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

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, F1 and F2, when F1 is smaller than F2 according to this ordering. In the one-sample case, we assume that F1 is known and provide a uniformly consistent estimator for F2. The problem of estimating F1 when F2 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 F1. In the two-sample case, we propose a new class of uniformly strongly consistent estimators for the two distribution functions, where n1 and n2 are the sample sizes. An extensive simulation study shows that for α = n1/(n1+n2), 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.