Asymptotic Inference For Locally Stationary Processes

The study of locally stationary processes contains theory and methods about a class of processes that describe random phenomena whose fluctuations occur both in time and space. We consider three aspects of locally stationary processes that have not been explore in the already vast literature on these nonstationary processes. We begin by studying the asymptotic efficiency of simple hypotheses tests via large deviation principles. We establish the analogues of classic results such as Stein's lemma, Chernoff bound and the more general Hoeffding bound. These results are based on a large deviation principle for the log-likelihood ratio test statistic between two locally stationary Gaussian processes which is obtained and presented in the first chapter. In the second chapter we consider the Bayesian estimation of two parameters of a locally stationary process: trend and time-varying spectral density functions, respectively. Under smoothness conditions on the latter function, we obtain the asymptotic normality and efficiency, with respect to a broad class of loss functions, of Bayesian estimators. In passing we also show the asymptotic equivalence between Bayesian estimators and the maximum likelihood estimate. Our concluding fourth chapter explores the time-varying spectral density estimation problem from the point of view of Le Cam's theory of statistical experiments. We establish that the estimation of a time-varying spectral density function can be asymptotically construed as a white noise problem with drift. This result is based on Le Cam's connection theorem