Nonparametric Goodness-of-Fit Testing Under Gaussian Models

There are two main problems in statistics, estimation theory and hypothesis testing. For the classical finite-parametric case, these problems were studied in parallel. On the other hand, many statistical problems are not parametric in the classical sense; the objects of estimation or testing are functions, images, and so on. These can be treated as unknown infinite-dimensional parameters that belong to specific functional sets. This approach to nonparametric estimation under asymptotically minimax setting was started in the 1960s-1970s and was developed very intensively for wide classes of functional sets and loss functions. Nonparametric estimation problems have generated a large literature. On the other hand, nonparametric hypotheses testing problems have not drawn comparable attention in the statistical literature. In this book, the authors develop a modern theory of nonparametric goodness-of-fit testing. The presentation is based on an asymptotic version of the minimax approach. The key element of the theory is the method of constructing of asymptotically least favorable priors for a wide enough class of nonparametric hypothesis testing problems. These provide methods for the construction of asymptotically optimal, rate optimal, and optimal adaptive test procedures. The book is addressed to mathematical statisticians who are interesting in the theory of nonparametric statistical inference. It will be of interest to specialists who are dealing with applied nonparametric statistical problems in signal detection and transmission, and technical and m other fields. The material is suitable for graduate courses on mathematical statistics. The book assumes familiarity with probability theory.