Estimation of \(P[Y<X]\) for generalized Pareto distribution. (English) Zbl 1177.62024
Summary: This paper deals with the estimation of \(R=P[Y<X]\) when \(X\) and \(Y\) are two independent generalized Pareto distributions with different parameters. The maximum likelihood estimator and its asymptotic distribution are obtained. An asymptotic confidence interval of \(P[Y<X]\) is constructed using the asymptotic distribution. Assuming that the common scale parameter is known, MLE, UMVUE, Bayes estimation of \(R\) and a confidence interval are obtained. The ML estimator of \(R\), the asymptotic distribution and Bayes estimation of \(R\) in the general case are also studied. Monte Carlo simulations are performed to compare the different proposed methods.
MSC:
| 62F10 | Point estimation |
| 62E20 | Asymptotic distribution theory in statistics |
| 62F15 | Bayesian inference |
| 62F25 | Parametric tolerance and confidence regions |
| 65C05 | Monte Carlo methods |
Keywords:
Bayes estimator; generalized Pareto distribution; maximum likelihood estimator; bootstrap confidence intervals; asymptotic distributionsReferences:
| [1] | Abd Elfattah, A.M., Marwa, O.M. Estimating of \(P(Y \&\#60; X)\) in the exponential case based on censored samples. Electronic Journal of Statistics, to appear, ISSN:1935-7524.; Abd Elfattah, A.M., Marwa, O.M. Estimating of \(P(Y \&\#60; X)\) in the exponential case based on censored samples. Electronic Journal of Statistics, to appear, ISSN:1935-7524. |
| [2] | Ahmad, K. E.; Fakhry, M. E.; Jaheen, Z. F., Empirical Bayes estimation of \(P(Y < X)\) and characterizations of the Burr-type X model, Journal of Statistical Planning and Inference, 64, 297-308 (1997) · Zbl 0915.62001 |
| [3] | Awad, A. M.; Azzam, M. M.; Hamadan, M. A., Some inference result in \(P(Y < X)\) in the bivariate exponential model, Communications in Statistics—Theory and Methods, 10, 2515-2524 (1981) |
| [4] | Baklizi, A.; El-Masri, A. Q., Shrinkage estimation of \(P(X < Y)\) in the exponential case with common location parameter, Metrika, 59, 2, 163-171 (2004) · Zbl 1079.62029 |
| [5] | Chen, M. H.; Shao, Q. M., Monte Carlo estimation of Bayesian credible and HPD intervals, Journal of Computational and Graphical Statistics, 8, 69-92 (1999) |
| [6] | Congdon, P., Bayesian Statistical Modeling (2001), Wiley: Wiley New York · Zbl 0967.62019 |
| [7] | Constantine, K.; Karson, M., The estimation of \(P(Y < X)\) in gamma case, Communications in Statistics—Computations and Simulations, 15, 365-388 (1986) · Zbl 0601.62043 |
| [8] | Devroye, L., A simple algorithm for generating random variates with a log-concave density, Computing, 33, 247-257 (1984) · Zbl 0561.65004 |
| [9] | Downtown, F., The estimation of \(P(Y < X)\) in the normal case, Technometrics, 15, 551-558 (1973) · Zbl 0262.62016 |
| [10] | Efron, B., 1982. The jackknife, the bootstrap and other resampling plans. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 38, SIAM, Phiadelphia, PA.; Efron, B., 1982. The jackknife, the bootstrap and other resampling plans. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 38, SIAM, Phiadelphia, PA. · Zbl 0496.62036 |
| [11] | Ferguson, T. S., Mathematical Statistics: A Decision Theoretic Approach (1967), Academic Press: Academic Press New York · Zbl 0153.47602 |
| [12] | Govidarajulu, Z., Tow sided confidence limits for \(P(X > Y)\) based on normal samples of \(X\) and \(Y\), Sankhya B, 29, 35-40 (1967) |
| [13] | Hall, P., Theoretical comparison of bootstrap confidence intervals, Annals of Statistics, 16, 927-953 (1988) · Zbl 0663.62046 |
| [14] | Kakade, C. S.; Shirke, D. T.; Kundu, D., Inference for \(P(Y < X)\) in exponentiated Gumbel distribution, Journal of Statistics and Applications, 3, 1-2, 121-133 (2008) · Zbl 1292.62024 |
| [15] | Kotz, S.; Lumelskii, Y.; Pensky, M., The Stress-Strength Model and its Generalizations: Theory and Applications (2003), World Scientific: World Scientific Singapore · Zbl 1017.62100 |
| [16] | Kremer, E., 1997. A characterization of the generalized Pareto-distribution with an application to reinsurance. Blätter der DGVFM 23 (1), 17-19.; Kremer, E., 1997. A characterization of the generalized Pareto-distribution with an application to reinsurance. Blätter der DGVFM 23 (1), 17-19. · Zbl 0881.62016 |
| [17] | Krishnamoorthy, K.; Mukherjee, S.; Guo, H., Inference on reliability in two-parameter exponential stress-strength model, Metrika, 65, 3, 261-273 (2007) · Zbl 1433.62061 |
| [18] | Kundu, D.; Gupta, R. D., Estimation of \(P [Y < X]\) for generalized exponential distribution, Metrika, 61, 291-308 (2005) · Zbl 1079.62032 |
| [19] | Kundu, D.; Gupta, R. D., Estimation of \(P [Y < X]\) for Weibull distributions, IEEE Transactions on Reliability, 55, 2, 270-280 (2006) |
| [20] | Lindley, D. V., Approximate Bayes method, Trabajos de Estadistica, 3, 281-288 (1980) |
| [21] | Mokhlis, N. A., Reliability of a stress-strength model with Burr type III distributions, Communications in Statistics—Theory and Methods, 34, 7, 1643-1657 (2005) · Zbl 1070.62091 |
| [22] | Nadarajah, S., Reliability for lifetime distributions, Mathematical and Computer Modelling, 37, 7-8, 683-688 (2003) · Zbl 1045.62106 |
| [23] | Nadarajah, S., Reliability for logistic distributions, Elektronnoe Modelirovanie, 26, 3, 65-82 (2004) |
| [24] | Nadarajah, S., Reliability for Laplace distributions, Mathematical Problems in Engineering, 2004, 2, 169-183 (2004) · Zbl 1070.62093 |
| [25] | Nadarajah, S., 2005a. Reliability for some bivariate beta distributions. Mathematical Problems in Engineering (1), 101-111.; Nadarajah, S., 2005a. Reliability for some bivariate beta distributions. Mathematical Problems in Engineering (1), 101-111. · Zbl 1069.62081 |
| [26] | Nadarajah, S., 2005b. Reliability for some bivariate gamma distributions. Mathematical Problems in Engineering (2), 151-163.; Nadarajah, S., 2005b. Reliability for some bivariate gamma distributions. Mathematical Problems in Engineering (2), 151-163. · Zbl 1094.62132 |
| [27] | Nadarajah, S.; Kotz, . S., Reliability for some bivariate exponential distributions, Mathematical Problems in Engineering, 2006, 1-14 (2006) · Zbl 1200.74143 |
| [28] | Owen, D. B.; Craswell, K. J.; Hanson, D. L., Non-parametric upper confidence bounds for \(P(Y < X)\) and confidence limits for \(P(Y < X)\) when \(X\) and \(Y\) are normal, Journal of the American Statistical Association, 59, 906-924 (1977) · Zbl 0127.10504 |
| [29] | Saraçoglu, B.; Kaya, M. F., Maximum likelihood estimation and confidence intervals of system reliability for Gompertz distribution in stress-strength models, Selçuk Journal of Applied Mathematics, 8, 2, 25-36 (2007) · Zbl 1142.62086 |
| [30] | Shi, G.; Atkinson, H. V.; Sellars, C. M.; Anderson, C. W., Application of the generalized Pareto distribution to the estimation of the size of the maximum inclusion in clean steels, Acta Materialia, 47, 5, 1455-1468 (1999) |
| [31] | Surles, J. G.; Padgett, W. J., Inference for \(P(Y < X)\) in the Burr type X model, Journal of Applied Statistical Sciences, 7, 225-238 (1998) · Zbl 0911.62092 |
| [32] | Surles, J. G.; Padgett, W. J., Inference for reliability and stress-strength for a scaled Burr-type X distribution, Lifetime Data Analysis, 7, 187-200 (2001) · Zbl 0984.62082 |
| [33] | Tong, H., A note on the estimation of \(P(Y < X)\) in the exponential case, Technometrics, 16, 625 (1974) · Zbl 0292.62021 |
| [34] | Tong, H., On the estimation of \(P(Y < X)\) for exponential families, IEEE Transactions on Reliability, 26, 54-56 (1977) · Zbl 0379.62025 |
| [35] | Woodward, W. A.; Kelley, G. D., Minimum variance unbiased estimation of \(P(Y < X)\) in the normal case, Technometrics, 19, 95-98 (1977) · Zbl 0352.62036 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
