On estimating the location parameter of the selected exponential population under the LINEX loss function.

*(English)*Zbl 1440.62082Suppose 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.

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.

Reviewer: Fraser Daly (Edinburgh)

##### MSC:

62F10 | Point estimation |

62F07 | Statistical ranking and selection procedures |

62F15 | Bayesian inference |

##### Keywords:

exponential populations; estimation after selection; equivariant estimators; LINEX loss function; inadmissible estimators; UMRU estimator##### Citations:

Zbl 1349.62054
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\textit{M. Arshad} and \textit{O. Abdalghani}, Braz. J. Probab. Stat. 34, No. 1, 167--182 (2020; Zbl 1440.62082)

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