Estimating Future from Past in Life Testing

Abstract

Let $\theta_p$ represent the unique 100$p$ per cent point of a continuous statistical population, while $x_r$ is the $r$th largest value of a sample of size $n$ from this population $(r = 1, \cdots, n)$. This paper considers estimation of $\theta_p$ on the basis of $x_{r(1)}, \cdots, x_{r(m)}$, where the $r(i)$ differ by $O(\sqrt{n + 1})$ and do not necessarily have values near $(n + 1)p$. Also considered is estimation of $x_R$ on the basis of $x_{r(1)}, \cdots, x_{r(m)}$, where the $r(i)$ differ by $O(\sqrt{n + 1})$ and do not necessarily have values near $R$. The results are of a nonparametric nature and based on expected value considerations. These estimation procedures may be useful for life-testing situations where time to failure is the variable and some of the items tested have not yet failed when observation is discontinued. Then $\theta_p$ and $x_R$ can be estimated for $p$ and $R$ values which extend a moderate way into the region where sample data is not available. Estimation of the $x_R$ value which would be obtained by continuing to observe the experiment represents a prediction of the future from the past. The results of this paper may be of value in the actuarial, population statistics, operations research, and other fields.