Statistical Hypothesis Testing

Abstract

In this chapter, hypothesis testing, which is another important method of statistical inference, is introduced. In the initial sections, the general guidelines for constructing hypotheses tests and the logic behind them are outlined. Then, the most common tests for normal populations are described and exemplified. To this purpose, a few hydrologic variables that are approximately symmetric are used as examples. It is well known, however, that hydrologic variables are skewed, a fact that imposes the use of the nonparametric tests, which do not assume from the beginning a distributional shape for the data being tested. As such, some sections of this chapter are devoted to describing some nonparametric tests, with particular emphasis to those that are essential to test the hypotheses of randomness, independence, homogeneity, and nonstationarity, generally required for hydrologic frequency analysis. In the final sections of this chapter, the most relevant goodness-of-fit tests, namely, the chi-square, Kolmogorov–Smirnov, Anderson–Darling, and PPCC (Probability Plot Correlation Coefficient) are described and exemplified, as well as the Grubbs–Beck test to detect and identify possible outliers in a sample. As usual in this textbook, the choice of methods and tests has been exercised on the basis of their practical usefulness for Statistical Hydrology. The subject of statistical hypothesis testing, however, is much broader and covers topics that are not included in this chapter, such as test optimality, sequential and multiple-hypotheses tests. Another important topic that is not covered in this chapter, that of likelihood-ratio tests, is described and employed in Chap. 12, in the context of nonstationary frequency analysis. The reader interested in tests and methods not covered in this chapter should consult Hoel et al. (1971) or Mood et al. (1974) or Ramachandran and Tsokos (2009), for books written at an intermediate level of complexity, or more in-depth texts such as Bickel and Doksum (1977) or Casella and Berger (1990).