Estimating location parameters of several exponential distributions with ordered restriction under Linex loss function
Shrajal Bajpai, Lakshmi Kanta Patra, Suchandan Kayal
TL;DR
This work studies component-wise estimation of ordered location parameters $\mu_i$ for $k$ exponential populations under the asymmetric Linex loss $L(t)=e^{pt}-pt-1$, presenting inadmissibility results for standard estimators and constructing explicit improved estimators that dominate the best affine equivariant estimator, the BAEE, and in some cases the MLE and RMLE. It covers three variance-structure scenarios (known $\sigma_i$, equal unknown $\sigma$, and unequal unknown $\sigma_i$), deriving conditional-risk-minimizing estimators and applying Brewster's improving technique to obtain dominating rules. Through extensive Monte Carlo simulations, the authors demonstrate substantial risk reductions (PRI) for the proposed estimators, with gains growing with the Linex parameter $p$ and depending on scale configurations. The framework is shown to extend naturally to Type-II censoring, progressive Type-II censoring, and record-value data, providing a unified approach for ordered-parameter life-testing problems with practical impact in reliability and survival analysis.
Abstract
Some improved estimators of the location parameters of several exponential distributions with ordered restriction are derived and compared numerically using Monte Carlo simulations. Note that the two-parameter exponential distribution is very useful in different areas like survival analysis, reliability engineering and biomedical research, where products have a guaranteed failure-free operating time before failures begin to occur. In the present manuscript, we address the component-wise estimation of location parameters of $k~(\ge 2)$ exponential distributions under an asymmetric Linex loss function. The location parameter represents a minimum guaranteed period before failure. At first, we consider the estimation of the location parameters with ordered scale parameters. Next, we address the estimation of ordered location parameters. For this, we take three different cases into account as follows: $(i)$ scale parameters are known, $(ii)$ scale parameters are unknown but equal, $(iii)$ scale parameters are unknown and unequal. In these cases, we establish general inadmissibility results. Further, using the general result, the inadmissibility of the best affine equivariant estimator is proved. The improved estimators are written in explicit forms. Additionally, we show that the results for several important life-testing schemes namely $(i)$ Type-II censoring, $(ii)$ progressive type-II censoring and $(iii)$ record value data can be obtained using i.i.d sample.Finally, for each case, the Monte Carlo simulation technique is used to compare the performance of the proposed estimators based on their risk values. The numerical results reveal a significant improvement of the proposed estimators.