Introduction

The General Idea. The basic ideas behind probability theory are as simple as those associated with making lists-the prospect of computing probabilities or thinking in a ‘probabilistic’ manner should not be intimidating. Conceptually, the steps required to compute the chance of any particular event are as follows.

Summary. In this chapter we introduce the ideas behind the art of testing statistical hypotheses (Section 6.1). The general concepts are described, terminology is introduced, and several elementary examples are discussed. We also examine some philosophical questions about testing and some extensions to cases in which it is difficult to build the statistical models needed for testing.

The significance level, power, bias, efficiency, and robustness of statistical tests are discussed in Section 6.2. The application of Monte Carlo simulation in problem testing is discussed in Section 6.3, and in Section 6.4 we examine how hypotheses are formulated and explore some of the limitations of statistical testing. The spatial correlation structure of the atmosphere often impacts testing problems. Strategies for copying with and using this structure are discussed in Section 6.5. A number of tests of the null hypothesis of equal means and variances are discussed in Sections 6.6 and 6.7. Tests designed to provide a global interpretation for a field of local decisions, called field significance tests, are presented in Section 6.8. Univariate and multivariate recurrence analysis are discussed in Sections 6.9 and 6.10.

The Concept of Statistical Tests

Introduction. Since we should now be somewhat comfortable with the ideas underlying hypothesis testing (see [1.2.7], [4.1.7–11], and the preamble to this part of the book), we only briefly characterize the testing paradigm here.

Statistical hypothesis testing is a formalized process that uses the information in a sample to decide whether or not to reject H0, the null hypothesis.

Introduction

The methods described in Section 6.4 enable the accuracy of the regional L-moment algorithm to be assessed for a given region. A wider study of the performance of the regional L-moment algorithm is also valuable, to establish circumstances under which the utility of the regional L-moment algorithm is particularly high (or low) and to facilitate comparison with other methods such as single-site frequency analysis.

Assessment of the performance of the regional L-moment algorithm for different regions can also influence how a given set of data should be analyzed. The regional L-moment algorithm involves estimation of both the index flood and the growth curve, and each of these estimators may have bias and variability, as discussed in Section 2.2. Two tradeoffs between bias and variability of the estimated regional growth curve are immediately apparent. A region with many sites will have low variability of its estimated regional growth curve, but the regional growth curve is likely to have more bias as an estimator of the at-site growth curves, because exact homogeneity is less likely to hold for a region with many sites than for one with few. A fitted distribution with few parameters will yield less variability in the regional growth curve than a distribution with many parameters, but the distribution with few parameters is less likely to contain a good approximation to the true regional frequency distribution of the region, and its regional growth curve is therefore likely to be more biased. Some subjective judgement is required to achieve a suitable compromise between these conflicting criteria; the heterogeneity and goodness-of-fit measures described in Chapters 4 and 5 should be helpful.