‘Interval estimation of location difference with incomplete data’

Abstract

Let (X, Y-0) be a random vector with continuous joint distribution function H which is symmetric in its arguments. Also, let (X1, Y1), ..., (Xn, Yn), (Xn + 1 v ) vI (Xn +s, ), ( , Yn + 1), ... I (. I Yn +t) be a random sample from (X, Y), where '.' denotes a missing observation. If H is a bivariate normal distribution function, a confidence interval for 0 can be obtained by simply inverting the test statistic used for testing Ho: 0 = 0 (Lin, 1973; Ekbohm, 1976; Bhoj, 1978). However, in practice real data often exhibit a slight tendency toward more extreme values than one would expect from normal samples (Stigler, 1977). Therefore, a distribution-free interval estimation procedure for 0 is desirable. Although, in theory, the estimator 6 of 0 proposed by Wei (1981) can be used to construct an asymptotically distribution-free confidence interval of 6, it is rather difficult to obtain its confidence bound. In this note an alternative distribution-free interval estimation procedure for 0 is proposed. The corresponding interval limit has a simple and explicit form and can be easily obtained. Under a contaminated bivariate normal model, the asymptotic relative efficiency comparisons with other known parametric confidence intervals for 0 are made to show the robustness property of the proposed procedure.