Asymptotic equivalence results for nonparametric regression experiments have always assumed that the variances of the observations are known. In practice, however the variance of each observation is generally considered to be an unknown nuisance parameter. We establish an asymptotic approximation to the nonparametric regression experiment when the value of the variance is an additional parameter to be estimated or tested. This asymptotically equivalent experiment has two components: the first contains all the information about the variance and the second has all the information about the mean. The result can be extended to regression problems where the variance varies slowly from observation to observation.

Tusnady's inequality is the key ingredient in the KMT/Hungarian coupling of the empirical distribution function with a Brownian bridge. We present an elementary proof of a result that sharpens the Tusnady inequality, modulo constants. Our method uses the beta integral representation of Binomial tails, simple Taylor expansion and some novel bounds for the ratios of normal tail probabilities.

This dissertation is composed of a study of estimation methods in classical and test theories and the elaboration and application of a cluster-robust variance estimator. Variance estimators derived from generalized estimating equations are known to be robust to most covariance structures and are therefore well suited for psychometric analysis of longitudinal test data. However, the approximate normal distribution of the test statistic for clustered binary experiments breaks down when the variation between cluster variances is large. The degrees of freedom for the test statistic are smaller than the number of clusters in unbalanced experiments and closer to an effective number of clusters, G*, which we estimate as the degrees of freedom using Satterthwaite approximation. We calculate a bias bound as a function of G* to improve the coverage percentages of the test statistic. Simulations generated by a beta-binomial model and a Markov chain model show that the bias-adjusted cluster-robust variance estimator improves the test statistic and achieves a coverage percentage of at least 94\% for highly heteroskedastic settings. For conservative confidence intervals in even more unbalanced situations, t-scores with G* degrees of freedom can be used. When compared to a quasibinomial generalized linear model and a wild bootstrap estimator, the bias-adjusted CRVE is closer to the asymptotic distribution for low effective numbers of clusters and yields almost equivalent results to the other two estimators across simulations. Consistency conditions based on cluster heterogeneity are shown to be sufficient for convergence of a chi-square test for testing multiple probabilities across each cluster. We show that the chi-square statistic can be used to test for parallel scales, equivalent items, or time effects in classical test analysis of longitudinal data. Finally, we discuss the use of generalized estimating equations and the multivariate cluster-robust variance estimator in Rasch analysis of repeated measures.

In Cho and White (2007) "Testing for Regime Switching" the authors obtain the asymptotic null distribution of a quasi-likelihood ratio (QLR) statistic. The statistic is designed to test the null hypothesis of one regime against the alternative of Markov switching between two regimes. Likelihood ratio statistics are used because the test involves nuisance parameters that are not identified under the null hypothesis, together with other nonstandard features. Cho and White focus on a quasi-likelihood, which ignores certain serial correlation properties but allows for a tractable factorization of the likelihood. While the majority of their paper focuses on asymptotic behavior under the null hypothesis, Theorem 1(b) states that the quasi-maximum likelihood estimator (QMLE) is consistent under the alternative hypothesis. Consistency of the QMLE requires that the expected quasi-log-likelihood attain a global maximum at the population parameter values. This requirement holds for some Markov regime-switching processes but, as we show below, not for an autoregressive process as analyzed in Cho and White.

This paper establishes the global asymptotic equivalence between a Poisson process with variable intensity and white noise with drift under sharp smoothness conditions on the unknown function. This equivalence is also extended to density estimation models by Poissonization. The asymptotic equivalences are established by constructing explicit equivalence mappings. The impact of such asymptotic equivalence results is that an investigation in one of these nonparametric models automatically yields asymptotically analogous results in the other models.