A random variable X is symmetric about 0 if X and -X have the same distribution. There is a large literature on the estimation of a distribution function (DF) under the symmetry restriction and tests for checking this symmetry assumption. Often the alternative describes some notion of skewness or one-sided bias. Various notions can be described by an ordering of the distributions of X and -X. One such important ordering is that $P(0<X<X\le x)-P(-x\le X<0)$ is increasing in $x<0$. The distribution of X is said to have a Type II positive bias in this case. If X has a density f, then this corresponds to the density ordering $f(-x)\le f(x)$ for $x<0$. It is known that the nonparametric maximum likelihood estimator (NPMLE) of the DF under this restriction is inconsistent. We provide a projection-type estimator that is similar to a consistent estimator of two DFs under uniform stochastic ordering, where the NPMLE also fails to be consistent. The weak convergence of the estimator has been derived which can be used for testing the null hypothesis of symmetry against this one-sided alternative. It also turns out that the same procedure can be used to estimate two cumulative incidence functions in a competing risks problem under the restriction that the cause specific hazard rates are ordered. We also provide some real life examples.
"Consistent estimation of distributions with type II bias with applications in competing risks problems." Ann. Statist. 32 (1) 245 - 267, February 2004. https://doi.org/10.1214/aos/1079120136