Testy statystyczne w procesie podejmowania decyzji

Abstract

Statistics emerged a separate discipline as a method of extracting information from the observed data and as the logic of decision making under uncertainty. Statistical knowledge is valuable for representatives of all professions. In today’s world, as never before, there is a need for statistical thinking. We are surrounded by challenges of various data banks which require better statistical methods, models, algorithms and processing systems. Statistics deals with collecting numerical information and its analysis and interpretation. Methods presented in the book attempt to answer the question of what these numerical information, treated as data, tell us about the population and the phenomena to which they refer. The answer depends not only on the very observations, but also on the prior knowledge. The book presents in a compact form different classes of statistical tests that can be used in the decision making process. The tests considered may be used in the analysis of economic, social, demographic, technical and medical phenomena. These test classes are characterized by different construction methods, and thus different test procedures. In Bayesian tests a parameter of random variable distribution is treated as a random variable, while in sequential tests the sample size is a random variable. In kernel tests, it is possible to use various kernel functions and various smoothing parameters, which affects heavily the results of the procedure used. In bootstrap tests, the inference procedure is based on the so-called bootstrap samples. Besides theoretical considerations, the results of the research concerning the properties of the verification procedures are presented, indicating the areas of their applications. The work consists of ten chapters. The starting point for consideration of the statistical tests described later in the book are the first three chapters. These include issues related to the classical and decision theoretical approach to the verification of statistical hypotheses. The relationship between statistical tests and decision-making is presented in the first chapter and the third one. The second chapter presents selected classical statistical tests, accounting for the conditions that must be fulfilled by the test to be used in practice. The next section presents the Bayesian tests in which the distribution parameter is treated as a random variable with known prior distribution. Using these tests we can make the decision of the acceptance of the hypothesis of a lower posterior risk, which is determined on the basis of the prior distribution and a fixed loss function. The Bayesian tests, considered, relate to the verification of statistical hypotheses about the parameters of random variables and indicate the possibility of using different sampling schemes. The bootstrap tests in the fifth chapter of the book, deserve attention due to the fact that they do not require information about the class distribution of the random variable investigated. The use of bootstrap methods to approximate distributions of test statistics allows for testing hypotheses about the parameters of the distribution of the population relying on small samples, which is a considerable advantage of these methods. Sequential tests considered in chapter six, is another group of non-classical tests. In these tests, the sample size is a random variable. Increasing the number of the elements of the random sample sequentially, we decide to accept one of the hypotheses verified with the accepted error probabilities of I and II. The advantage of the use of the tests belonging to this class is even twice smaller the expected value of the sample size needed to make a decision in comparison with classical tests for identical mistakes of I and II. In the seventh chapter the class of kernel tests is considered. The kernel method, derived from the estimation of the probability density function is a typical non-parametric approach to statistical inference procedures. In this chapter we discuss the procedures for the verification of hypotheses about the distribution of the random variable including its normality, the goodness-of-fit tests for two or more distributions, hypothesis about the form of the regression function and hypotheses of the independence of random variables. The eighth chapter is devoted to the multidimensional approach in the verification of statistical hypotheses, as most research in the economic and social studies is multidimensional. The tests for the hypotheses about the expected value of vectors and hypotheses concerning the covariance matrix are analyzed, both the classical and those constructed on the basis of random matrix theory limit theorems. The ninth chapter concerns the procedures of statistical inference applicable in situations where the data are time cross-sectional and there is no information for certain periods or moments of time. This chapter presents the most important class of tests for censored data. Tests for goodness-of-fit of two or more populations, or the goodness-of-fit with the hypothetical distribution are considered. A special group of tests used in time series analysis is explored in the tenth chapter, with particular emphasis on the analysis of stationarity and non-stationarity of the stochastic process and parameters verification for the VAR models.