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.
Description
Thesis (Ph.D.)-- Wichita State University, College of Liberal Arts and Sciences, Dept. of Mathematics, Statistics and Physics