Two contributions to order restricted inferences

Abstract

We are proposing two separate problems from order restricted inferences. The rst is a one-sided test for stochastic ordering of two distribution functions that protects against false positive conclusions because of model assumptions. In a traditional hypothesis test there is no protection against the fact that both the null and alternative hypotheses could be false. We modify the classical test to allow for any of the following multiple decisions to be made: (1) Decide the equality of distribution functions cannot be rejected, (2) Decide the distribution functions are ordered in one direction, (3) Decide the distribution functions are ordered in the opposite direction, and (4) Decide both (2) and (3) hold at the same time. Via simulations and examples we show that this procedure provides protection against false positive conclusions while reducing the power of the test minimally when the ordering is correct. The second problem is an improved estimation of a decreasing density of a random variable. The standard Grenander (1956) nonparametric maximum likelihood estimator of a decreasing density has an n??1=3 convergence rate with an unfamiliar asymptotic distribution whereas nonparametric kernel estimators have an n??2=5 convergence rate with a normal asymptotic distribution. We propose a hybrid estimator that will utilize both concepts of estimation and guarantees monotonicity of the estimator. The hybrid estimator will also substantially improve upon past estimators of f(0). We show this analytically through simulations.