Simultaneous Confidence Intervals for All Pairwise Differences between the Coefficients of Variation of Multiple Birnbaum–Saunders Distributions
Abstract
:1. Introduction
2. Methods
2.1. The PB Approach
Algorithm 1 The PB approach 

2.2. The GCI Approach
 The observed value of $Q(Y,y,\eta ,\vartheta )$ denoted as $Q(y,y,\eta ,\vartheta )$ is free of nuisance parameter $\vartheta $.
 The probability distribution of $Q(Y,y,\eta ,\vartheta )$ is free of unknown parameters.
Algorithm 2 The GCI approach 

2.3. The MOVER Approach
2.3.1. The MOVER Based on ACI Approach
Algorithm 3 The MOVER based on ACI approach 
2.3.2. The MOVER Based on GCI Approach
2.4. The BayCrI Approach
 (1)
 Calculate $a({r}_{i})$ and ${b}^{+}({r}_{i})$.
 (2)
 Simulate ${u}_{i}$ and ${v}_{i}$ from $U(0,a({r}_{i}))$ and $U(0,{b}^{+}({r}_{i}))$, where $U(s,t)$ refer to a uniform distribution with parameters s and t, then compute ${\rho}_{i}={v}_{i}/{u}_{i}^{{r}_{i}}$.
 (3)
 If ${u}_{i}\le [\pi {({\rho}_{i}{\mathit{x}}_{ij})]}^{1/({r}_{i}+1)}$, set ${\tilde{\beta}}_{i}={\rho}_{i}$, otherwise go back to step (2).
Algorithm 5 The BayCrI and HPD approaches 

3. Simulation Study Settings and Results
4. Empirical Application of the Methods with Three Real Datasets
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Scenarios  (n_{1}, n_{2}, …, n_{k})  (a_{1}, a_{2}, …, a_{k}) 

k = 3  
1–6  (30_{3})  (0.5_{3}),(0.5, 1.0_{2}), (1.0_{3}), (0.5, 1.0, 2.0), (1.0, 1.5, 2.0), (1.5, 2.0_{2}) 
7–12  (30_{2}, 50)  (0.5_{3}),(0.5, 1.0_{2}), (1.0_{3}), (0.5, 1.0, 2.0), (1.0, 1.5, 2.0), (1.5, 2.0_{2}) 
13–18  (50_{3})  (0.5_{3}), (0.5, 1.0_{2}), (1.0_{3}), (0.5, 1.0, 2.0), (1.0, 1.5, 2.0), (1.5, 2.0_{2}) 
19–24  (50_{2}, 100)  (0.5_{3}), (0.5, 1.0_{2}), (1.0_{3}), (0.5, 1.0, 2.0), (1.0, 1.5, 2.0), (1.5, 2.0_{2}) 
25–30  (100_{3})  (0.5_{3}), (0.5, 1.0_{2}), (1.0_{3}), (0.5, 1.0, 2.0), (1.0, 1.5, 2.0), (1.5, 2.0_{2}) 
k = 5  
31–36  (30_{2}, 50_{3})  (0.5_{3}, 1.0, 2.0), (0.5_{2}, 1.0_{2}, 1.5), (0.5,1.0_{3}, 1.5), (0.5, 1.0_{2}, 2.0_{2}),(1.0_{3}, 1.5_{2}), (1.0, 1.5, 2.0_{3}) 
37–42  (30_{2}, 50_{2}, 100)  (0.5_{3}, 1.0, 2.0), (0.5_{2}, 1.0_{2}, 1.5), (0.5, 1.0_{3}, 1.5), (0.5, 1.0_{2}, 2.0_{2}),(1.0_{3}, 1.5_{2}), (1.0, 1.5, 2.0_{3}) 
43–48  (50_{5})  (0.5_{3}, 1.0, 2.0), (0.5_{2}, 1.0_{2}, 1.5), (0.5,1.0_{3}, 1.5), (0.5, 1.0_{2}, 2.0_{2}),(1.0_{3}, 1.5_{2}), (1.0, 1.5, 2.0_{3}) 
49–54  (30, 50_{2}, 100_{2})  (0.5_{3}, 1.0, 2.0), (0.5_{2}, 1.0_{2}, 1.5), (0.5,1.0_{3}, 1.5), (0.5, 1.0_{2}, 2.0_{2}),(1.0_{3}, 1.5_{2}), (1.0, 1.5, 2.0_{3}) 
55–60  (50_{2}, 100_{3})  (0.5_{3}, 1.0, 2.0), (0.5_{2}, 1.0_{2}, 1.5), (0.5,1.0_{3}, 1.5), (0.5, 1.0_{2}, 2.0_{2}),(1.0_{3}, 1.5_{2}), (1.0, 1.5, 2.0_{3}) 
k = 10  
61–66  (30_{5}, 50_{5})  (0.5_{3}, 1.0_{7}), (0.5_{3}, 1.0_{4}, 1.5_{3}), (0.5_{3}, 1.0_{2},1.5_{3}, 2.0_{2}), (1.0_{4}, 1.5_{3},2.0_{3}), (1.0_{3}, 1.5_{3}, 2.0_{4}), (1.0_{2}, 1.5_{2}, 2.0_{6}) 
67–72  (30_{5}, 50_{3}, 100_{2})  (0.5_{3}, 1.0_{7}), (0.5_{3}, 1.0_{4}, 1.5_{3}), (0.5_{3}, 1.0_{2},1.5_{3}, 2.0_{2}), (1.0_{4}, 1.5_{3},2.0_{3}), (1.0_{3}, 1.5_{3}, 2.0_{4}), (1.0_{2}, 1.5_{2}, 2.0_{6}) 
73–78  (30_{3}, 50_{4}, 100_{3})  (0.5_{3}, 1.0_{7}), (0.5_{3}, 1.0_{4}, 1.5_{3}), (0.5_{3}, 1.0_{2},1.5_{3}, 2.0_{2}), (1.0_{4}, 1.5_{3},2.0_{3}), (1.0_{3}, 1.5_{3}, 2.0_{4}), (1.0_{2}, 1.5_{2}, 2.0_{6}) 
79–84  (50_{6}, 100_{4})  (0.5_{3}, 1.0_{7}), (0.5_{3}, 1.0_{4}, 1.5_{3}), (0.5_{3}, 1.0_{2},1.5_{3}, 2.0_{2}), (1.0_{4}, 1.5_{3},2.0_{3}), (1.0_{3}, 1.5_{3}, 2.0_{4}), (1.0_{2}, 1.5_{2}, 2.0_{6}) 
Scenarios  Coverage Probability (Average Length)  

PB  GCI  MOVER1  MOVER2  BayCrI  HPD  
1  0.928  0.946  0.944  0.952  0.945  0.950 
(0.3391)  (0.4030)  (0.3613)  (0.4154)  (0.4006)  (0.3968)  
2  0.922  0.950  0.939  0.953  0.948  0.946 
(0.5169)  (0.5836)  (0.5452)  (0.5979)  (0.5789)  (0.5736)  
3  0.935  0.953  0.943  0.958  0.951  0.946 
(0.5892)  (0.6591)  (0.6197)  (0.6756)  (0.6536)  (0.6481)  
4  0.916  0.949  0.932  0.953  0.947  0.944 
(0.5237)  (0.5624)  (0.5356)  (0.5740)  (0.5554)  (0.5500)  
5  0.929  0.946  0.932  0.952  0.944  0.938 
(0.6222)  (0.6490)  (0.6272)  (0.6605)  (0.6398)  (0.6339)  
6  0.930  0.952  0.936  0.958  0.948  0.942 
(0.6301)  (0.6304)  (0.6183)  (0.6382)  (0.6187)  (0.6135)  
7  0.928  0.951  0.945  0.954  0.948  0.951 
(0.3175)  (0.3711)  (0.3367)  (0.3809)  (0.3690)  (0.3652)  
8  0.920  0.946  0.934  0.950  0.945  0.943 
(0.4719)  (0.5288)  (0.4973)  (0.5403)  (0.5247)  (0.5200)  
9  0.923  0.946  0.934  0.950  0.946  0.941 
(0.5503)  (0.6096)  (0.5769)  (0.6225)  (0.6048)  (0.5994)  
10  0.924  0.949  0.937  0.952  0.947  0.944 
(0.4721)  (0.5145)  (0.4879)  (0.5241)  (0.5094)  (0.5041)  
11  0.927  0.951  0.940  0.957  0.949  0.944 
(0.5739)  (0.6072)  (0.5864)  (0.6165)  (0.5998)  (0.5941)  
12  0.922  0.947  0.933  0.952  0.945  0.938 
(0.5830)  (0.5877)  (0.5771)  (0.5937)  (0.5780)  (0.5731)  
13  0.942  0.953  0.952  0.956  0.951  0.954 
(0.2711)  (0.3018)  (0.2830)  (0.3078)  (0.3004)  (0.2978)  
14  0.931  0.953  0.944  0.956  0.952  0.949 
(0.4098)  (0.4436)  (0.4263)  (0.4502)  (0.4411)  (0.4372)  
15  0.942  0.954  0.950  0.957  0.954  0.952 
(0.4670)  (0.5036)  (0.4857)  (0.5114)  (0.5005)  (0.4964)  
16  0.929  0.952  0.943  0.956  0.951  0.948 
(0.4080)  (0.4288)  (0.4168)  (0.4342)  (0.4249)  (0.4210)  
17  0.940  0.951  0.944  0.955  0.949  0.945 
(0.4826)  (0.4969)  (0.4873)  (0.5023)  (0.4921)  (0.4878)  
18  0.940  0.950  0.943  0.955  0.950  0.944 
(0.4817)  (0.4831)  (0.4781)  (0.4869)  (0.4774)  (0.4734)  
19  0.937  0.951  0.949  0.954  0.951  0.952 
(0.2483)  (0.2727)  (0.2578)  (0.2770)  (0.2718)  (0.2691)  
20  0.932  0.950  0.944  0.953  0.948  0.947 
(0.3646)  (0.3918)  (0.3780)  (0.3967)  (0.3896)  (0.3861)  
21  0.934  0.952  0.945  0.954  0.949  0.946 
(0.4279)  (0.4572)  (0.4429)  (0.4626)  (0.4544)  (0.4503)  
22  0.933  0.950  0.943  0.954  0.950  0.946 
(0.3606)  (0.3829)  (0.3708)  (0.3871)  (0.3803)  (0.3766)  
23  0.936  0.953  0.943  0.957  0.951  0.945 
(0.4386)  (0.4572)  (0.4479)  (0.4613)  (0.4534)  (0.4493)  
24  0.934  0.951  0.945  0.953  0.949  0.946 
(0.4370)  (0.4424)  (0.4381)  (0.4449)  (0.4377)  (0.4340)  
25  0.943  0.951  0.951  0.953  0.950  0.948 
(0.1964)  (0.2079)  (0.2015)  (0.2101)  (0.2072)  (0.2054)  
26  0.935  0.947  0.943  0.949  0.946  0.942 
(0.2970)  (0.3096)  (0.3039)  (0.3120)  (0.3084)  (0.3058)  
27  0.933  0.942  0.939  0.944  0.940  0.939 
(0.3379)  (0.3520)  (0.3459)  (0.3548)  (0.3505)  (0.3476)  
28  0.936  0.948  0.946  0.950  0.949  0.946 
(0.2902)  (0.2993)  (0.2953)  (0.3012)  (0.2975)  (0.2948)  
29  0.938  0.949  0.945  0.950  0.948  0.945 
(0.3402)  (0.3481)  (0.3451)  (0.3500)  (0.3460)  (0.3431)  
30  0.940  0.949  0.944  0.951  0.947  0.944 
(0.3352)  (0.3386)  (0.3373)  (0.3399)  (0.3364)  (0.3336) 
Scenarios  Coverage Probability (Average Length)  

PB  GCI  MOVER1  MOVER2  BayCrI  HPD  
31  0.928  0.948  0.941  0.952  0.946  0.946 
(0.3816)  (0.4174)  (0.3941)  (0.4255)  (0.4142)  (0.4102)  
32  0.933  0.949  0.943  0.953  0.948  0.947 
(0.4279)  (0.4689)  (0.4452)  (0.4783)  (0.4658)  (0.4617)  
33  0.928  0.948  0.939  0.953  0.947  0.944 
(0.4792)  (0.5221)  (0.4981)  (0.5317)  (0.5181)  (0.5134)  
34  0.933  0.952  0.943  0.956  0.951  0.948 
(0.4691)  (0.5013)  (0.4809)  (0.5092)  (0.4966)  (0.4917)  
35  0.930  0.948  0.938  0.952  0.947  0.942 
(0.5348)  (0.5765)  (0.5537)  (0.5862)  (0.5717)  (0.5664)  
36  0.932  0.952  0.941  0.956  0.951  0.944 
(0.5320)  (0.5503)  (0.5362)  (0.5566)  (0.5436)  (0.5385)  
37  0.933  0.952  0.946  0.955  0.950  0.949 
(0.3516)  (0.3905)  (0.3668)  (0.3978)  (0.3880)  (0.3839)  
38  0.936  0.951  0.945  0.954  0.950  0.949 
(0.3974)  (0.4377)  (0.4142)  (0.4459)  (0.4348)  (0.4307)  
39  0.928  0.950  0.941  0.953  0.947  0.944 
(0.4510)  (0.4934)  (0.4699)  (0.5017)  (0.4899)  (0.4852)  
40  0.928  0.948  0.939  0.951  0.946  0.943 
(0.4421)  (0.4767)  (0.4564)  (0.4837)  (0.4724)  (0.4675)  
41  0.928  0.949  0.939  0.953  0.948  0.944 
(0.5079)  (0.5494)  (0.5272)  (0.5575)  (0.5450)  (0.5397)  
42  0.924  0.949  0.936  0.952  0.947  0.942 
(0.5064)  (0.5280)  (0.5141)  (0.5333)  (0.5215)  (0.5165)  
43  0.929  0.949  0.944  0.952  0.947  0.944 
(0.3554)  (0.3799)  (0.3655)  (0.3856)  (0.3771)  (0.3737)  
44  0.931  0.952  0.945  0.954  0.950  0.948 
(0.4053)  (0.4343)  (0.4190)  (0.4405)  (0.4313)  (0.4275)  
45  0.930  0.947  0.940  0.950  0.946  0.942 
(0.4414)  (0.4721)  (0.4567)  (0.4789)  (0.4687)  (0.4646)  
46  0.932  0.951  0.943  0.955  0.949  0.945 
(0.4313)  (0.4499)  (0.4385)  (0.4553)  (0.4458)  (0.4418)  
47  0.936  0.952  0.945  0.955  0.950  0.946 
(0.4845)  (0.5119)  (0.4981)  (0.5189)  (0.5079)  (0.5037)  
48  0.934  0.947  0.941  0.952  0.946  0.942 
(0.4739)  (0.4826)  (0.4754)  (0.4869)  (0.4772)  (0.4731)  
49  0.936  0.950  0.944  0.953  0.949  0.947 
(0.3090)  (0.3365)  (0.3197)  (0.3417)  (0.3346)  (0.3313)  
50  0.935  0.950  0.945  0.953  0.949  0.947 
(0.3586)  (0.3886)  (0.3718)  (0.3944)  (0.3865)  (0.3828)  
51  0.935  0.951  0.944  0.954  0.949  0.947 
(0.3969)  (0.4282)  (0.4114)  (0.4343)  (0.4257)  (0.4218)  
52  0.932  0.947  0.941  0.950  0.946  0.943 
(0.3874)  (0.4152)  (0.3995)  (0.4204)  (0.4125)  (0.4083)  
53  0.932  0.951  0.942  0.954  0.950  0.947 
(0.4540)  (0.4871)  (0.4703)  (0.4929)  (0.4837)  (0.4791)  
54  0.925  0.951  0.939  0.953  0.950  0.944 
(0.4476)  (0.4676)  (0.4564)  (0.4714)  (0.4632)  (0.4586)  
55  0.935  0.948  0.944  0.950  0.946  0.945 
(0.2818)  (0.2994)  (0.2897)  (0.3028)  (0.2979)  (0.2952)  
56  0.941  0.950  0.947  0.952  0.948  0.947 
(0.3169)  (0.3355)  (0.3257)  (0.3395)  (0.3338)  (0.3310)  
57  0.937  0.949  0.944  0.951  0.947  0.946 
(0.3586)  (0.3786)  (0.3687)  (0.3826)  (0.3767)  (0.3733)  
58  0.940  0.953  0.948  0.955  0.951  0.948 
(0.3484)  (0.3644)  (0.3558)  (0.3677)  (0.3621)  (0.3587)  
59  0.935  0.948  0.943  0.951  0.947  0.944 
(0.4010)  (0.4218)  (0.4122)  (0.4256)  (0.4193)  (0.4155)  
60  0.938  0.954  0.949  0.955  0.953  0.948 
(0.3922)  (0.4053)  (0.3994)  (0.4079)  (0.4022)  (0.3985) 
Scenarios  Coverage Probability (Average Length)  

PB  GCI  MOVER1  MOVER2  BayCrI  HPD  
61  0.929  0.950  0.943  0.954  0.949  0.946 
(0.4572)  (0.5083)  (0.4800)  (0.5188)  (0.5048)  (0.5003)  
62  0.932  0.951  0.942  0.955  0.949  0.947 
(0.4707)  (0.5146)  (0.4895)  (0.5245)  (0.5102)  (0.5057)  
63  0.929  0.949  0.940  0.954  0.948  0.944 
(0.4704)  (0.5059)  (0.4839)  (0.5148)  (0.5013)  (0.4965)  
64  0.931  0.949  0.938  0.953  0.948  0.943 
(0.5451)  (0.5789)  (0.5580)  (0.5876)  (0.5725)  (0.5671)  
65  0.929  0.949  0.937  0.953  0.947  0.942 
(0.5481)  (0.5756)  (0.5575)  (0.5836)  (0.5689)  (0.5636)  
66  0.929  0.950  0.937  0.954  0.947  0.942 
(0.5459)  (0.5637)  (0.5491)  (0.5705)  (0.5560)  (0.5510)  
67  0.927  0.949  0.940  0.953  0.947  0.946 
(0.4323)  (0.4795)  (0.4531)  (0.4886)  (0.4762)  (0.4717)  
68  0.929  0.950  0.941  0.953  0.948  0.946 
(0.4417)  (0.4850)  (0.4605)  (0.4936)  (0.4816)  (0.4770)  
69  0.928  0.950  0.940  0.953  0.948  0.946 
(0.4440)  (0.4817)  (0.4594)  (0.4895)  (0.4775)  (0.4728)  
70  0.924  0.948  0.936  0.952  0.947  0.942 
(0.5208)  (0.5565)  (0.5361)  (0.5643)  (0.5508)  (0.5454)  
71  0.923  0.948  0.935  0.952  0.946  0.941 
(0.5244)  (0.5530)  (0.5351)  (0.5601)  (0.5469)  (0.5416)  
72  0.924  0.950  0.937  0.954  0.948  0.942 
(0.5225)  (0.5412)  (0.5269)  (0.5472)  (0.5341)  (0.5291)  
73  0.933  0.949  0.944  0.952  0.948  0.947 
(0.3930)  (0.4303)  (0.4093)  (0.4376)  (0.4278)  (0.4238)  
74  0.931  0.947  0.941  0.951  0.946  0.945 
(0.3987)  (0.4339)  (0.4142)  (0.4410)  (0.4313)  (0.4273)  
75  0.931  0.948  0.940  0.951  0.946  0.944 
(0.4019)  (0.4308)  (0.4135)  (0.4371)  (0.4276)  (0.4235)  
76  0.926  0.949  0.938  0.952  0.948  0.944 
(0.4795)  (0.5111)  (0.4941)  (0.5171)  (0.5068)  (0.5018)  
77  0.928  0.951  0.940  0.954  0.949  0.945 
(0.4787)  (0.5068)  (0.4911)  (0.5123)  (0.5019)  (0.4970)  
78  0.926  0.951  0.940  0.953  0.949  0.944 
(0.4778)  (0.4975)  (0.4849)  (0.5022)  (0.4919)  (0.4872)  
79  0.936  0.951  0.946  0.953  0.950  0.947 
(0.3606)  (0.3864)  (0.3733)  (0.3912)  (0.3845)  (0.3811)  
80  0.936  0.951  0.945  0.953  0.949  0.946 
(0.3678)  (0.3911)  (0.3792)  (0.3957)  (0.3888)  (0.3853)  
81  0.932  0.949  0.943  0.951  0.947  0.944 
(0.3669)  (0.3864)  (0.3763)  (0.3905)  (0.3838)  (0.3804)  
82  0.935  0.950  0.944  0.953  0.949  0.945 
(0.4253)  (0.4446)  (0.4351)  (0.4487)  (0.4412)  (0.4372)  
83  0.933  0.949  0.941  0.951  0.947  0.943 
(0.4263)  (0.4428)  (0.4344)  (0.4465)  (0.4393)  (0.4354)  
84  0.932  0.949  0.942  0.951  0.947  0.943 
(0.4204)  (0.4319)  (0.4254)  (0.4349)  (0.4277)  (0.4240) 
Lamphun  56  49  49  57  46  30  27  33  33  47 
49  114  129  132  138  130  106  80  69  46  
40  43  111  210  107  96  64  53  55  119  
137  
Mae Hong Son  93  79  94  84  63  42  45  72  68  74 
81  87  94  96  95  95  86  96  105  67  
94  110  174  233  163  133  209  171  170  239  
245  
Nan  47  50  55  61  64  47  59  82  63  65 
81  103  122  158  177  158  112  84  80  51  
47  66  100  146  114  52  54  33  46  111  
124 
Distributions  Lognormal  BS  Exponential  Gamma  Weibull 

Lamphun  315.4453  314.6908  335.0575  316.917  319.2145 
Mae Hong Son  330.1128  329.9111  358.0465  332.3196  336.2999 
Nan  309.3649  308.9072  338.9008  310.9628  314.1231 
Distributions  Lognormal  BS  Exponential  Gamma  Weibull 

Lamphun  318.3132  317.5587  336.4915  319.7850  322.0825 
Mae Hong Son  332.9808  332.7791  359.4805  335.1876  339.1679 
Nan  312.2328  311.7752  340.3348  313.8308  316.9911 
Group  n  Min.  Median  Mean  Max.  SD  CV 

Lamphun  31  27  57  79.1936  210  44.0503  0.5562 
Mae Hong Son  31  42  94  114.7419  245  56.7955  0.4950 
Nan  31  33  66  84.2581  177  38.8964  0.4616 
Comparison  PB  GCI  MOVER1  MOVER2  BayCrI  HPD 

${\theta}_{1}{\theta}_{2}$  [−0.0574–0.2279]  [−0.1169–0.3045]  [−0.0938–0.2746]  [−0.11410.3058]  [−0.1023–0.2853]  [−0.1025–0.2853] 
${\theta}_{1}{\theta}_{3}$  [−0.0237–0.2361]  [−0.0812–0.3158]  [−0.0720–0.2904]  [−0.08260.3225]  [−0.0931–0.3266]  [−0.1001–0.3166] 
${\theta}_{2}{\theta}_{3}$  [−0.1188–0.1447]  [−0.1609–0.2028]  [−0.1458–0.1834]  [−0.1629–0.2115]  [−0.1662–0.2023]  [−0.1558–0.2086] 
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Puggard, W.; Niwitpong, S.A.; Niwitpong, S. Simultaneous Confidence Intervals for All Pairwise Differences between the Coefficients of Variation of Multiple Birnbaum–Saunders Distributions. Symmetry 2022, 14, 2666. https://doi.org/10.3390/sym14122666
Puggard W, Niwitpong SA, Niwitpong S. Simultaneous Confidence Intervals for All Pairwise Differences between the Coefficients of Variation of Multiple Birnbaum–Saunders Distributions. Symmetry. 2022; 14(12):2666. https://doi.org/10.3390/sym14122666
Chicago/Turabian StylePuggard, Wisunee, SaAat Niwitpong, and Suparat Niwitpong. 2022. "Simultaneous Confidence Intervals for All Pairwise Differences between the Coefficients of Variation of Multiple Birnbaum–Saunders Distributions" Symmetry 14, no. 12: 2666. https://doi.org/10.3390/sym14122666