Order Restricted Statistical Inference Of Parameter Models

In this paper, problems of parameters' statistical inference under order restriction are studied mainly. The origins of order restricted statistical inference are usually dated back to the early 1950s. At about the same time, a number of researchers started working independently on similar problems. These resesrchers included Brunk D. and his coworkers and Chacko V.J. in the United States, Bartholomev D. in the United Kingdom et al. The field developed rapidly during the 1960s and early 1970s and formed the more perfect theory of estimation and testing under a variety of assumptions. By now, it is still a fertile area for research in statistical analyses. While a signficient volume of new developments has taken place, it has been more and more used to deal with some practical problems. Such as, clinical trials, bioassays, biomed-ical sciences, genetics, and bioinformatics are all among the interdisciplinary fields that have a growing need for statistical inference. For example, in dose-response studies, the therapeutic variables may increase with dose levels at first and then decrease with the further increasing of dose levels due to adverse effect. Thus, we can use the theory of order restricted statistical inference to find the configuration of therapeutic variables vary with the dose level, the no-observed-adverse-effect level, the maximum therapeutic effect matching the least dose level. The fact that the utilization of ordering information increases the efficiency of statistical inference procedures is well documented. Therefore, research on this area is of important practical significience.The relative content about order restricted statistical inference is introduced in Chapter 1 in order to prepare for the later proofs. The paper mainly contains:(I) A generalized ordered restricted information criterion — GORIC criterionThe detection of the configuration of parameters is one of the most important problems in statistical studies. It is well known that the Akaike's information criterion (AIC) is a key tool for this problem . However, it is not appropriate for the order restricted maximum likehood estimator, sincethe normality or the asymptotic normality of the estimator is not valid. Based on the idea of the AIC and the theory of order restricted inference , Anraku proposed an ordered restricted information (ORIC) to distinguish the configuration of parameters under the restriction of simple order, and showed that it performs well than the usual AIC by a monte carlo study.However, in many application problems, our interesting parameters may present an umbrella order trend instead of simple order trend. For example, in dose-response studies, the therapeutic variables may increase with dose levels at first and then decrease with the further increasing of dose levels due to adverse effect. So, an information criterion for this umbrella ordering is in great demand. For this purpose, we generalize the ORIC to the case of umbrella ordering in this paper and call it GORIC. This GORIC can be used to detect a changepoint in a sequence of parameters with an umbrella ordering trend.(II) The maximum likelihood estimates and the likelihood ratio tests of two multinomial probability vectors under the Stochastic orderings when missing data existsThe estimation and testing problem of paramrters are the most important part of order restricted statistical inference. Stochastic ordering of distributions is an important concept in the theory of statistical inference. It is used to compare probability distributions. There are a lot of problems involving this ordering, for example , some stochastic scheduling, closed queueing network, reliability problems, and survival analysis. Several different types of stochastic ordering have been defined in the literature. Here, only the (usual) stochastic ordering and the likelihood ratio ordering are considered. There has been a considerable amount of work done on inference problems concerning stochastic ordering. Brunk, Frank, Hanson and Hogg (1966) obtained nonparametric maximum likelihood estimates of two stochastically ordered distribution func-tions and studied their properties. Based on this result, Robertson and Wright (1981), Lee and Wolfe (1976), Pranck (1984), Dykstra, Madsen and Fairbanks (1983) discussed the corresponding testing problems.The statistical inference of two multinomial probability vectors under the stochastic orderings have been studied by many researchers. In the context of sampling from continuous distributions, maximum likelihood estimates of stochastically ordered distributions were first obtained by Brunk et al. (1966). Using Fenchel duality, Barlow and Brunk (1972) derived the maximum likelihood estimates of two multinomial probability vectors nder the Stochastic orderings. Both of these two contributions are considered to be seminal. Robertson and Wright (1981) presented the likehood ratio tests of two multinomial probability vectors under the Stochastic orderings. Lucas and Wright (1986) extend this work to multivariate Stochastic orderings. Dykstra et al. (1995) considered the estimates and likehood ratio tests of two multinomial probability vectors under the likelihood ratio orderings. All these results are obtained for complete data. But in many situations, the data are incomplete. So it is of importance for us to consider the statistical inference of two multinomial probability vectors under the Stochastic orderings when missing data exists.(1) When the two multinomial probability vectors satisfying the (usual) stochastic ordering, the maximum likelihood estimates can be obtained by EM algorithm, but the results depend on initial value, and in some situation it is not the expected one. The maximum likelihood estimates can also be obtainedfrom formula method, that is, to obtain maximum point from the observed likelihood function. But the solution is not single, and the complete likelihood function not always achieve the maxima when substitute the missing data by its mean. Here, we propose a problem - solving technique ,and call it M — M method. Based on the maximum likeliood estimate by M — M method, we consider the likelihood ratio tests and derive the asymptotic distribution of the likelihood ratio statistic.(2) When the two multinomial probability vectors satisfying the likelihood ratio ordering, we only present the asymptotic distribution of the likelihood ratio statistic based on the the observed likelihood function.(Ill) Under the Pitman criterion, we show that the restricted maximum likelihood estimator improves upon the usual maximum likelihood estimatorLet Xij, j — 1, ? ? ? ,rii, be a random sample from the i-th normal populations i = 1,2, ? ? ?, k. Let X = {X\, X2, ? ? ?, Xk) be the sample means, then X ~ Nk(6, D), where 6 = (B\, 62, ■ ■ ■, Ok) is the unknown parameter vector, D = diag(a1/rii, o\ln 0, i — 1, 2, ? ? ? ,p}. By X* denotes the restricted maximum likelihood estimator of 6 under the above linear restrictions, the problem of comparing the estimators X and X* is interesting. Some results have been obtained about this problem. Brunk (1965) proved that for any order...