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On estimating the location parameter of the selected exponential population under the LINEX loss function. (English) Zbl 1440.62082
Suppose that \(X_1,\ldots,X_k\) are sample minima based on IID samples from \(k\) independent exponential populations; that is, \(X_i=\min\{X_{i1},\ldots,X_{in_i}\}\), where \(X_{ij}\) has an exponential distribution with unknown location parameter \(\mu_i\) and known scale parameter \(\sigma_i\). The authors consider the problem of estimation of \(\mu_{[k]}=\max\{\mu_1,\ldots,\mu_k\}\) under the linear-exponential loss function given by \[ L(\mu,\hat{\mu})=e^{a(\hat{\mu}-\mu)}-a(\hat{\mu}-\mu)-1\,, \] where \(a\not=0\) is a parameter of the loss function. An exponential population, chosen to correspond to this maximal location parameter, is first selected, and the corresponding parameter is then estimated.
The authors consider three estimators. One is based on the MLEs for the individual \(\mu_i\), another based on the UMVUEs, and the third is based on the minimum risk equivalent estimators; this latter estimator is shown to arise in a Bayesian setting with a non-informative prior distribution. The uniformly minimum risk unbiased estimator (UMRUE) is also derived. A procedure for improving a location-equivariant estimator is proposed, and domination results between the estimators under consideration are established. Finally, a simulation study is used to investigate performance of these estimators.

MSC:
62F10 Point estimation
62F07 Statistical ranking and selection procedures
62F15 Bayesian inference
Citations:
Zbl 1349.62054
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References:
[1] Abughalous, M. M. and Bansal, N. K. (1994). On the problem of selecting the best population in life testing models. Communications in Statistics Theory and Methods 23, 1471-1481. · Zbl 0825.62159
[2] Abughalous, M. M. and Miescke, K. J. (1989). On selecting the largest success probability with unequal sample sizes. Journal of Statistical Planning and Inference 21, 53-68. · Zbl 0666.62024
[3] Arshad, M. and Misra, N. (2015a). Selecting the exponential population having the larger guarantee time with unequal sample sizes. Communications in Statistics Theory and Methods 44, 4144-4171. · Zbl 1333.62037
[4] Arshad, M. and Misra, N. (2015b). Estimation after selection from uniform populations with unequal sample sizes. American Journal of Mathematical and Management Sciences 34, 367-391.
[5] Arshad, M. and Misra, N. (2016). Estimation after selection from exponential populations with unequal scale parameters. Statistical Papers 57, 605-621. · Zbl 1349.62054
[6] Arshad, M. and Misra, N. (2017). On estimating the scale parameter of the selected uniform population under the entropy loss function. Brazilian Journal of Probability and Statistics 31, 303-319. · Zbl 1370.62014
[7] Arshad, M., Misra, N. and Vellaisamy, P. (2015). Estimation after selection from gamma populations with unequal known shape parameters. Journal of Statistical Theory and Practice 9, 395-418. · Zbl 1425.62037
[8] Brewster, J. F. and Zidek, Z. V. (1974). Improving on equivarient estimators. The Annals of Statistics 2, 21-38. · Zbl 0275.62006
[9] Gangopadhyay, A. K. and Kumar, S. (2005). Estimating average worth of the selected subset from two-parameter exponential populations. Communications in Statistics Theory and Methods 34, 2257-2267. · Zbl 1079.62028
[10] Kumar, S., Mahapatra, A. K. and Vellaisamy, P. (2009). Reliability estimation of the selected exponential populations. Statistical Probability Letters 79, 1372-1377. · Zbl 1163.62015
[11] Meena, K. R., Arshad, M. and Gangopadhyay, A. K. (2018). Estimating the parameter of selected uniform population under the squared log error loss function. Communications in Statistics Theory and Methods 47, 1679-1692. · Zbl 1392.62063
[12] Misra, N., Anand, R. and Singh, H. (1998). Estimation after subset selection from exponential populations: Location parameter case. American Journal Mathematical Managemment Sciences 18, 291-326. · Zbl 0927.62021
[13] Misra, N. and Arshad, M. (2014). Selecting the best of two gamma populations having unequal shape parameters. Statistical Methodology 18, 41-63. · Zbl 07035566
[14] Misra, N. and Dhariyal, I. D. (1994). Non-minimaxity of natural decision rules under heteroscedasticity. Statistics & Decisions 12, 79-89. · Zbl 0802.62025
[15] Misra, N. and Singh, G. N. (1993). On the UMVUE for estimating the parameter of the selected exponential population. Journal of Indian Statistical Association 31, 61-69.
[16] Misra, N. and van der Meulen, E. C. (2001). On estimation following selection from nonregular distributions Vol. 30, 2543-2561. · Zbl 1009.62528
[17] Nematollahi, N. and Jozani, M. J. (2016). On risk unbiased estimation after selection. Brazilian Journal of Probability and Statistics 30, 91-106. · Zbl 1381.62051
[18] Nematollahi, N. and Motamed-Shariati, F. (2012). Estimation of the parameter of the selected uniform population under the entropy loss function. Journal of Statistical Planning and Inference 142, 2190-2202. · Zbl 1408.62039
[19] Nematollahi, N. and Pagheh, A. (2017). Estimation of the location parameter and the average worth of the selected subset of two parameter exponential populations under LINEX loss function. Communications in Statistics Theory and Methods 46, 3901-3914. · Zbl 1368.62047
[20] Parsian, A. and Farsipour, N. S. (1999). Estimation of the mean of the selected population under asymmetric loss function. Metrika 50, 89-107. · Zbl 0990.62023
[21] Risko, K. J. (1985). Selecting the better binomial population with unequal sample sizes. Communications in Statistics Theory and Methods 14, 123-158. · Zbl 0572.62028
[22] Varian, H. R. (1975). A Bayesian approach to real estate assessment. In Studies in Bayesian Econometric and Statistics in Honor of Leonard J. Savage, 195-208.
[23] Vellaisamy, P. (1996). A note on the estimation of the selected scale parameters. Journal of Statistical Planning and Inference 55, 39-46. · Zbl 0860.62022
[24] Vellaisamy, P. (2009). A note on unbiased estimation following selection. Statistical Methodology 6, 389-396. · Zbl 1463.62103
[25] Vellaisamy, P., Kumar, S. and Sharma, D. (1988). Estimating the mean of the selected uniform population. Communications in Statistics Theory and Methods 17, 3447-3475. · Zbl 0696.62092
[26] Vellaisamy, P. and Punnen, A. P. (2002). Improved estimators for the selected location parameters. Statistical Papers 43, 291-299. · Zbl 1020.62016
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