On the Estimation of Symmetric Distributions under Peakedness Order Constraints

Abstract

Consider distribution functions F and G and suppose that F is more peaked about a than G is about b. The problem of estimating F or G, or both, when F and G are symmetric, arises quite naturally in applications. The empirical distribution functions F_n and G_m will not necessarily satisfy the order constraint imposed by the experimental conditions. Rojo and Batun-Cutz [Series in Biostatistics vol. 3, Advances in Statistical Modeling and Inference, (2007) 649–670] proposed some estimators that are strongly uniformly consistent when both m and n tend to infinity. However the estimators fail to be consistent when only either m or n tend to infinity. Here estimators are proposed that circumvent these problems and the asymptotic distribution of the estimators is delineated. A simulation study compares these estimators in terms of Mean Squared Error and Bias behavior with their competitors.