Statistical Inference of the Rayleigh Distribution Based on Progressively Type II Censored Competing Risks Data
Abstract
:1. Introduction
2. Model Description and Parameter Estimation
2.1. Maximum Likelihood Estimation
2.2. Bayes Estimation
3. Interval Estimation
3.1. Asymptotic Confidence Intervals
3.1.1. Observed Fisher Information Matrix
3.1.2. Expected Fisher Information Matrix
3.2. HPD Credible Intervals
4. Simulation Study
Algorithm 1 Generating the progressively type II censored samples. |
|
4.1. Point Estimates
4.2. Interval Estimates
5. Data Analysis
6. Further Discussion
7. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
- Balakrishnan, N.; Cramer, E. The Art of Progressive Censoring, 2nd ed.; Birkhäuser: New York, NY, USA, 2014; pp. 313–325. [Google Scholar]
- Madi, M.T.; Raqab, M.Z. Bayesian inference for the generalized exponential distribution based on progressively censored data. Commun. Stat. 2009, 38, 2016–2029. [Google Scholar] [CrossRef]
- Rastogi, M.K.; Tripathi, Y.M. Estimating the parameters of a Burr distribution under progressive type-II censoring. Stat. Methodol. 2012, 9, 381–391. [Google Scholar] [CrossRef]
- Andersen, P.K.; Abildstrom, S.Z.; Rosthoj, S. Competing risks as a multi-state model. Stat. Methods Med. Res. 2002, 11, 203–215. [Google Scholar] [CrossRef] [PubMed]
- Cramer, E.; Schmiedt, A.B. Progressively type-II censored competing risks data from Lomax distributions. Comput. Stat. Data Anal. 2011, 55, 1285–1303. [Google Scholar] [CrossRef]
- Ahmadi, K.; Rezaei, M.; Yousefzadeh, F. Point predictors of the latent failure times of censored units in progressively type-II censored competing risks data from the exponential distributions. J. Stat. Comput. Simul. 2015, 86, 1620–1634. [Google Scholar] [CrossRef]
- Dey, A.K.; Jha, A.; Dey, S. Bayesian analysis of modified Weibull distribution under progressively censored competing risk model. arXiv 2016, arXiv:1605.06585. [Google Scholar]
- Hashemi, R.; Amiri, L. Analysis of progressive type-II censoring in the Weibull model for competing risks data with binomial removals. Appl. Math. Sci. 2000, 5, 1073–1087. [Google Scholar]
- Wu, S.J.; Chen, D.H.; Chen, S.T. Bayesian inference for Rayleigh distribution under progressive censored sample. Appl. Stoch. Models Bus. Ind. 2010, 22, 269–279. [Google Scholar] [CrossRef]
- Ali Mousa, M.A.M.; Al-Sagheer, S.A. Statistical inference for the Rayleigh model based on progressively type-II censored data. Statistics 2006, 40, 149–157. [Google Scholar] [CrossRef]
- Chen, M.; Shao, Q. Monte carlo estimation of bayesian credible and hpd intervals. J. Comput. Graph. Stat. 1999, 8, 69–92. [Google Scholar]
- Balakrishnan, N.; Sandhu, R.A. A simple simulational algorithm for generating progressive type-II censored samples. Am. Stat. 1995, 49, 229–230. [Google Scholar]
- Lawless, J. Statistical Models and Methods for Lifetime Data; John Wiley & Sons: Hoboken, NJ, USA, 2003. [Google Scholar]
Scheme 1 | (n = 25, m = 20) |
R | [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5] |
Scheme 2 | (n = 36, m = 24) |
R | [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12] |
Scheme 3 | (n = 40, m = 30) |
R | [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10] |
Scheme 4 | (n = 40, m = 30) |
R | [10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] |
Scheme 5 | (n = 40, m = 30) |
R | [5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5] |
MLE | |||||||||
---|---|---|---|---|---|---|---|---|---|
Bias | MSE | Bias | MSE | ||||||
0.8 | 1 | 25 | 20 | 0.8099 | 0.0099 | 0.0149 | 1.0251 | 0.0251 | 0.0438 |
36 | 24 | 0.8064 | 0.0064 | 0.0117 | 1.0234 | 0.0234 | 0.0330 | ||
40 | 30 | 0.8034 | 0.0034 | 0.0099 | 1.0239 | 0.0239 | 0.0257 | ||
40 | 30 | 0.8021 | 0.0021 | 0.0096 | 1.0158 | 0.0158 | 0.0253 | ||
40 | 30 | 0.8022 | 0.0022 | 0.0097 | 1.0176 | 0.0176 | 0.0241 | ||
0.8 | 0.6 | 25 | 20 | 0.8238 | 0.0238 | 0.0286 | 0.6042 | 0.0042 | 0.0071 |
36 | 24 | 0.8175 | 0.0175 | 0.0240 | 0.6037 | 0.0037 | 0.0046 | ||
40 | 30 | 0.8169 | 0.0169 | 0.0176 | 0.6034 | 0.0034 | 0.0050 | ||
40 | 30 | 0.8161 | 0.0161 | 0.0177 | 0.6016 | 0.0016 | 0.0078 | ||
40 | 30 | 0.8174 | 0.0174 | 0.0166 | 0.6015 | 0.0015 | 0.0050 | ||
0.5 | 0.6 | 25 | 20 | 0.5049 | 0.0049 | 0.0056 | 0.6137 | 0.0137 | 0.0153 |
36 | 24 | 0.5031 | 0.0031 | 0.0051 | 0.6112 | 0.0112 | 0.0127 | ||
40 | 30 | 0.5023 | 0.0023 | 0.0036 | 0.6099 | 0.0099 | 0.0093 | ||
40 | 30 | 0.5020 | 0.0020 | 0.0037 | 0.6054 | 0.0054 | 0.0080 | ||
40 | 30 | 0.5022 | 0.0022 | 0.0041 | 0.6088 | 0.0088 | 0.0095 |
n | m | SEL | EL | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | ||||||||
Non-informative | |||||||||||||||
0.8 | 1 | 25 | 20 | 0.8329 | 0.0329 | 0.0168 | 1.0885 | 0.0885 | 0.0829 | 0.8193 | 0.0193 | 0.0156 | 1.0465 | 0.0465 | 0.0513 |
36 | 24 | 0.8281 | 0.0281 | 0.0137 | 1.0933 | 0.0933 | 0.0893 | 0.8107 | 0.0107 | 0.0129 | 1.0330 | 0.0330 | 0.0333 | ||
40 | 30 | 0.8260 | 0.0260 | 0.0112 | 1.0590 | 0.0590 | 0.0369 | 0.8089 | 0.0089 | 0.0095 | 1.0329 | 0.0329 | 0.0298 | ||
40 | 30 | 0.8208 | 0.0208 | 0.0106 | 1.0429 | 0.0429 | 0.0308 | 0.8056 | 0.0056 | 0.0098 | 1.0229 | 0.0229 | 0.0242 | ||
40 | 30 | 0.8221 | 0.0221 | 0.0110 | 1.0524 | 0.0524 | 0.0341 | 0.8064 | 0.0064 | 0.0093 | 1.0342 | 0.0342 | 0.0313 | ||
Non-informative | |||||||||||||||
0.8 | 0.6 | 25 | 20 | 0.8696 | 0.0696 | 0.0407 | 0.6237 | 0.0237 | 0.0087 | 0.8428 | 0.0428 | 0.0353 | 0.6086 | 0.0086 | 0.0084 |
36 | 24 | 0.8569 | 0.0569 | 0.0358 | 0.6194 | 0.0194 | 0.0073 | 0.8387 | 0.0387 | 0.0293 | 0.6065 | 0.0065 | 0.0064 | ||
40 | 30 | 0.8491 | 0.0491 | 0.0238 | 0.6133 | 0.0133 | 0.0057 | 0.8319 | 0.0319 | 0.0204 | 0.6075 | 0.0075 | 0.0051 | ||
40 | 30 | 0.8486 | 0.0486 | 0.0259 | 0.6115 | 0.0115 | 0.0054 | 0.8281 | 0.0281 | 0.0202 | 0.6028 | 0.0028 | 0.0049 | ||
40 | 30 | 0.8493 | 0.0493 | 0.0261 | 0.6128 | 0.0128 | 0.0055 | 0.8293 | 0.0293 | 0.0213 | 0.6036 | 0.0036 | 0.0050 | ||
Non-informative | |||||||||||||||
0.5 | 0.6 | 25 | 20 | 0.5201 | 0.0201 | 0.0069 | 0.6515 | 0.0515 | 0.0224 | 0.5109 | 0.0109 | 0.0061 | 0.6249 | 0.0249 | 0.0137 |
36 | 24 | 0.5192 | 0.0192 | 0.0062 | 0.6440 | 0.0440 | 0.0167 | 0.5091 | 0.0091 | 0.0049 | 0.6228 | 0.0228 | 0.0178 | ||
40 | 30 | 0.5118 | 0.0118 | 0.0042 | 0.6260 | 0.0260 | 0.0098 | 0.5066 | 0.0066 | 0.0038 | 0.6209 | 0.0209 | 0.0104 | ||
40 | 30 | 0.5115 | 0.0115 | 0.0040 | 0.6290 | 0.0290 | 0.0108 | 0.5039 | 0.0039 | 0.0035 | 0.6144 | 0.0144 | 0.0090 | ||
40 | 30 | 0.5117 | 0.0117 | 0.0047 | 0.6354 | 0.0354 | 0.0119 | 0.5047 | 0.0047 | 0.0039 | 0.6202 | 0.0202 | 0.0098 |
n | m | SEL | EL | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | ||||||||
Informative-I | |||||||||||||||
0.8 | 1 | 25 | 20 | 0.8077 | 0.0077 | 0.0102 | 1.0492 | 0.0492 | 0.0245 | 0.7887 | −0.0113 | 0.0088 | 1.0240 | 0.0240 | 0.0211 |
36 | 24 | 0.8052 | 0.0052 | 0.0086 | 1.0469 | 0.0469 | 0.0214 | 0.7892 | −0.0108 | 0.0085 | 1.0221 | 0.0221 | 0.0179 | ||
40 | 30 | 0.8029 | 0.0029 | 0.0078 | 1.0391 | 0.0391 | 0.0176 | 0.7914 | −0.0086 | 0.0080 | 1.0166 | 0.0166 | 0.0149 | ||
40 | 30 | 0.8018 | 0.0018 | 0.0075 | 1.0375 | 0.0375 | 0.0181 | 0.7920 | −0.0080 | 0.0065 | 1.0093 | 0.0093 | 0.0147 | ||
40 | 30 | 0.8019 | 0.0019 | 0.0076 | 1.0453 | 0.0453 | 0.0187 | 0.7918 | −0.0082 | 0.0067 | 1.0150 | 0.0150 | 0.0141 | ||
Informative-II | |||||||||||||||
0.8 | 0.6 | 25 | 20 | 0.8129 | 0.0129 | 0.0141 | 0.6156 | 0.0156 | 0.0071 | 0.7849 | −0.0151 | 0.0130 | 0.6041 | 0.0041 | 0.0064 |
36 | 24 | 0.8099 | 0.0099 | 0.0133 | 0.6159 | 0.0159 | 0.0065 | 0.7867 | −0.0133 | 0.0116 | 0.6028 | 0.0028 | 0.0048 | ||
40 | 30 | 0.8124 | 0.0124 | 0.0106 | 0.6131 | 0.0131 | 0.0044 | 0.7900 | −0.0100 | 0.0093 | 0.6021 | 0.0021 | 0.0045 | ||
40 | 30 | 0.8108 | 0.0108 | 0.0108 | 0.6111 | 0.0111 | 0.0047 | 0.7903 | −0.0097 | 0.0100 | 0.6008 | 0.0008 | 0.0040 | ||
40 | 30 | 0.8118 | 0.0118 | 0.0113 | 0.6128 | 0.0128 | 0.0048 | 0.7884 | −0.0116 | 0.0102 | 0.6011 | 0.0011 | 0.0042 | ||
Informative-III | |||||||||||||||
0.5 | 0.6 | 25 | 20 | 0.4976 | −0.0024 | 0.0041 | 0.6324 | 0.0324 | 0.0131 | 0.4906 | −0.0094 | 0.0036 | 0.6121 | 0.0121 | 0.0088 |
36 | 24 | 0.4983 | −0.0017 | 0.0037 | 0.6273 | 0.0273 | 0.0108 | 0.4915 | −0.0085 | 0.0033 | 0.6109 | 0.0109 | 0.0088 | ||
40 | 30 | 0.4983 | −0.0017 | 0.0032 | 0.6196 | 0.0196 | 0.0085 | 0.4917 | −0.0083 | 0.0031 | 0.6095 | 0.0095 | 0.0073 | ||
40 | 30 | 0.4991 | −0.0009 | 0.0027 | 0.6193 | 0.0193 | 0.0074 | 0.4923 | −0.0077 | 0.0028 | 0.6053 | 0.0053 | 0.0066 | ||
40 | 30 | 0.4985 | −0.0015 | 0.0028 | 0.6211 | 0.0211 | 0.0071 | 0.4919 | −0.0081 | 0.0027 | 0.6075 | 0.0075 | 0.0063 |
Scheme | , | , | , | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 0.4645 | 0.93 | 0.7655 | 0.94 | 0.6301 | 0.93 | 0.3384 | 0.95 | 0.2946 | 0.93 | 0.4428 | 0.94 |
0.4603 | 0.93 | 0.7795 | 0.94 | 0.6371 | 0.93 | 0.3392 | 0.94 | 0.2920 | 0.94 | 0.4530 | 0.94 | |
0.4770 | 0.93 | 0.8100 | 0.94 | 0.7025 | 0.92 | 0.3392 | 0.94 | 0.2962 | 0.93 | 0.4872 | 0.93 | |
0.3971 | 0.95 | 0.6367 | 0.96 | 0.5152 | 0.94 | 0.3193 | 0.95 | 0.2536 | 0.94 | 0.4133 | 0.94 | |
2 | 0.4232 | 0.95 | 0.6893 | 0.95 | 0.5948 | 0.93 | 0.3065 | 0.95 | 0.2679 | 0.94 | 0.4032 | 0.95 |
0.4208 | 0.94 | 0.6851 | 0.95 | 0.5743 | 0.94 | 0.3074 | 0.95 | 0.2687 | 0.95 | 0.4004 | 0.94 | |
0.4142 | 0.93 | 0.7149 | 0.94 | 0.5954 | 0.93 | 0.3077 | 0.94 | 0.2676 | 0.93 | 0.4169 | 0.94 | |
0.3689 | 0.95 | 0.5920 | 0.96 | 0.4747 | 0.94 | 0.2877 | 0.95 | 0.2336 | 0.94 | 0.3751 | 0.95 | |
3 | 0.3759 | 0.95 | 0.6157 | 0.95 | 0.5606 | 0.95 | 0.2974 | 0.95 | 0.2377 | 0.95 | 0.3564 | 0.95 |
0.4066 | 0.95 | 0.6480 | 0.95 | 0.5056 | 0.94 | 0.2746 | 0.96 | 0.2583 | 0.96 | 0.3842 | 0.95 | |
0.3715 | 0.94 | 0.6139 | 0.96 | 0.5119 | 0.93 | 0.2728 | 0.94 | 0.2365 | 0.94 | 0.3581 | 0.94 | |
0.3356 | 0.95 | 0.5303 | 0.96 | 0.4270 | 0.95 | 0.2605 | 0.95 | 0.2104 | 0.95 | 0.3318 | 0.95 | |
4 | 0.3732 | 0.95 | 0.6069 | 0.94 | 0.5060 | 0.94 | 0.2762 | 0.94 | 0.2377 | 0.95 | 0.3528 | 0.94 |
0.3731 | 0.95 | 0.6118 | 0.95 | 0.5042 | 0.95 | 0.2742 | 0.94 | 0.2399 | 0.95 | 0.3512 | 0.94 | |
0.3712 | 0.95 | 0.6175 | 0.94 | 0.5339 | 0.94 | 0.2677 | 0.95 | 0.2401 | 0.94 | 0.3547 | 0.94 | |
0.3329 | 0.96 | 0.5289 | 0.96 | 0.4229 | 0.95 | 0.2549 | 0.95 | 0.2123 | 0.94 | 0.3260 | 0.95 | |
5 | 0.3751 | 0.95 | 0.6021 | 0.95 | 0.5027 | 0.94 | 0.2732 | 0.94 | 0.2376 | 0.95 | 0.3593 | 0.95 |
0.3752 | 0.95 | 0.6081 | 0.95 | 0.5101 | 0.95 | 0.2712 | 0.95 | 0.2341 | 0.95 | 0.3539 | 0.94 | |
0.3740 | 0.94 | 0.6162 | 0.94 | 0.5080 | 0.94 | 0.2706 | 0.94 | 0.2355 | 0.94 | 0.3628 | 0.94 | |
0.3343 | 0.96 | 0.5297 | 0.96 | 0.4195 | 0.96 | 0.2578 | 0.94 | 0.2092 | 0.93 | 0.3316 | 0.95 |
Scheme 1 | (n = 33, m = 20) |
R | [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13] |
Data | (11,2) (35,2) (49,2) (170,2) (329,2) (381,2) (708,2) (958,2) (1062,2) (1167,1) |
(1594,2) (1925,1) (1990,1) (2223,1) (2327,2) (2400,1) (2451,2) (2471,1) (2551,1) | |
(2568,1) | |
Scheme 2 | (n = 33, m = 24) |
R | [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9] |
Data | (11,2) (35,2) (49,2) (170,2) (329,2) (381,2) (708,2) (958,2) (1062,2) (1167,1) |
(1594,2) (1925,1) (1990,1) (2223,1) (2327,2) (2400,1) (2451,2) (2471,1) (2551,1) | |
(2568,1) (2694,1) (2702,2) (2761,2) (2831,2) | |
Scheme 3 | (n = 33, m = 27) |
R | [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6] |
Data | (11,2) (35,2) (49,2) (170,2) (329,2) (381,2) (708,2) (958,2) (1062,2) (1167,1) |
(1594,2) (1925,1) (1990,1) (2223,1) (2327,2) (2400,1) (2451,2) (2471,1) (2551,1) | |
(2568,1) (2694,1) (2702,2) (2761,2) (2831,2) (3034,1) (3059,2) (3112,1) | |
Scheme 4 | (n = 33, m = 27) |
R | [6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] |
Data | (708,2) (958,2) (1062,2) (1167,1) (1594,2) (1925,1) (1990,1) (2223,1) (2327,2) |
(2400,1) (2451,2) (2471,1) (2551,1) (2568,1) (2694,1) (2702,2) (2761,2) (2831,2) | |
(3034,1) (3059,2) (3112,1) (3214,1) (3478,1) (3504,1) (4329,1) (6976,1) (7846,1) | |
Scheme 5 | (n = 33, m = 27) |
R | [3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3] |
Data | (170,2) (329,2) (381,2) (708,2) (958,2) (1062,2) (1167,1) (1594,2) (1925,1) (1990,1) |
(2223,1) (2327,2) (2400,1) (2451,2) (2471,1) (2551,1) (2568,1) (2694,1) (2702,2) | |
(2761,2) (2831,2) (3034,1) (3059,2) (3112,1) (3214,1) (3478,1) (3504,1) |
Scheme | ||||||
---|---|---|---|---|---|---|
1 | 3947.022 | 4217.053 | 3020.257 | 3222.73 | 3385.922 | 2453.236 |
2 | 3571.484 | 3847.889 | 3004.723 | 2766.46 | 2928.25 | 2314.567 |
3 | 2798.409 | 2898.552 | 2830.38 | 2320.318 | 2376.531 | 2338.514 |
4 | 2872.625 | 2937.995 | 2893.822 | 3745.445 | 3893.618 | 3792.537 |
5 | 2575.975 | 2647.641 | 2599.073 | 2673.215 | 2753.547 | 2699.037 |
Scheme | Method 1 | Method 2 | Method 3 | |||
---|---|---|---|---|---|---|
1 | 2735.096 | 1823.398 | 2735.422 | 1823.615 | 2879.202 | 1760.525 |
2 | 2333.327 | 1399.996 | 2333.233 | 1399.940 | 2659.170 | 1376.380 |
3 | 1653.724 | 1136.935 | 1653.661 | 1136.892 | 1641.680 | 1006.269 |
4 | 1365.534 | 2321.408 | 1365.534 | 2321.408 | 1568.784 | 2169.245 |
5 | 1349.353 | 1453.150 | 1349.352 | 1453.148 | 1258.528 | 1435.572 |
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Liao, H.; Gui, W. Statistical Inference of the Rayleigh Distribution Based on Progressively Type II Censored Competing Risks Data. Symmetry 2019, 11, 898. https://doi.org/10.3390/sym11070898
Liao H, Gui W. Statistical Inference of the Rayleigh Distribution Based on Progressively Type II Censored Competing Risks Data. Symmetry. 2019; 11(7):898. https://doi.org/10.3390/sym11070898
Chicago/Turabian StyleLiao, Hongyi, and Wenhao Gui. 2019. "Statistical Inference of the Rayleigh Distribution Based on Progressively Type II Censored Competing Risks Data" Symmetry 11, no. 7: 898. https://doi.org/10.3390/sym11070898