Estimating the parameter of selected uniform population under the squared log error loss function

ABSTRACT

Let π₁, …, π_k be k (⩾ 2) independent populations, where π_i denotes the uniform distribution over the interval (0, θ_i) and θ_i > 0 (i = 1, …, k) is an unknown scale parameter. The population associated with the largest scale parameter is called the best population. For selecting the best population, We use a selection rule based on the natural estimators of θ_i, i = 1, …, k, for the case of unequal sample sizes. Consider the problem of estimating the scale parameter θ_L of the selected uniform population when sample sizes are unequal and the loss is measured by the squared log error (SLE) loss function. We derive the uniformly minimum risk unbiased (UMRU) estimator of θ_L under the SLE loss function and two natural estimators of θ_L are also studied. For k = 2, we derive a sufficient condition for inadmissibility of an estimator of θ_L. Using these condition, we conclude that the UMRU estimator and natural estimator are inadmissible. Finally, the risk functions of various competing estimators of θ_L are compared through simulation.

KEYWORDS:

• Inadmissibility
• natural estimators
• selection rule
• squared log error loss function
• UMRU estimator
• uniform distributions.

MATHEMATICS SUBJECT CLASSIFICATION:

• 62F07
• 62F10
• 62C15
• 62C20

Acknowledgments

The authors are thankful to the anonymous referees and the editor for their valuable suggestions which have improved presentation of the original manuscript. The second author would like to thank UGC [UGC-BSR Startup Grant: F.30-90/2015(BSR), dated 30/03/2015] New Delhi, India for providing financial support.