstrap-resampling-based method B 1 generally yielded shorter ECWs than others (e.g., see Tables 4 and 7); the CIs corresponding to bivariate t-distribution are generally wider than those corresponding to bivariate normal distribution; the ECWs decrease as the correlation coefficient ρ increases. Second, the RNCPs of all the considered CIs lie in the interval [0.4,0.6] (e.g., see Tables 5 and 8), which show that our derived CIs generally demonstrate symmetry. Third, when \({\sigma _{1}^{2}}={\sigma _{2}^{2}}\), the CIs based on statistics T 3, T 4 and T 5 behave unsatisfactory (e.g., see Tables 9 and 10) because their corresponding ECPs are almost less than the pre-specified confidence level 95 %. Fourth, powers corresponding to W a and B 1 are larger than others (e.g., see Tables 12 and 13). From the above findings, we would recommend the usage of the Bootstrap-resampling-based CI (i.e., B 1) because its coverage probability is generally close to the pre-chosen confidence level, it consistently yields the shortest interval width even when sample size is small, it usually guarantees its ratios of the MNCPs to the non-coverage probabilities lying in [0.4, 0.6], and its power is usually larger than others.
Table 9

ECPs of various confidence intervals with different ρ and δ, μ 1, μ 2, (n,n 1,n 2)=(5,5,2), when \({\sigma _{1}^{2}}={\sigma _{2}^{2}}=4\)

Bivariate normal distribution

ρ

δ

μ 1

μ 2

T 3

T 4

T 5

W s

W a

B 1

B 2

B 3

B 4

-0.9

-0.25

0

0.25

0.935

0.960

0.906

0.920

0.880

0.952

0.954

0.947

0.954

 

0

1

1

0.944

0.956

0.894

0.920

0.869

0.946

0.947

0.933

0.947

 

0.5

2

1.5

0.944

0.967

0.902

0.931

0.883

0.951

0.953

0.942

0.951

-0.5

-0.25

0

0.25

0.941

0.961

0.903

0.910

0.861

0.942

0.943

0.939

0.943

 

0

1

1

0.937

0.958

0.900

0.915

0.862

0.950

0.952

0.949

0.951

 

0.5

2

1.5

0.941

0.962

0.898

0.925

0.882

0.952

0.957

0.952

0.957

-0.1

-0.25

0

0.25

0.933

0.958

0.900

0.903

0.838

0.944

0.945

0.945

0.946

 

0

1

1

0.939

0.966

0.907

0.912

0.853

0.952

0.951

0.954

0.953

 

0.5

2

1.5

0.943

0.975

0.924

0.943

0.892

0.961

0.959

0.960

0.959

0

-0.25

0

0.25

0.936

0.964

0.914

0.913

0.860

0.949

0.949

0.950

0.950

 

0

1

1

0.925

0.959

0.906

0.908

0.861

0.941

0.941

0.940

0.940

 

0.5

2

1.5

0.932

0.968

0.913

0.924

0.887

0.952

0.952

0.951

0.951

0.1

-0.25

0

0.25

0.922

0.960

0.918

0.911

0.858

0.948

0.948

0.948

0.947

 

0

1

1

0.923

0.963

0.909

0.906

0.859

0.944

0.946

0.944

0.944

 

0.5

2

1.5

0.928

0.969

0.913

0.935

0.889

0.946

0.947

0.947

0.946

0.5

-0.25

0

0.25

0.927

0.968

0.923

0.904

0.843

0.950

0.947

0.934

0.947

 

0

1

1

0.928

0.964

0.923

0.913

0.857

0.942

0.944

0.935

0.947

 

0.5

2

1.5

0.924

0.978

0.933

0.947

0.901

0.960

0.958

0.943

0.960

0.9

-0.25

0

0.25

0.913

0.947

0.974

0.929

0.880

0.951

0.951

0.777

0.951

 

0

1

1

0.908

0.952

0.976

0.930

0.883

0.947

0.955

0.781

0.951

 

0.5

2

1.5

0.913

0.942

0.974

0.974

0.944

0.946

0.953

0.778

0.954

Bivariate t-distribution

-0.9

-0.25

0

0.25

0.922

0.972

0.908

0.929

0.870

0.952

0.953

0.946

0.956

 

0

1

1

0.915

0.973

0.914

0.935

0.868

0.948

0.943

0.937

0.948

 

0.5

2

1.5

0.930

0.978

0.914

0.937

0.873

0.948

0.950

0.941

0.951

-0.5

-0.25

0

0.25

0.929

0.976

0.921

0.939

0.869

0.942

0.941

0.940

0.945

 

0

1

1

0.931

0.975

0.925

0.935

0.872

0.943

0.942

0.943

0.946

 

0.5

2

1.5

0.922

0.971

0.910

0.924

0.868

0.953

0.951

0.950

0.955

-0.1

-0.25

0

0.25

0.932

0.973

0.922

0.925

0.856

0.951

0.951

0.955

0.954

 

0

1

1

0.926

0.971

0.924

0.923

0.859

0.941

0.942

0.946

0.947

 

0.5

2

1.5

0.924

0.972

0.918

0.921

0.859

0.950

0.948

0.954

0.955

0

-0.25

0

0.25

0.919

0.973

0.921

0.918

0.852

0.944

0.944

0.949

0.949

 

0

1

1

0.925

0.972

0.923

0.925

0.864

0.940

0.940

0.947

0.947

 

0.5

2

1.5

0.939

0.977

0.924

0.926

0.857

0.950

0.950

0.954

0.954

0.1

-0.25

0

0.25

0.930

0.971

0.929

0.928

0.857

0.954

0.954

0.956

0.956

 

0

1

1

0.929

0.982

0.927

0.928

0.857

0.949

0.949

0.950

0.951

 

0.5

2

1.5

0.934

0.979

0.924

0.930

0.859

0.952

0.953

0.957

0.957

0.5

-0.25

0

0.25

0.929

0.973

0.947

0.940

0.864

0.944

0.950

0.942

0.951

 

0

1

1

0.920

0.976

0.937

0.928

0.861

0.943

0.944

0.936

0.946

 

0.5

2

1.5

0.939

0.970

0.942

0.930

0.868

0.945

0.947

0.942

0.951

0.9

-0.25

0

0.25

0.923

0.969

0.978

0.943

0.880

0.939

0.938

0.797

0.939

 

0

1

1

0.920

0.966

0.977

0.952

0.887

0.939

0.942

0.795

0.949

 

0.5

2

1.5

0.931

0.965

0.979

0.944

0.878

0.953

0.944

0.804

0.947

Table 10

ECW of various confidence interals with different ρ and δ, μ 1, μ 2, (n,n 1,n 2)=(5,5,2), when \({\sigma _{1}^{2}}={\sigma _{2}^{2}}=4\)

Bivariate normal distribution

ρ

δ

μ 1

μ 2

T 3

T 4

T 5

W s

W a

B 1

B 2

B 3

B 4

-0.9

-0.25

0

0.25

6.350

7.032

5.019

3.821

3.148

5.150

5.370

5.149

5.368

 

0

1

1

6.389

7.038

5.047

3.833

3.162

5.151

5.370

5.151

5.370

 

0.5

2

1.5

6.447

7.052

5.060

3.947

3.290

5.152

5.370

5.152

5.370

-0.5

-0.25

0

0.25

5.883

6.473

4.610

3.503

2.894

4.800

4.881

4.799

4.880

 

0

1

1

5.885

6.436

4.606

3.510

2.903

4.800

4.881

4.799

4.879

 

0.5

2

1.5

5.877

6.413

4.606

3.655

3.078

4.802

4.883

4.802

4.882

-0.1

-0.25

0

0.25

5.282

5.891

4.187

3.198

2.651

4.333

4.337

4.333

4.338

 

0

1

1

5.318

5.898

4.186

3.213

2.670

4.335

4.340

4.334

4.338

 

0.5

2

1.5

5.270

5.888

4.183

3.397

2.893

4.336

4.340

4.336

4.339

0

-0.25

0

0.25

5.114

5.733

4.046

3.096

2.571

4.190

4.190

4.189

4.189

 

0

1

1

5.147

5.729

4.076

3.139

2.614

4.190

4.190

4.190

4.190

 

0.5

2

1.5

5.123

5.763

4.069

3.337

2.849

4.191

4.191

4.189

4.189

0.1

-0.25

0

0.25

4.869

5.519

3.921

3.004

2.500

4.033

4.037

4.032

4.037

 

0

1

1

4.870

5.550

3.899

3.004

2.504

4.032

4.037

4.033

4.038

 

0.5

2

1.5

4.849

5.636

3.926

3.254

2.795

4.031

4.036

4.033

4.037

0.5

-0.25

0

0.25

3.805

5.050

3.412

2.608

2.188

3.202

3.360

3.202

3.360

 

0

1

1

3.811

5.019

3.398

2.624

2.213

3.201

3.360

3.199

3.357

 

0.5

2

1.5

3.857

5.211

3.401

2.955

2.583

3.200

3.359

3.200

3.360

0.9

-0.25

0

0.25

1.776

5.606

2.702

2.133

1.832

1.537

2.505

1.537

2.505

 

0

1

1

1.766

5.561

2.676

2.147

1.853

1.539

2.503

1.538

2.503

 

0.5

2

1.5

1.784

5.548

2.689

2.554

2.303

1.537

2.505

1.536

2.504

Bivariate t-distribution

-0.9

-0.25

0

0.25

35.039

42.148

28.140

21.360

17.207

30.479

31.779

31.062

32.486

 

0

1

1

35.226

42.660

28.523

21.569

17.374

30.470

31.763

31.048

32.470

 

0.5

2

1.5

34.854

42.020

28.032

21.260

17.135

30.472

31.771

31.038

32.484

-0.5

-0.25

0

0.25

32.156

38.993

25.809

19.534

15.765

28.402

28.881

28.936

29.495

 

0

1

1

33.177

39.103

26.338

19.953

16.106

28.417

28.901

28.961

29.518

 

0.5

2

1.5

31.999

38.876

25.558

19.403

15.677

28.393

28.870

28.941

29.480

-0.1

-0.25

0

0.25

28.753

36.668

23.542

17.849

14.456

25.621

25.643

26.126

26.164

 

0

1

1

28.672

36.649

23.652

17.809

14.435

25.637

25.661

26.146

26.184

 

0.5

2

1.5

29.087

35.900

23.651

17.894

14.523

25.622

25.645

26.140

26.175

0

-0.25

0

0.25

27.123

35.382

22.633

17.113

13.892

24.786

24.786

25.284

25.284

 

0

1

1

27.852

35.371

23.033

17.424

14.146

24.797

24.797

25.292

25.292

 

0.5

2

1.5

27.607

34.434

22.581

17.116

13.919

24.786

24.786

25.288

25.288

0.1

-0.25

0

0.25

26.299

34.969

22.037

16.679

13.565

23.842

23.869

24.322

24.332

 

0

1

1

26.797

35.384

22.411

16.960

13.787

23.854

23.882

24.349

24.365

 

0.5

2

1.5

26.420

34.911

22.164

16.798

13.679

23.864

23.891

24.357

24.372

0.5

-0.25

0

0.25

20.192

32.428

19.137

14.443

11.860

18.938

19.877

19.369

20.262

 

0

1

1

20.217

32.478

19.118

14.526

11.942

18.950

19.891

19.385

20.271

 

0.5

2

1.5

20.314

30.975

18.783

14.325

11.783

18.928

19.869

19.361

20.257

0.9

-0.25

0

0.25

9.426

36.100

15.345

11.627

9.744

9.094

14.818

9.355

15.174

 

0

1

1

9.491

34.843

15.055

11.622

9.750

9.090

14.804

9.352

15.167

 

0.5

2

1.5

9.569

35.234

15.210

11.735

9.875

9.098

14.813

9.353

15.176

Table 11

RNCP of various confidence intervals with different ρ and δ, μ 1, μ 2, (n,n 1,n 2)=(5,5,2), when \({\sigma _{1}^{2}}={\sigma _{2}^{2}}=4\)

Bivariate normal distribution

ρ

δ

μ 1

μ 2

T 3

T 4

T 5

W s

W a

B 1

B 2

B 3

B 4

-0.9

-0.25

0

0.25

0.4697

0.5652

0.4787

0.4000

0.5187

0.4583

0.5217

0.5185

0.5217

 

0

1

1

0.4464

0.5968

0.4190

0.4304

0.4151

0.4815

0.4340

0.4478

0.4340

 

0.5

2

1.5

0.4386

0.6170

0.4796

0.6324

0.4796

0.4898

0.4792

0.5000

0.4800

-0.5

-0.25

0

0.25

0.4915

0.5577

0.5258

0.4396

0.5258

0.5000

0.5088

0.5246

0.5088

 

0

1

1

0.4444

0.5577

0.4800

0.4824

0.4800

0.4706

0.4286

0.4423

0.4490

 

0.5

2

1.5

0.4915

0.5814

0.4950

0.6081

0.4902

0.4286

0.4545

0.4898

0.4773

-0.1

-0.25

0

0.25

0.4776

0.6042

0.4800

0.4330

0.4800

0.4912

0.4727

0.4630

0.4630

 

0

1

1

0.4918

0.5714

0.4839

0.4773

0.4839

0.4583

0.4490

0.4783

0.4681

 

0.5

2

1.5

0.5862

0.6563

0.5200

0.6724

0.5132

0.4750

0.4878

0.4878

0.4878

0

-0.25

0

0.25

0.5077

0.5641

0.5233

0.4598

0.5233

0.5385

0.5385

0.5600

0.5600

 

0

1

1

0.5333

0.5769

0.5000

0.4891

0.5000

0.5085

0.5085

0.5000

0.5000

 

0.5

2

1.5

0.5000

0.5957

0.5116

0.6053

0.5057

0.4167

0.4167

0.4286

0.4286

0.1

-0.25

0

0.25

0.5256

0.5652

0.5000

0.4205

0.5000

0.5000

0.5192

0.5000

0.5094

 

0

1

1

0.4545

0.5625

0.4778

0.4681

0.4725

0.5179

0.5273

0.5179

0.5000

 

0.5

2

1.5

0.5694

0.6486

0.5057

0.6212

0.5057

0.5741

0.5660

0.5556

0.5556

0.5

-0.25

0

0.25

0.5139

0.6604

0.4805

0.4167

0.4805

0.4510

0.4630

0.4615

0.4815

 

0

1

1

0.4930

0.6667

0.5513

0.5057

0.5584

0.4746

0.5088

0.5077

0.5283

 

0.5

2

1.5

0.5067

0.7027

0.5455

0.6604

0.5373

0.4878

0.5238

0.5439

0.5250

0.9

-0.25

0

0.25

0.5057

0.8286

0.5556

0.4028

0.5769

0.5000

0.4694

0.4798

0.4800

 

0

1

1

0.4624

0.8333

0.5000

0.5000

0.5000

0.5185

0.4565

0.5227

0.5000

 

0.5

2

1.5

0.4943

0.7733

0.4074

0.6538

0.4231

0.4630

0.5319

0.4775

0.5435

Bivariate t-distribution

-0.9

-0.25

0

0.25

0.5195

0.6977

0.4891

0.5000

0.4930

0.4750

0.4375

0.4444

0.4318

 

0

1

1

0.4706

0.6905

0.5349

0.5152

0.5231

0.4717

0.5690

0.5469

0.5769

 

0.5

2

1.5

0.5362

0.7436

0.5000

0.5469

0.5556

0.5192

0.4800

0.5085

0.4898

-0.5

-0.25

0

0.25

0.5915

0.6818

0.4684

0.4426

0.4426

0.4915

0.5085

0.5000

0.5000

 

0

1

1

0.4928

0.7143

0.4800

0.4531

0.4462

0.4912

0.4576

0.4737

0.4815

 

0.5

2

1.5

0.5256

0.7021

0.5056

0.5526

0.5526

0.4167

0.3878

0.3529

0.3696

-0.1

-0.25

0

0.25

0.3971

0.5526

0.4937

0.4667

0.4667

0.5102

0.5000

0.5333

0.5217

 

0

1

1

0.5270

0.7250

0.5395

0.5325

0.5325

0.4667

0.4655

0.4630

0.4717

 

0.5

2

1.5

0.4605

0.5750

0.4444

0.4810

0.4810

0.5000

0.4717

0.5106

0.5000

0

-0.25

0

0.25

0.5309

0.6341

0.5000

0.4819

0.4878

0.5088

0.5088

0.4902

0.4902

 

0

1

1

0.5067

0.6389

0.4805

0.4865

0.4800

0.5667

0.5667

0.5660

0.5660

 

0.5

2

1.5

0.5574

0.7097

0.5132

0.5068

0.5000

0.5200

0.5200

0.5435

0.5435

0.1

-0.25

0

0.25

0.5714

0.5294

0.5556

0.5139

0.5139

0.5532

0.5652

0.5814

0.5814

 

0

1

1

0.5211

0.7813

0.5833

0.5833

0.5833

0.5098

0.4902

0.5000

0.4898

 

0.5

2

1.5

0.4925

0.6563

0.4800

0.5000

0.5000

0.5208

0.5106

0.5116

0.5116

0.5

-0.25

0

0.25

0.5493

0.6744

0.4717

0.4833

0.4833

0.4821

0.5000

0.5000

0.4800

 

0

1

1

0.4625

0.7083

0.4444

0.4861

0.4861

0.4386

0.4912

0.4688

0.4630

 

0.5

2

1.5

0.5161

0.6744

0.5172

0.5286

0.5286

0.5455

0.5283

0.5085

0.5306

0.9

-0.25

0

0.25

0.5455

0.8803

0.4348

0.5088

0.5088

0.4677

0.4677

0.4926

0.4754

 

0

1

1

0.5570

0.8534

0.5652

0.6042

0.6042

0.5000

0.5000

0.5194

0.4706

 

0.5

2

1.5

0.4348

0.8333

0.5714

0.4821

0.4821

0.5000

0.5357

0.4898

0.5370

Table 12

Power of various confidence intervals with different ρ and δ, μ 1, \(\mu _{2}, {\sigma _{1}^{2}}\) and (n,n 1,n 2)=(5,2,2) and \({\sigma _{2}^{2}}=4\)

ρ

\({\sigma _{1}^{2}}\)

δ

μ 1

μ 2

T 1

T 2

T g

W s

W a

B 1

B 2

B 3

B 4

-0.9

1

-0.25

0

0.25

6.40

4.35

5.10

7.30

12.60

5.20

5.10

6.40

5.05

  

0.5

2

1.5

7.25

5.20

5.50

9.10

13.70

6.50

6.35

7.80

6.35

 

8

-0.25

0

0.25

5.80

3.30

5.25

7.60

13.30

5.50

5.10

6.10

4.95

  

0.5

2

1.5

6.20

4.00

7.65

8.25

14.40

5.90

6.30

7.65

6.40

-0.5

1

-0.25

0

0.25

6.50

4.00

6.60

7.75

13.10

4.70

4.65

5.45

4.80

  

0.5

2

1.5

7.75

4.90

7.60

10.15

15.85

6.10

5.95

6.40

5.70

 

8

-0.25

0

0.25

6.60

4.55

7.00

9.55

15.30

5.80

5.40

6.20

5.70

  

0.5

2

1.5

5.80

4.10

6.05

8.25

13.85

5.50

5.70

5.85

5.45

-0.1

1

-0.25

0

0.25

6.95

3.55

8.45

6.90

12.70

4.65

4.50

4.70

4.65

  

0.5

2

1.5

7.70

4.85

7.55

9.90

15.75

6.80

6.80

6.85

6.65

 

8

-0.25

0

0.25

7.25

3.95

7.90

9.75

15.60

6.25

6.15

6.25

6.20

  

0.5

2

1.5

6.60

3.50

7.25

8.75

15.20

5.25

5.35

5.15

5.10

0

1

-0.25

0

0.25

8.10

4.60

7.10

8.20

13.40

5.45

5.45

5.45

5.45

  

0.5

2

1.5

8.35

4.70

8.50

11.50

17.90

6.55

6.55

6.65

6.65

 

8

-0.25

0

0.25

7.45

3.50

8.90

9.10

15.25

5.45

5.45

5.40

5.40

  

0.5

2

1.5

7.30

3.65

7.45

10.55

16.80

6.10

6.10

6.10

6.10

0.1

1

-0.25

0

0.25

7.05

3.95

9.85

8.40

13.85

5.45

5.60

5.60

5.70

  

0.5

2

1.5

7.55

4.45

8.45

11.55

16.90

5.85

6.15

5.90

5.95

 

8

-0.25

0

0.25

6.30

3.85

8.70

8.05

14.20

4.75

4.85

5.00

5.05

  

0.5

2

1.5

7.65

4.05

9.60

9.70

16.40

5.85

6.00

6.25

6.30

0.5

1

-0.25

0

0.25

7.30

4.15

9.35

6.95

12.90

5.10

4.85

6.15

4.90

  

0.5

2

1.5

8.40

4.75

8.15

12.70

19.35

6.00

5.95

7.10

6.15

 

8

-0.25

0

0.25

8.80

4.20

7.80

9.80

15.40

5.30

5.15

6.80

5.30

  

0.5

2

1.5

9.10

4.05

8.40

11.55

16.45

6.65

6.95

8.50

7.15

0.9

1

-0.25

0

0.25

7.30

5.25

8.10

7.50

13.60

5.10

5.35

7.20

5.40

  

0.5

2

1.5

8.45

5.35

8.55

18.00

26.95

7.55

7.70

8.25

7.75

 

8

-0.25

0

0.25

8.95

5.40

5.35

7.25

13.45

5.80

5.90

7.10

6.10

  

0.5

2

1.5

11.45

5.30

6.25

12.30

18.20

10.05

8.00

9.60

7.95

Table 13

Power of various confidence intervals with different ρ and δ, μ 1, μ 2, (n,n 1,n 2)=(5,5,2), when \({\sigma _{1}^{2}}={\sigma _{2}^{2}}=4\)

Bivariate normal distribution

ρ

δ

μ 1

μ 2

T 3

T 4

T 5

W s

W a

B 1

B 2

B 3

B 4

-0.9

-0.25

0

0.25

1.5

2.5

4.3

7.5

12.4

5.2

4.8

6.6

5.0

 

0.5

2

1.5

3.2

4.1

6.4

10.8

15.6

7.5

7.4

9.4

7.4

-0.5

-0.25

0

0.25

3.9

3.0

5.7

8.5

12.9

5.6

5.1

5.6

5.2

 

0.5

2

1.5

4.0

3.0

6.5

9.9

14.4

6.8

6.9

7.3

6.8

-0.1

-0.25

0

0.25

3.6

2.9

6.2

9.5

14.8

5.8

5.8

5.9

6.0

 

0.5

2

1.5

5.5

4.9

8.7

11.3

16.4

8.3

8.2

7.9

7.9

0

-0.25

0

0.25

4.4

3.3

6.9

9.8

14.7

5.7

5.7

5.9

5.9

 

0.5

2

1.5

4.7

4.0

7.6

10.8

16.7

7.9

7.9

7.6

7.6

0.1

-0.25

0

0.25

3.5

2.9

5.5

8.2

13.3

5.8

5.7

5.7

5.7

 

0.5

2

1.5

5.1

4.3

8.1

11.6

16.2

7.6

7.3

7.5

7.4

0.5

-0.25

0

0.25

4.7

3.3

5.9

9.6

14.7

6.7

6.5

8.5

6.3

 

0.5

2

1.5

5.3

5.1

8.4

13.1

17.9

11.1

10.8

13.2

10.6

0.9

-0.25

0

0.25

3.9

3.5

4.7

9.7

15.4

10.7

6.5

27.5

6.4

 

0.5

2

1.5

9.1

6.0

8.2

13.7

18.0

27.9

11.4

27.3

11.2

Bivariate t-distribution

-0.9

-0.25

0

0.25

1.2

2.1

4.0

6.7

11.6

4.9

5.1

5.9

4.7

 

0.5

2

1.5

1.5

2.0

4.0

6.1

11.4

4.9

5.0

6.1

4.3

-0.5

-0.25

0

0.25

2.0

1.5

4.2

6.2

12.2

4.8

5.1

5.1

4.9

 

0.5

2

1.5

2.0

1.8

5.0

6.8

12.7

6.3

6.3

6.4

5.9

-0.1

-0.25

0

0.25

2.9

2.8

6.0

8.3

15.2

7.1

7.0

6.7

6.4

 

0.5

2

1.5

2.0

1.9

5.0

7.0

12.7

4.4

4.4

4.1

4.0

0

-0.25

0

0.25

2.5

2.0

4.1

6.7

12.4

5.0

5.0

4.5

4.5

 

0.5

2

1.5

2.2

1.9

4.6

6.5

12.8

6.1

6.1

5.9

5.9

0.1

-0.25

0

0.25

2.4

2.1

4.4

7.0

12.0

5.2

5.1

5.0

5.0

 

0.5

2

1.5

2.9

2.7

5.6

7.4

13.2

5.3

5.1

4.9

5.0

0.5

-0.25

0

0.25

1.3

2.0

4.4

6.1

11.4

5.0

5.2

6.4

5.2

 

0.5

2

1.5

1.7

2.0

4.7

6.1

11.4

4.9

5.1

5.9

4.7

0.9

-0.25

0

0.25

1.3

2.8

3.4

5.0

10.4

5.0

4.8

5.4

4.4

 

0.5

2

1.5

2.1

2.2

2.7

5.1

11.7

5.7

5.8

5.8

5.2

An worked example

In this subsection, the data introduced in Section for the action of two doses of formoterol solution aerosol are used to illustrate the proposed methodologies. In this example, we are interested in CI construction of the difference of two FEV1 values for two doses of formoterol solution aerosol. Under the previously given notation, we have n=7, n 1=9, n 2=8, \(\hat \delta =a\overline {x}_{1}^{(n)}+\left (1-a\right)\overline {x}_{1}^{(n_{1})}-b\overline {x}_{2}^{(n)}-(1-b)\overline {x}_{2}^{(n_{2})}=-0.0840\) (or \(\hat {\delta }=\sum _{j=1}^{n+n_{1}}x_{1j}/(n+n_{1})-\sum _{j=1}^{n+n_{2}}x_{2j}/(n+n_{2})=0.0228\)). Various 95 % CIs for δ under Σ unknown assumption are presented in Table 14. Examination of Table 14 shows that the actions of two doses of formaterol solutions aerosol are the same because all the derived CIs include zero.
Table 14

Various 95 % confidence intervals for δ=μ 1μ 2 based on formoterol solution aerosol

 

T 1

T 2

T 3

T 4

T 5

T g

Lower

-0.2751

-0.4764

-0.472

-0.5542

-0.4431

-0.4883

Upper

0.1071

0.5220

0.3741

0.5999

0.4888

0.5039

Width

0.3822

0.9984

0.8461

1.1541

0.9319

0.9922

 

W s

W a

B 1

B 2

B 3

B 4

Lower

-0.5940

-0.5787

-0.5408

-0.5938

-0.5259

-0.5681

Upper

0.6495

0.6334

0.3995

0.4394

0.4309

0.4058

Width

1.2435

1.2121

0.9403

1.0332

0.9568

0.9739

Discussion

Although testing equivalence of two correlated means with incomplete data has been studied, there is little work done on their interval estimators. To address the issue, this paper proposes various interval estimators of the difference of two correlated means for Σ known and unknown cases based on the large sample method, hybrid method and Bootstrap-resampling method. Extensive simulation studies are conducted to evaluate the finite performance of the proposed CIs in terms of the empirical coverage probability, empirical interval width and ratio of the mesial non-coverage probability to the non-coverage probability (RNCP). Empirical results evidence that the Bootstrap-resampling-based CIs B 1, B 2, B 4 behave satisfactorily for small to moderate sample sizes in the sense that their coverage probabilities could be well controlled around the pre-specified nominal confidence level and their RNCPs almost lie in the interval [0.4, 0.6]. However, confidence intervals based on the large sample method and hybrid method behave unsatisfactory for small sample sizes because the distributions of statistics T 1,,T 5 are asymptotical, and these asymptotical distributions are proper only when N i . When Σ is unknown, using GEE method to estimate variance is less efficient.

It is interesting to investigate confidence interval construction of the difference of two means with incomplete correlated data under missing at random and non-ignorable missing data mechanism assumptions of bivariate variables. We are working on the topics.

Conclusion

According to the aforementioned findings, we can draw the following conclusions. The Bootstrap-resampling-based CI B 1 is a desirable interval estimator for the difference of two means with incomplete correlated data.

Declarations

Acknowledgements

The research of Hui-Qiong LI was supported by the Natural Science Foundation of China (11201412, 11561075). The work of the second author was partially supported by the grants from the National Science Foundation of China (11225103).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.

Authors’ Affiliations

(1)
Department of Statistics, Yunnan University, Kunming, China
(2)
School of Statistics, Beijing Normal University, Beijing, China

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© Li et al. 2016

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