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 $\ge z_{\alpha /2}^2{\hat{\theta }}^2/d^2+2$,

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

$\underset{d\to 0}{lim}P(\theta \in I_{\tau })=1-\alpha ,$

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