Estimation of change points in failure rate models

In a life distribution, the failure rate function $\gamma$(x) plays an important role. A point $\theta$ is called a change point of $\gamma$(x) if either(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\gamma(\rm x) = \cases{\rm a(x),& $\rm 0 \le x \le \theta$\cr \rm b(x),& $\rm x \ge \theta$\cr}\leqno{\rm(A)}$$(TABLE/EQUATION ENDS)or(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\gamma(\theta) = \rm sup\sb{x \u3e 0}\ \gamma(x)\left(\inf\sbsp{x \u3e 0}{}\ \gamma(x)\right).\leqno{\rm(B)}$$(TABLE/EQUATION ENDS)The estimation of $\theta$ when a(x) $\equiv$ a, b(x) $\equiv$ b, where a and b are two unknown constants, under model (A) has been done by Nguyen et al. (1984) and Yao (1986). We consider the general case of estimation of $\theta$ under model (A) when a(x) and b(x) are two unknown functions satisfying certain regularity conditions. An estimator $\\theta$ of $\theta$ based on $\acute\gamma(\theta-$) and $\acute\gamma(\theta+$) is proposed for any two functions a(x) and b(x). Also for the case where a(x) and b(x) are both increasing (decreasing), an estimator of $\theta$ is developed using isotonic regression technique. Under model (B) an estimator of $\theta$ is constructed using an estimator of $\gamma$(x). The asymptotic properties of these estimators are obtained. Simulations are carried out for all the proposed estimators for a number of different functions $\gamma$(x) following both model (A) and (B) to examine the behavior of the suggested estimators in each situation. These estimators are compared with other available estimators whenever the competitors are applicable for the special functions $\gamma$(x) being considered