Isotonic Statistical Inferenee in Linear Models

Abstract

Many situations occur in statistical inference where the prior information of an ordinal kind exists, i.e. when the data arc arranged in ordered groups, the mean value of a random variable is assumed to change monotonically with the ordering of the groups. For example, in a dosage response experiment, the probability of response is usually an increasing function of dose level. It is then reasonable to take account of the order restrictions in making inferences about the group means, such as point or interval estimations or significance tests.

It is possible to make better estimates or perform more powerful tests when the information of the prior knowledge is fully utilized than when it is ignored. Taking shapes or order restrictions into account can improve the efficiency of statistical analysis by reducing the error or expected error of estimates or by increasing the power of test procedures, provided that the hypothesized order restriction actually holds. The theory of isotonic inference aims to provide means for doing this. In many fileds the isotonic inference are used, such that the economics, the bioassay and the agriculture. In Section 1.1 of this chapter we introduced the review of relevant

literature. The purpose of the dissertation was given in Section 1.2

1. I- Review of the Literature

Extensive literature was existed on estimation and testing of isotonic and order-restricted inferences for one way linear model. These studies were dealing with estimation of fixed effect. Little attention had been paid to hypothesis testing in specially for the parameters of one way model.