### Abstract:

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 orderingo f the distributionso f X and -X. One such importanto rderingi s
that P(O < X < x) - P(-x < 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) < f(x) for x > 0. It is
known that the nonparametricm aximum 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.