The concept of return period in stationary univariate frequency analysis is prone to misconceptions and misuses that are well known but still widespread. In this study we highlight how nonstationary and multivariate extensions of such a concept are affected by additional misconceptions, thus easily resulting in further ill-posed procedures and misleading conclusions. We also show that the concepts of probability of exceedance and risk of failure over a given design life period provide more coherent, general and well devised tools for risk assessment and communication.

[1] This study attempts to reconcile the conflicting results reported in the literature concerning the behavior of peak‐over‐threshold (POT) daily rainfall extremes and their distribution. By using two worldwide data sets, the impact of threshold selection and record length on the upper tail behavior of POT observations is investigated. The rainfall process is studied within the framework of generalized Pareto (GP) exceedances according to the classical extreme value theory (EVT), with particular attention paid to the study of the GP shape parameter, which controls the heaviness of the upper tail of the GP distribution. A twofold effect is recognized. First, as the threshold decreases, and nonextreme values are progressively incorporated in the POT samples, the variance of the GP shape parameter reduces and the mean converges to positive values denoting a tendency to heavy tail behavior. Simultaneously, the EVT asymptotic hypotheses are less and less realistic, and the GP asymptote tends to be replaced by the Weibull penultimate asymptote whose upper tail is exponential but apparently heavy. Second, for a fixed high threshold, the variance of the GP shape parameter reduces as the record length (number of years) increases, and the mean values tend to be positive, thus denoting again the prevalence of heavy tail behavior. In both cases, i.e., threshold selection and record length effect, the heaviness of the tail may be ascribed to mechanisms such as the blend of extreme and nonextreme values, and fluctuations of the parent distributions. It is shown how these results provide a link between previous studies and pave the way for more comprehensive analyses which merge empirical, theoretical, and operational points of view. This study also provides several ancillary results, such as a set of formulae to correct the bias of the GP shape parameter estimates due to short record lengths accounting for uncertainty, thus avoiding systematic underestimation of extremes which results from the analysis of short time series.

The presence of serial correlation in hydro-meteorological time series often makes the detection of deterministic gradual or abrupt changes with tests such as Mann–Kendall (MK) and Pettitt problematic. In this study we investigate the adverse impact of serial correlation on change point analyses performed by the Pettitt test. Building on methods developed for the MK test, different prewhitening procedures devised to remove the serial correlation are examined, and the effects of the sample size and strength of serial dependence on their performance are tested by Monte Carlo experiments involving the first-order autoregressive [AR(1)] process, fractional Gaussian noise (fGn), and fractionally integrated autoregressive [ARFIMA(1,d,0)] model. Results show that (1) the serial correlation affects the Pettitt test more than tests for slowly varying monotonic trends such as the MK test both for short-range and long-range persistence; (2) the most efficient prewhitening procedure based on AR(1) involves the simultaneous estimation of step change and lag-1 autocorrelation ρ, and bias correction of ρ estimates; (3) as expected, the effectiveness of the prewhitening procedure strongly depends upon the model selected to remove the serial correlation; (4) prewhitening procedures allow for a better control of the type I error resulting in rejection rates reasonably close to the nominal values. As ancillary results, (5) we show the ineffectiveness of the original formulation of the so-called trend-free prewhitening (TFPW) method and provide analytical results supporting a corrected version called TFPWcu; and (6) we propose an improved two-stage bias correction of ρ estimates for AR(1) signals.

Investigating the nature of trends in time series is one of the most common analyses performed in hydro-climate research. However, trend analysis is also widely abused and misused, often overlooking its underlying assumptions, which prevent its application to certain types of data. A mechanistic application of graphical diagnostics and statistical hypothesis tests for deterministic trends available in ready-to-use software can result in misleading conclusions. This problem is exacerbated by the existence of questionable methodologies that lack a sound theoretical basis. As a paradigmatic example, we consider the so-called Şen’s ‘innovative’ trend analysis (ITA) and the corresponding formal trend tests. Reviewing each element of ITA, we show that (1) ITA diagrams are equivalent to well-known two-sample quantile-quantile (q–q) plots; (2) when applied to finite-size samples, ITA diagrams do not enable the type of trend analysis that it is supposed to do; (3) the expression of ITA confidence intervals quantifying the uncertainty of ITA diagrams is mathematically incorrect; and (4) the formulation of the formal tests is also incorrect and their correct version is equivalent to a standard parametric test for the difference between two means. Overall, we show that ITA methodology is affected by sample size, distribution shape, and serial correlation as any parametric technique devised for trend analysis. Therefore, our results call into question the ITA method and the interpretation of the corresponding empirical results reported in the literature.

[1] Moving from univariate to multivariate frequency analysis, this study extends the Klemeš' critique of the widespread belief that the increasingly refined mathematical structures of probability functions increase the accuracy and credibility of the extrapolated upper tails of the fitted distribution models. In particular, we discuss key aspects of multivariate frequency analysis applied to hydrological data such as the selection of multivariate design events (i.e., appropriate subsets or scenarios of multiplets that exhibit the same joint probability to be used in design applications) and the assessment of the corresponding uncertainty. Since these problems are often overlooked or treated separately, and sometimes confused, we attempt to clarify properties, advantages, shortcomings, and reliability of results of frequency analysis. We suggest a selection method of multivariate design events with prescribed joint probability based on simple Monte Carlo simulations that accounts for the uncertainty affecting the inference results and the multivariate extreme quantiles. It is also shown that the exploration of the p-level probability regions of a joint distribution returns a set of events that is a subset of the p-level scenarios resulting from an appropriate assessment of the sampling uncertainty, thus tending to overlook more extreme and potentially dangerous events with the same (uncertain) joint probability. Moreover, a quantitative assessment of the uncertainty of multivariate quantiles is provided by introducing the concept of joint confidence intervals. From an operational point of view, the simulated event sets describing the distribution of the multivariate p-level quantiles can be used to perform multivariate risk analysis under sampling uncertainty. As an example of the practical implications of this study, we analyze two case studies already presented in the literature.