Bayesian Estimation of the Stress–Strength Parameter for Bivariate Normal Distribution Under an Updated Type-II Hybrid Censoring
Abstract
:1. Introduction
2. Model and Notations
2.1. Type-II Censoring Scheme
2.2. Updated Type-II Hybrid Censoring Scheme
- 1
- If , ; both x and y are observed. Denote .
- 2
- If , ; only x is observed and y is truncated at and recorded as +. Denote .
- 3
- If , ; x is truncated at , and y is only recorded as −. Denote .
- 4
- If ; ; both x and y are not observed and they are truncated at . Denote .
2.3. The Likelihood of Updated Type-II Hybrid Data
- The likelihood of is the same as the pdf in Equation (1). That is,
- The likelihood of can be integrated.
- The likelihood of can be integrated in several ways.The first term is the normal cdf with mean and variance , and the second term is the bivariate normal cdf, which can be given by some statistical libraries, such as the ‘pmvnorm’ function in the R ‘mvtnorm’ package.
- The likelihood of can be integrated in several ways.Alternatively, the density can be calculated by a bivariate normal cdf.
3. Bayesian Framework
3.1. A Markov Chain Monte Carlo Process
- 1
- sample
- 2
- sample
- 3
- sample
3.2. Bayes Estimation
3.2.1. Bayes Estimate of
3.2.2. Mean Value Monte Carlo Method
4. Monte Carlo Simulation Study
5. Data Analysis
6. Concluding Remarks
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
- 2.
- The likelihood of can be integrated.
- 3.
- The likelihood of can be integrated in several ways.Alternatively,
- 4.
- The likelihood of can be integrated in several ways.Alternatively,
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, Target Parameters: | |||||||
---|---|---|---|---|---|---|---|
n | r | ||||||
30 | 21 | 12.09 | 34.66 | 6.25 | 30.52 | 3.62 | 9.58 |
(−0.25178) | (0.45034) | (−0.1592) | (0.41183) | (0.07531) | (0.27343) | ||
50 | 35 | 9.59 | 33.81 | 5.44 | 23.26 | 3.49 | 10.49 |
(−0.24495) | (0.48732) | (−0.18328) | (0.39908) | (0.09353) | (0.29814) | ||
100 | 70 | 7.73 | 24.24 | 4.99 | 13.53 | 2.81 | 9.88 |
(−0.24406) | (0.44873) | (−0.19901) | (0.32298) | (0.10475) | (0.30055) | ||
200 | 140 | 7.08 | 21.73 | 4.67 | 11.48 | 2.50 | 10.01 |
(−0.2485) | (0.44442) | (−0.20574) | (0.31519) | (0.12037) | (0.30878) | ||
500 | 350 | 6.70 | 2.09 | 4.64 | 9.47 | 2.56 | 10.65 |
(−0.25198) | (0.44904) | (−0.21074) | (0.29898) | (0.14311) | (0.32308) | ||
, Target Parameters: | |||||||
30 | 27 | 7.03 | 12.27 | 3.82 | 11.04 | 3.05 | 2.07 |
(−0.09469) | (0.15244) | (−0.06817) | (0.21988) | (0.00744) | (0.06944) | ||
50 | 45 | 4.58 | 7.84 | 2.72 | 7.45 | 2.15 | 1.40 |
(−0.10791) | (0.14755) | (−0.09377) | (0.1915) | (0.00054) | (0.07419) | ||
100 | 90 | 2.80 | 4.81 | 2.06 | 4.68 | 1.13 | 0.95 |
(−0.10582) | (0.14387) | (−0.11208) | (0.16793) | (0.01296) | (0.07356) | ||
200 | 180 | 2.12 | 3.76 | 1.82 | 3.50 | 0.68 | 0.89 |
(−0.11042) | (0.15438) | (−0.11807) | (0.16146) | (0.01701) | (0.08077) | ||
500 | 450 | 1.45 | 3.06 | 1.69 | 2.88 | 0.35 | 0.75 |
(−0.10493) | (0.15942) | (−0.12333) | (0.15947) | (0.02025) | (0.08115) | ||
, Target Parameters: | |||||||
30 | 21 | 33.26 | 63.51 | 7.76 | 30.51 | 8.54 | 9.30 |
(−0.50789) | (0.69634) | (−0.20115) | (0.39244) | (0.08704) | (0.2808) | ||
50 | 35 | 29.79 | 65.49 | 7.167 | 22.55 | 7.61 | 9.95 |
(−0.49977) | (0.73351) | (−0.22556) | (0.3699) | (0.10369) | (0.2986) | ||
100 | 70 | 27.53 | 51.16 | 6.95 | 10.91 | 5.86 | 9.81 |
(−0.50081) | (0.6837) | (−0.24529) | (0.27187) | (0.125) | (0.3038) | ||
200 | 140 | 26.47 | 46.91 | 6.76 | 7.79 | 4.81 | 9.79 |
(−0.50244) | (0.67149) | (−0.25065) | (0.24689) | (0.14515) | (0.3075) | ||
500 | 350 | 25.56 | 47.12 | 7.02 | 5.85 | 4.47 | 10.41 |
(−0.50097) | (0.68074) | (−0.26143) | (0.2286) | (0.17711) | (0.32049) | ||
, Target Parameters: | |||||||
30 | 27 | 7.04 | 12.27 | 3.82 | 11.04 | 3.05 | 2.07 |
(−0.22941) | (0.25317) | (−0.06257) | (0.22424) | (−0.0157) | (0.07838) | ||
50 | 45 | 9.49 | 11.26 | 2.67 | 6.88 | 2.19 | 1.11 |
(−0.24389) | (0.24296) | (−0.08879) | (0.17862) | (−0.00003) | (0.08239) | ||
100 | 90 | 7.84 | 8.53 | 2.19 | 4.15 | 1.39 | 0.94 |
(−0.24403) | (0.24386) | (−0.11503) | (0.1524) | (−0.00113) | (0.08572) | ||
200 | 180 | 7.49 | 7.84 | 1.91 | 3.13 | 0.81 | 0.96 |
(−0.25423) | (0.2556) | (−0.12061) | (0.14907) | (0.00019) | (0.09178) | ||
500 | 450 | 6.79 | 7.47 | 1.84 | 2.44 | 0.37 | 0.94 |
(−0.25325) | (0.26354) | (−0.12903) | (0.14488) | (0.00017) | (0.09445) |
, Target Parameters: | |||||||
---|---|---|---|---|---|---|---|
n | r | ||||||
30 | 21 | 9.26 | 26.56 | 5.75 | 26.27 | 1.98 | 10.10 |
(−0.17821) | (0.36582) | (−0.12596) | (0.37264) | (0.04714) | (0.27063) | ||
50 | 35 | 6.68 | 25.04 | 4.74 | 19.34 | 1.69 | 10.67 |
(−0.17942) | (0.39978) | (−0.15544) | (0.3586) | (0.04742) | (0.29580) | ||
100 | 70 | 5.11 | 17.49 | 3.99 | 11.46 | 1.25 | 9.88 |
(−0.18255) | (0.36765) | (−0.16994) | (0.29621) | (0.05441) | (0.29728) | ||
200 | 140 | 4.24 | 15.01 | 3.43 | 9.69 | 1.03 | 9.63 |
(−0.1832) | (0.36108) | (−0.17116) | (0.28792) | (0.07129) | (0.30107) | ||
500 | 350 | 3.89 | 13.89 | 3.39 | 7.85 | 0.92 | 0.10067 |
(−0.18925) | (0.36309) | (−0.17826) | (0.27014) | (0.08022) | (0.31324) | ||
, Target Parameters: | |||||||
30 | 27 | 6.65 | 10.92 | 4.03 | 9.89 | 2.34 | 2.52 |
(−0.0589) | (0.12883) | (−0.05056) | (0.20095) | (0.00325) | (0.06697) | ||
50 | 45 | 4.11 | 7.11 | 2.36 | 6.41 | 1.63 | 1.66 |
(−0.06743) | (0.11896) | (−0.07536) | (0.17033) | (0.00076) | (0.07046) | ||
100 | 90 | 2.04 | 4.12 | 1.74 | 4.01 | 0.87 | 1.04 |
(−0.06258) | (0.12135) | (−0.08946) | (0.14986) | (0.01464) | (0.06984) | ||
200 | 180 | 1.33 | 2.89 | 1.35 | 2.97 | 0.48 | 0.87 |
(−0.06632) | (0.1284) | (−0.09459) | (0.1465) | (0.01392) | (0.07599) | ||
500 | 450 | 0.74 | 2.31 | 1.14 | 2.31 | 0.24 | 0.71 |
(−0.0624) | (0.13451) | (−0.09633) | (0.14108) | (0.01863) | (0.07740) | ||
, Target Parameters: | |||||||
30 | 21 | 45.55 | 75.21 | 7.49 | 25.50 | 6.67 | 8.58 |
(−0.61387) | (0.77865) | (−0.19284) | (0.32413) | (0.05777) | (0.27256) | ||
50 | 35 | 41.29 | 75.45 | 7.01 | 17.73 | 5.62 | 9.23 |
(−0.60485) | (0.8081) | (−0.21895) | (0.29807) | (0.07702) | (0.28955) | ||
100 | 70 | 40.28 | 62.66 | 6.79 | 8.03 | 3.38 | 9.08 |
(−0.61411) | (0.76446) | (−0.24012) | (0.20838) | (0.07305) | (0.29297) | ||
200 | 140 | 39.83 | 59.28 | 6.51 | 5.08 | 2.47 | 9.00 |
(−0.62047) | (0.7583) | (−0.24456) | (0.18596) | (0.08025) | (0.29572) | ||
500 | 350 | 40.14 | 58.65 | 6.72 | 3.43 | 1.62 | 9.35 |
(−0.62986) | (0.76098) | (−0.25534) | (0.1662) | (0.08718) | (0.30396) | ||
, Target Parameters: | |||||||
30 | 27 | 14.44 | 18.25 | 4.59 | 10.2 | 2.42 | 1.25 |
(−0.27116) | (0.29286) | (−0.03828) | (0.19868) | (−0.00912) | (0.07857) | ||
50 | 45 | 12.27 | 13.66 | 2.51 | 6.21 | 1.81 | 1.03 |
(−0.29195) | (0.29456) | (−0.06423) | (0.16253) | (0.00189) | (0.08442) | ||
100 | 90 | 10.54 | 11.27 | 1.79 | 3.65 | 0.98 | 0.92 |
(−0.29355) | (0.294) | (−0.08849) | (0.13375) | (−0.00224) | (0.08744) | ||
200 | 180 | 10.57 | 10.99 | 1.43 | 2.45 | 0.57 | 0.97 |
(−0.30723) | (0.31027) | (−0.09409) | (0.12944) | (−0.00342) | (0.09356) | ||
500 | 450 | 10.06 | 10.43 | 1.18 | 1.83 | 0.29 | 0.95 |
(−0.31097) | (0.31556) | (−0.09943) | (0.12336) | (−0.00633) | (0.09588) |
(3.545, 3.472) | (4.165, 3.088) | (3.536, 3.722) | (4.039, 2.19) | (3.183, 3.195) |
(+, +) | (+, −) | (+, −) | (3.724, 2.737) | (4.008, 3.093) |
(3.705, 3.902) | (3.581, 3.534) | (+, −) | (3.11, +) | (3.981, 3.968) |
(4.162, 3.504) | (4.05, 3.521) | (3.278, 3.621) | (2.975, +) | (3.72, 3.187) |
(4.213, 2.993) | (3.806, 3.757) | (+, −) | (2.403, 4.089) | (3.224, 2.807) |
(3.745, 4.091) | (+, −) | (3.666, 4.074) | (+, −) | (+, −) |
(3.727, 3.801) | (3.912, 3.889) | (+, −) | (+, −) | (3.554, 3.123) |
(+, −) | (+, −) | (+, −) | (2.942, 3.726) | (3.862, 4.238) |
(3.753, 4.168) | (+, −) | (3.44, 3.641) | (3.812, 3.936) | (2.897, 4.175) |
(3.435, 4.176) | (3.951, 3.199) | (+, −) | (3.14, 4.167) | (3.328, 3.437) |
(3.736, 3.106) | (+, −) | (3.832, 3.985) | (4.162, 3.872) | (3.8, 4.048) |
(4.091, 3.262) | (3.158, 3.616) | (+, −) | (4.137, 2.996) | (3.462, 3.456) |
(4.034, 2.264) | (4.034, 4.26) | (+, +) | (3.826, 3.796) | (+, −) |
(4.189, 4.083) | (3.709, 3.525) | (3.147, 3.709) | (3.917, 4.094) | (3.522, 3.973) |
(+, −) | (3.444, +) | (3.649, 3.53) | (3.253, 2.995) | (3.939, 3.555) |
(3.857, 3.187) | (3.978, 3.26) | (4.261, 2.626) | (4.228, 2.951) | (3.636, +) |
(4.027, 3.179) | (3.86, 3.907) | (3.906, +) | (4.244, 3.975) | (+, −) |
(+, −) | (2.853, 3.159) | (3.909, 3.416) | (4.224, 3.537) | (+, −) |
(4.257, 4.223) | (2.959, 3.974) | (+, −) | (3.96, 3.728) | (+, −) |
(3.123, 3.675) | (4.131, 3.919) | (+, −) | (3.47, 3.844) | (4.053, 3.689) |
(3.969, 4.256) | (4.088, 3.673) | (3.407, 3.666) | (4.143, 4.084) | (4.189, 3.775) |
(4.169, 3.515) | (+, −) | (4.002, 3.419) | (+, −) | (3.921, 3.553) |
(4.107, 3.611) | (+, −) | (4.225, 3.636) | (3.637, 3.79) | (+, +) |
(3.885, 3.255) | (3.226, 4.174) | (3.981, 4.05) | (4.207, 3.308) | (3.749, 3.66) |
(4.059, 3.728) | (3.24, 4.231) | (4.006, 4.134) | (4.088, 3.435) | (+, −) |
(3.994, 3.715) | (4.023, 2.988) | (4.015, 2.648) | (3.432, 3.12) | (+, −) |
(4.001, 2.3) | (3.457, 3.855) | (3.154, 3.802) | (3.696, 2.739) | (3.947, 3.402) |
(3.7, 3.337) | (+, −) | (4.18, 3.36) | (3.308, 3.722) | (3.185, +) |
(+, −) | (+, −) | (4.021, 3.735) | (+, −) | (3.003, 3.183) |
(3.307, 3.918) | (3.35, 3.736) | (+, −) | (+, −) | (3.91, 3.474) |
(3.344, 3.343) | (3.698, 2.683) | (+, −) | (+, −) | (3.223, 2.442) |
(+, −) | (3.852, 3.46) | (3.484, 4.176) | (2.821, 3.796) | (+, −) |
(+, −) | (+, −) | (3.81, 3.176) | (+, −) | (3.705, 3.699) |
(4.246, 3.277) | (4.1, 3.26) | (4.163, 3.712) | (3.612, 4.218) | (3.571, 2.442) |
(3.8, 3.51) | (3.026, 2.779) | (3.641, 3.056) | (+, −) | (3.614, +) |
(+, −) | (3.418, 2.882) | (4.151, 3.851) | (+, −) | (3.474, 4.234) |
(4.216, 3.526) | (3.26, +) | (3.705, 4.029) | (+, −) | (3.246, 3.262) |
(3.51, 3.233) | (3.79, 2.668) | (+, −) | (4.121, 2.694) | (3.875, 3.474) |
(2.995, 3.814) | (3.512, 4.084) | (+, −) | (+, −) | (3.554, 3.772) |
(3.198, 2.464) | (4.213, 4.085) | (+, −) | (+, −) | (3.479, 3.758) |
Parameter | Markov Chain 1 | Markov Chain 2 | Markov Chain 3 | |||
---|---|---|---|---|---|---|
3.9312 | (3.8492, 4.0046) | 3.9262 | (3.8492, 4.0046) | 3.9319 | (3.8727, 4.0205) | |
3.5370 | (3.4580, 3.6047) | 3.5348 | (3.4530, 3.6103) | 3.5416 | (3.4582, 3.6161) | |
0.5282 | (0.4802, 0.5862) | 0.5321 | (0.4793, 0.5907) | 0.5251 | (0.4778, 0.5788) | |
0.5341 | (0.4869, 0.5917) | 0.5375 | (0.4814, 0.5987) | 0.5316 | (0.4815, 0.5899) | |
−0.172 | (−0.318, −0.026) | −0.117 | (−0.319, 0.1032) | −0.148 | (−0.317, 0.0741) | |
0.2722 | (0.2032, 0.3455) | 0.2664 | (0.1687, 0.3435) | 0.2673 | (0.1823, 0.3415) |
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Lin, Y.-J.; Lio, Y.; Tsai, T.-R. Bayesian Estimation of the Stress–Strength Parameter for Bivariate Normal Distribution Under an Updated Type-II Hybrid Censoring. Mathematics 2025, 13, 792. https://doi.org/10.3390/math13050792
Lin Y-J, Lio Y, Tsai T-R. Bayesian Estimation of the Stress–Strength Parameter for Bivariate Normal Distribution Under an Updated Type-II Hybrid Censoring. Mathematics. 2025; 13(5):792. https://doi.org/10.3390/math13050792
Chicago/Turabian StyleLin, Yu-Jau, Yuhlong Lio, and Tzong-Ru Tsai. 2025. "Bayesian Estimation of the Stress–Strength Parameter for Bivariate Normal Distribution Under an Updated Type-II Hybrid Censoring" Mathematics 13, no. 5: 792. https://doi.org/10.3390/math13050792
APA StyleLin, Y.-J., Lio, Y., & Tsai, T.-R. (2025). Bayesian Estimation of the Stress–Strength Parameter for Bivariate Normal Distribution Under an Updated Type-II Hybrid Censoring. Mathematics, 13(5), 792. https://doi.org/10.3390/math13050792