Estimating a distribution function subject to a stochastic order restriction: a comparative study
O. Davidov and G. Iliopoulos
In this article we compare four nonparametric estimators of a distribution function, estimated under a stochastic order restriction. The estimators are compared by simulation using four criteria: (1) the estimation of cumulative distribution functions; (2) the estimation of quantiles; (3) the estimation of moments and other functionals; and (4) as tools for testing for stochastic order. Our simulation study shows that estimators based on the pointwise maximum likelihood estimator outperform all other estimators when
the underlying distributions are "close" to each other. The gain in efficiency may be as high as 25%. If the distribution functions are far apart then the pointwise maximum likelihood estimator may not be the best. However, the efficiency loss using the pointwise maximum likelihood estimator relative to the best estimator in each case is generally low (about 5%). We also find that the test based on the pointwise maximum likelihood estimator is the most powerful in the majority of cases although the gain in power relative to other tests is generally small.
Key words and phrases: empirical distribution function, order restricted inference, rank tests, nonparametric estimation, usual stochastic order.