Abstract: A random variable X is said to have a symmetric distribution function (DF) about zero if X and −X have the same distribution. The estimation of such a distribution and tests for symmetry are widely studied in the literature. Some of the alternatives to symmetry describe some notion of skewness or one-sided bias in terms of an ordering of the distributions of X and −X. One such ordering is characterized by r_−X(𝓍) ≤ r_X(𝓍) for all 𝓍 > 0 where r_−X(𝓍) and r_X(𝓍) are the hazard rates of −X and X, respectively. This is equivalent to the ratio P(X > 𝓍)/P(X < −𝓍) being nondecreasing in . In this paper we derive the nonparametric maximum likelihood estimator (NPMLE) of F under this constraint and show that it is inconsistent. We then construct a new estimator and establish its consistency and weak convergence. We also develop a test for symmetry against this one-sided alternative and study the finite sample performance of this new estimator. We show through simulations that it outperforms the NPMLE in terms of mean squared error for all the distributions under consideration. We also show how to apply this approach to compare the conditional distributions (conditional on the risks) of two competing risks in a competing risks model.

Key words and phrases: Competing risks, consistency, nonparametric likelihood estimator, restricted estimation, weak convergence.