Simultaneous estimation of several CDF’s: homogeneity constraint

Abstract

ABSTRACTLet {xij(1 ⩽ j ⩽ ni)|i = 1, 2, …, k} be k independent samples of size nj from respective distributions of functions Fj(x)(1 ⩽ j ⩽ k). A classical statistical problem is to test whether these k samples came from a common distribution function, F(x) whose form may or may not be known. In this paper, we consider the complementary problem of estimating the distribution functions suspected to be homogeneous in order to improve the basic estimator known as “empirical distribution function” (edf), in an asymptotic setup. Accordingly, we consider four additional estimators, namely, the restricted estimator (RE), the preliminary test estimator (PTE), the shrinkage estimator (SE), and the positive rule shrinkage estimator (PRSE) and study their characteristic properties based on the mean squared error (MSE) and relative risk efficiency (RRE) with tables and graphs. We observed that for k ⩾ 4, the positive rule SE performs uniformly better than both shrinkage and the unrestricted estimator, while PTEs works reasonably well for k < 4.