Large Sample Study of Empirical Distributions in a Random-Multiplicative Censoring Model

Abstract

Consider an incomplete data problem with the following specifications. There are three independent samples (Xl, , Xm), (Z1, , Zn) and (UW,..., U0). The first two samples are drawn from a common lifetime distribution function G, while the third sample is drawn from the uniform distribution over the interval (0, 1). In this paper we derive the large sample properties of Gm no the nonparametric maximum likelihood esti- mate of G based on the observed data Xl, . . ., Xm and Y,, . . ., Yn, where Yi - ZiUi, i = 1, . . ., n. (The Z's and U's are unobservable.) In particular we show that if m and n approach infinity at a suitable rate, then suptIGm,n(t) - G(t)m 0 (a.s.), nm +n(Gm - G) converges weakly to a Gaussian process and the estimate Gm,n is asymptotically efficient in a nonparametric sense.