Estimation And Tests In Several Linear Statistical Models

The thesis is concerned with the theory, methods and other related problemsof parameter estimation and hypothesis testing in several linear statistical models.For the panel models, the problem of the minimal su?cient (MS) statisticsis considered. The MS statistics of the parameters are obtained and some rela-tionships between these MS statistics and several kinds of estimates in the panelmodels which are often used in practice are established. Based on a special caseof the panel models, the completeness of the MS statistics is proved and the min-imum variance unbiased estimates of the parameters are obtained in this model.By using the concepts of generalized p-value and generalized confidence interval,the problems of hypothesis testing and interval estimation for the unknown pa-rameters are researched in the panel models. For the regression coe?cients, theproblems of hypothesis testing and interval estimation under single situation areconsidered, and exact tests and confidence intervals are obtained. For the variancecomponents, the problems of hypothesis testing and interval estimation for theirarbitrary linear combination are discussed, and exact tests and confidence inter-vals are established. The methods of obtaining exact tests and confidence intervalshave the features that are convenient to compute and are easy to apply to smallsample problems, based on the generalized p-values and generalized confidence intervals. In addition, we investigate the statistical properties of these exact testsand confidence intervals from the analytical and numerical perspectives, respec-tively.For general random models with balanced data, a class of invariant quadraticestimators is established based on the analysis of variance estimate (ANOVAE)of variance components. In the sense of mean square error, it is proved that thisclass is uniformly superior to ANOVAE under certain conditions. On the basisof this class, we obtain two nonnegative estimators of variance components, andshow that they are uniformly superior to ANOVAE and restricted maximum likeli-hood estimate (REMLE) in the sense of mean square error, respectively. By usingthe concepts of generalized p-value and generalized confidence interval, the prob-lems of hypothesis testing and interval estimation for the variance components areconsidered in general random models with balanced data. Exact tests and con-fidence intervals for a single variance component corresponding to random e?ectare developed. Exact tests and confidence intervals are also established for com-paring the random-e?ects variance components and the sum of random-e?ectsvariance components in two independent general random models with balanceddata. Furthermore, we investigate the statistical properties of the resulting testsand confidence intervals. Some simulation results on the type I error probabilityand power of the proposed test are reported. The simulation results indicate thatexact test is very satisfactory for controlling type I error probability. For unbalanced two-way random models, the problem of hypothesis testingfor the exposure level is considered. Tests and confidence intervals are developedbased on the generalized approach (generalized p-value and generalized confidenceinterval), which generalizes the conclusions of balanced two-way random modelsfrom Mu et al.(2007). The frequency properties of the resulting confidence inter-vals are investigated. Some simulation results on the type I error probability andpower of the proposed tests are reported. The simulation results indicate thatexact tests could control type I error probability validly. Furthermore, the prob-lems of hypothesis testing and interval estimation for the intraclass correlationcoe?cients are researched. Tests and confidence intervals are established based onthe generalized approach and modified large-sample approach, respectively, whichgeneralizes the results under the balanced case from Gilder et al.(2007). Moreover,we compare these two approaches from the numerical perspective. The compari-son results indicate that the modified large-sample approach performs better thanthe generalized approach in the coverage probability and expected length, but thegeneralized approach could control type I error probability validly.For the growth curve models, the problem of estimating the unknown param-eters is considered under the situations of the covariance matrix having specialcovariance structures. The explicit expressions of estimators for the unknown pa-rameters are given based on the concept of unbiased estimating equation. Thefirst-order moment and mean square error matrix of the proposed estimators are investigated. Furthermore, we compare the proposed estimator with the existingapproaches from the numerical perspective. The comparison results indicate thatnew estimator obtains much improvement in the sense of mean square error undermost situations. Finally, these methods are generalized to the general extendedgrowth curve model with special covariance structures.