## Brazilian Journal of Probability and Statistics

- Braz. J. Probab. Stat.
- Volume 31, Number 2 (2017), 303-319.

### On estimating the scale parameter of the selected uniform population under the entropy loss function

#### Abstract

Let $\pi_{1},\ldots,\pi_{k}$ be $k$ ($\geq2$) independent populations, where $\pi_{i}$ denotes the uniform distribution over the interval $(0,\theta_{i})$ and $\theta_{i}>0$ ($i=1,\ldots,k$) is an unknown scale parameter. Let $\theta_{[1]}\leq\cdots\leq\theta_{[k]}$ be the ordered values of $\theta_{1},\ldots,\theta_{k}$. The population $\pi_{(k)}$ ($\pi_{(1)}$) associated with the unknown parameter $\theta_{[k]}$ ($\theta_{[1]}$) is called the best (worst) population. For selecting the best population, we consider a general class of selection rules based on the natural estimators of $\theta_{i},i=1,\ldots,k$. Under the entropy loss function, we consider the problem of estimating the scale parameter $\theta_{S}$ of the population selected using a fixed selection rule from this class. We derive the uniformly minimum risk unbiased estimator of $\theta_{S}$ and two natural estimators of $\theta_{S}$ are also considered. We derive a general result for improving a scale invariant estimator of $\theta_{S}$ under the entropy loss function. A simulation study on the performances of various competing estimators of $\theta_{S}$ is also reported. Finally, we provide similar results for the problem of estimating the scale parameter of selected population when the selection goal is that of selecting the worst uniform population.

#### Article information

**Source**

Braz. J. Probab. Stat., Volume 31, Number 2 (2017), 303-319.

**Dates**

Received: July 2015

Accepted: February 2016

First available in Project Euclid: 14 April 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bjps/1492156964

**Digital Object Identifier**

doi:10.1214/16-BJPS314

**Mathematical Reviews number (MathSciNet)**

MR3635907

**Zentralblatt MATH identifier**

1370.62014

**Keywords**

Estimation after selection uniform population inadmissible estimators entropy loss function natural selection rule

#### Citation

Arshad, Mohd.; Misra, Neeraj. On estimating the scale parameter of the selected uniform population under the entropy loss function. Braz. J. Probab. Stat. 31 (2017), no. 2, 303--319. doi:10.1214/16-BJPS314. https://projecteuclid.org/euclid.bjps/1492156964