Testing For Heteroscedasticity And Varying Dispersion In Nonlinear Models

In ordinary regression analysis, the homoscedasticity of observed data is a basic assumption. Under this assumption, it can be feasible to make routine statistical inference. If the variances of observations are heterogeneous and unknown, then the regression analysis will meet many troubles. For generalized linear and nonlinear models, it is still a conventional hypothesis that observed data should have nominal dispersion, otherwise statistical inference would be more difficult. However, the rationality of these assumptions for variance of observations is doubtable. Therefore, it is necessary to test for heteroscedasticity and varying dispersion for the data, which is an important step of dealing with regression problems and plays an important role in theory and practice. This thesis studies the tests for heteroscedasticity and varying dispersion in various nonlinear regression models.Chapter 2 devotes to studying the test for heteroscedasticity in ordinary nonlinear models. We first discuss the test for variance heterogeneity in weighted nonlinear models using the method of variance parameterization due to Cook & Weisberg (1983). The likelihood ratio statistic, score statistic and their adjustments are obtained. Secondly, we respectively detect the tests for heteroscedasticity in nonlinear models through randomized regression coefficients and variance weight functions, and two score test statistics are obtained. The properties of all test statistics are investigated through Monte Carlo simulations, which display that these tests have nice powers.Situations in which data are collected sequentially over time may raise to serial correlation in errors. Antedependence is one of common serial correlations. As in ordinary regression models, the problem of the heteroscedasticity test still exists in nonlinear models with correlated errors, but, the test for correlation also needs to be considered. Chapter 3 studies the tests for heteroscedasticity and correlation in nonlinear models with correlated errors. First, we develop the individual tests, composite test for heteroscedasticity and antedependence and their adjusted forms in nonlinear regression models with AD(p) errors. Secondly, we proposed a score test and its adjustment for heteroscedasticity in nonlinear models with ARIMA(0,l,0) errors (i.e. random walk errors). Furthermore, an approximate local power function is derived.The generalized nonlinear models(GNLMs), i.e. exponential family nonlinear models are the natural generalization of ordinary nonlinear models. For generalized nonlinear models, since the variance functions are always nonconstant (except the normal case), it is not necessary to detect nonconstant variance. However, the variance problems still exist for GNLMs, which become the tests for varying dispersion, that includes the tests for variation of dispersion parameters and the significance of random factors to the data. Chapter 4 systematicallyinvestigates the tests for varying dispersion in GNLMs. We first present the tests for varying dispersion in common discrete exponential family nonlinear models (binomial, Poisson and negative binomial models etc). For these models, because dispersion parameters are constant, the tests for variation of dispersion parameters are not necessary. We obtain score test statistics based on random coefficient models and random effect models. Nonconstant dispersion parameters and influence of random factors may result in varying dispersion in continuous GNLMs(normal, Gamma and inverse Gaussian models etc). Therefore, we discuss the tests for varying dispersion in continuous GNLMs based on the above statements. Several score test statistics are obtained.Analysis of longitudinal data is a hot topic in recent years and is an important method to investigate statistical characteristic displayed in the data, which is measured repeatedly over time or space for each study participant . The likely sources modelling covariance structure of longitudinal data include random effects, serial correlation and measurement errors. Chapter 5 detects the tests for heteroscedasticity in nonlinear models based on longitudinal data. Several types of heterogeneity of variance are characterized and several score test statistics are developed in nonlinear models with random effects. In the nonlinear models with autocorre-lated errors, and with both random effects and autocorrelated errors respectively, we discuss the tests for homogeneity of variance and between-individual autocorrelation coefficients for the first time. Many test statistics are obtained. These tests are illustrated with an example and Monte Carlo simulations.Chapter 6 investigates the tests for varying dispersion in exponential family nonlinear models based on longitudinal data. We first develop the tests for varying dispersion in two special generalized nonlinear models of longitudinal data: (1) logistic nonlinear models in binomial data; (2) log-nonlinear models in counting Poisson data. Secondly, we discuss the tests for varying dispersion in generalized nonlinear models of longitudinal data. These test statistics are all constructed through testing nonconstant dispersion parameters or influence of random effects. If random effects exist, it is necessary to test the homogeneity of variance of random effects across every individual. However, there is little work on this subject in the literature. In this chapter, we study the test for variance heterogeneity of random effects in Poisson-gamma models as overdispersion occurs.In summary, we have thoroughly and systematically studied the tests for heteroscedasticity and varying dispersion in ordinary nonlinear models, generalized nonlinear models and random-effect models or autocorrelated models based on longitudinal data. The emphases of the thesis are placed on the heterogeneous tests related to several nonlinear models and longitudinal data. Furthermore, the homogeneous tests for autocorrelation coefficients in normal nonlinear models, and, between-individual variance in generalized nonlinear models, based on longitudianal data are investigated for the first time. A series of new test statistics are obtained, and illustrative examples and Monte Carlo simulations show that the test techniques are very well.