New and improved estimators of distribution functions under second-order stochastic dominance

Abstract

Second-order stochastic ordering plays a fundamental role in many scientific areas including economics and finance. This article is concerned with the estimation of two continuous distribution functions, F 1 and F 2, when F 1 is smaller than F 2 according to this ordering. In the one-sample case, we assume that F 1 is known and provide a uniformly consistent estimator for F 2. The problem of estimating F 1 when F 2 is known was considered in Rojo and El Barmi [J. Rojo and H. El Barmi, Estimation of distribution functions under second order stochastic dominance, Statist. Sinica 13 (2003), pp. 903–926]. For this case, we show that their estimator continues to be uniformly strongly consistent without the restrictive conditions that they impose on F 1. In the two-sample case, we propose a new class of uniformly strongly consistent estimators for the two distribution functions, where n 1 and n 2 are the sample sizes. An extensive simulation study shows that for α = n 1/(n 1+n 2), the new estimators outperform those proposed by Rojo and El Barmi [J. Rojo and H. El Barmi, Estimation of distribution functions under second order stochastic dominance, Statist. Sinica 13 (2003), pp. 903–926] for the two-sample case in terms of mean squared error at most of the quantiles of the distributions that we consider. An example is discussed to illustrate the theoretical results.