On Estimation and Inference under Order Restrictions.

Abstract

The aim of statistical analysis and inference is to draw meaningful conclusions. In the case where there is prior knowledge of stochastic orderings or inequalities, it is desirable to incorporate this information in the estimation. This avoids possible unrealistic estimates, and may also lead to gain in efficiency.

In this dissertation we first present the constrained nonparametric maximum likelihood estimator (C-NPMLE) of the survivor functions in one- and two-sample settings. Dykstra (1982) also considered C-NPMLE for such problems, however, as we show, Dykstra's method has an error and does not always give the C-NPMLE. We corrected this error and simulation shows improvement in efficiency compared to Dykstra's estimator. Confidence intervals based on bootstrap methods are proposed. Uniqueness and consistency of the proposed estimators is established.

Second, we propose a new estimator, the pointwise C-NPMLE, which is defined at each time t by the estimates of the survivor functions subject to constraints at t only. The estimator is shown to be non-increasing in t, and the consistency and the asymptotic distribution of the estimators are presented. In the development of this estimator and the characterization of its properties, we transform the problem into one that uses the profile likelihood; we adapt the pool-adjacent-violators algorithm, in which pooling is defined in a special way. Different methods to construct confidence intervals are also proposed. The estimator is shown to have good properties compared to other potential estimators.

Finally, we propose a new method to construct confidence intervals (CIs) for G independent normal means under the linear ordering constraint. The method is based on defining intermediate random variables that are related to the original observations and using the CIs of the means of these intermediate random variables to restrict the original CIs from the separate groups. This method is extended to the case with three or more groups and the simulation studies show that the proposed CIs have coverage rates close to nominal levels with reduced average widths.