Sequential interval estimation for the exponential hazard rate when the loss function is strictly convex

Abstract

Let $X_1 ,X_2, \cdots, X_n$ be independent and identically distributed random variables having common exponential density with unknown mean $\mu$. In the sequential confidence interval estimation for the exponential hazard rate $\theta=1/\mu$, when the loss function is strictly convex, the following stopping rule is proposed with the half length $d$ of prescribed confidence interval $I_n$ for the parameter $\theta$;

$\tau$=smallest integer n such that\ n $\geq {z_{\alpha/2}^2 \widehat{\theta}^2}/d^2 + 2$,

where $\widehat{\theta}=(n-1){\overline X_n}^{-1}/n$ is the minimum risk estimator for $\theta$ and $z_{\alpha/2}$ is defined by $P(|Z|\leq \alpha/2)=1-\alpha \ (\alpha \in (0,1))$ with $Z\sim N(0,1)$. For the confidence intervals $I_n$ which is required to satisfy $P(\theta \in I_n)\geq 1-\alpha$, These estimated intervals $I_{\tau}$ have the asymptotic consistency of the sequential procedure;

$\lim_{d\rightarrow 0}P(\theta \in I_\tau)= 1-\alpha,$

where $\alpha\in(0,1)$ is given.