Table 2 Scenarios of random effects distributions and their true parameters

Scenario 1-3: Normal distribution (N)

N(mean,variance)

1

N(0,0.1581²)

0.000

0.025

20.0%

2

N(0,0.2582²)

0.000

0.067

40.0%

3

N(0,0.6325²)

0.000

0.400

80.0%

Scenario 4-6: Skew-normal distribution with moderate positive skewness (pSN1)

SN(location,scale,slant)

4

S N(−0.1724,0.2547,5)

0.000

0.025

20.0%

5

S N(−0.2812,0.4159,5)

0.000

0.067

40.0%

6

S N(−0.6883,1.0198,5)

0.000

0.400

80.0%

Scenario 7-9: Skew-normal distribution with large positive skewness (pSN2)

SN(location,scale,slant)

7

S N(−0.1734,0.2557,20)

0.000

0.025

20.0%

8

S N(−0.2829,0.4192,20)

0.000

0.067

40.0%

9

S N(−0.6923,1.0258,20)

0.000

0.400

80.0%

Scenario 10-12: Skew-normal distribution with moderate negative skewness (nSN1)

SN(location,scale,slant)

10

S N(0.1715,0.2546,−5)

0.000

0.025

20.0%

11

S N(0.2803,0.4154,−5)

0.000

0.067

40.0%

12

S N(0.6874,1.0195,−5)

0.000

0.400

80.0%

Scenario 13-15: Skew-normal distribution with large negative skewness (nSN2)

SN(location,scale,slant)

13

S N(0.1725,0.2556,−20)

0.000

0.025

20.0%

14

S N(0.2820,0.4191,−20)

0.000

0.067

40.0%

15

S N(0.6914,1.0255,−20)

0.000

0.400

80.0%

Scenario 16-18: Shifted exponential distribution (EXP)

EXP(rate,shift)

16

E X P(5.1507,−0.1348)

0.000

0.025

20.0%

17

E X P(3.1542,−0.2202)

0.000

0.067

40.0%

18

E X P(1.2877,−0.5377)

0.000

0.400

80.0%

Scenario 19-21: Shifted log-normal distribution (LN)

LN(mean,variance,shift)

19

L N(0,0.1578²,−1)

0.000

0.025

20.0%

20

L N(0,0.2569²,−1)

0.000

0.067

40.0%

21

L N(0,0.6147²,−1)

0.000

0.400

80.0%