Estimation of Bonferroni and Total Time on Test Curve using Optimally Selected Order Statistics in Large Samples

K.M. Hassanein and E.F. Brown (2003). Estimation of Bonferroni and Total Time on Test Curve using Optimally Selected Order Statistics in Large Samples. Journal of Statistical Research, Vol. 37, No. 1, pp. 31-42.

Abstract

Let $X$ be a non-negative r.v. with continuous cdf $F_0\left(\frac{x-\mu}{\sigma}\right)$ , where $\mu$ and $\sigma$ are the location and scale parameters respectively. This paper deals with the estimation of the Bonferroni and Total Time on Test (TTT) function defined as $B_p(\mu,\sigma)=\frac{1}{p[\mu+\sigma E(z)]}\left\{p\mu+\sigma\int_0^pQ_0(t)dt\right\}$ and $T_p(\mu,\sigma)=pB_p(\mu,\sigma)+(1-p)\frac{\mu+\sigma Q_0(p)}{\mu+\sigma E(z)}$ , respectively for $0\leq p \leq 1$ , ( $Q_0(t)$ is the quantile function of $F_0(z)$ ) based on a few optimally selected order statistics when the sample size $n$ is large. We use the asymptotically best linear unbiased estimators (ABLUE) of $(\mu,\sigma)$ based on an arbitrary set of $k\leq n$ order statistics $(x_{(r_1)},x_{(r_2)},\cdots,x_{(r_k)})$ with ranks $(r_1,\cdots,r_k)$ satisfying $1\leq r_1\leq r_2\leq \cdots\leq r_k\leq n$ to define the estimate of the Bonferroni and TTT function and study the asymptotic efficiency property of these estimates. We propose how to obtain the optimum order statistics maximising asymptotic relative efficiency (ARE) of the estimator (relative to complete sample estimates). General theory as well as specific case studies are given for the Gamma, Exponential and Pareto distributions.