Andrew V. Carter

Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift

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.

Asymptotic Behavior of a t Test Robust to Cluster Heterogeneity

For a cluster-robust t-statistic under cluster heterogeneity we establish that the cluster-robust t-statistic has a gaussian asymptotic null distribution and develop the effective number of clusters, which scales down the actual number of clusters, as a guide to the behavior of the test statistic. The implications for hypothesis testing in applied work are that the number of clusters, rather than the number of observations, should be reported as the sample size, and the effective number of clusters should be reported to guide inference. If the effective number of clusters is large, testing based on critical values from a normal distribution is appropriate.

Markov Regime-Switching Tests: Asymptotic Critical Values

Abstract Empirical research with Markov regime-switching models often requires the researcher not only to estimate the model but also to test for the presence of more than one regime. Despite the need for both estimation and testing, methods of estimation are better understood than are methods of testing. We bridge this gap by explaining, in detail, how to apply the newest results in the theory of regime testing, developed by Cho and White [Cho, J. S., and H. White 2007. “Testing for Regime Switching.” Econometrica 75 (6): 1671–1720.]. A key insight in Cho and White is to expand the null region to guard against false rejection of the null hypothesis due to a small group of extremal values. Because the resulting asymptotic null distribution is a function of a Gaussian process, the critical values are not obtained from a closed-form distribution such as the χ². Moreover, the critical values depend on the covariance of the Gaussian process and so depend both on the specification of the model and the specification of the parameter space. To ease the task of calculating critical values, we describe the limit theory and detail how the covariance of the Gaussian process is linked to the specification of both the model and the parameter space. Further, we show that for linear models with Gaussian errors, the relevant parameter space governs a standardized index of regime separation, so one need only refer to the tabulated critical values we present. While the test statistic under study is designed to detect regime switching in the intercept, the test can be used to detect broader alternatives in which slope coefficients and error variances may also switch over regimes.

Asymptotic approximation of nonparametric regression experiments with unknown variances

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.

Tusnády’s inequality revisited

Tusnadys 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.

Asymptotically Sufficient Statistics in Nonparametric Regression Experiments with Correlated Noise

We find asymptotically sufficient statistics that could help simplify inference in nonparametric regression problems with correlated errors. These statistics are derived from a wavelet decomposition that is used to whiten the noise process and to effectively separate high-resolution and low-resolution components. The lower-resolution components contain nearly all the available information about the mean function, and the higher-resolution components can be used to estimate the error covariances. The strength of the correlation among the errors is related to the speed at which the variance of the higher-resolution components shrinks, and this is considered an additional nuisance parameter in the model. We show that the NPR experiment with correlated noise is asymptotically equivalent to an experiment that observes the mean function in the presence of a continuous Gaussian process that is similar to a fractional Brownian motion. These results provide a theoretical motivation for some commonly proposed wavelet estimation techniques.