Recurrent-extreme waiting times under combined nonstationarity and serial dependence

Classical hydrologic design quantifies extremes through the return period (the geometric waiting time to the first exceedance of a design threshold) and the binomial risk of exceedance within a design life. Obeysekera and Salas (2016) rebuilt this frequency apparatus for annually varying exceedance probabilities, using the Poisson binomial law for exceedance counts, a nonstationary geometric return period, and an expected-number-of-exceedances design criterion, and listed what remained unresolved. Later work resolved the independent-year part of the k-th-exceedance waiting-time theory (Salas and Obeysekera 2019), made exact computation of Poisson binomial probabilities available in the statistics software literature, and treated the stationary-dependence case (Serinaldi and Lombardo 2020), whose paper explicitly leaves the joint case — exceedance indicators that are simultaneously non-identically distributed (time-varying annual exceedance probabilities) and serially dependent (clustered or persistent) — without tractable distributions. The open problem is this surviving core: a validated probabilistic theory of the timing and frequency of recurrent exceedances under nonstationarity together with serial dependence, propagation of the uncertainty of the exceedance-probability trajectory into the resulting risk metrics, and a coherent, theory-grounded choice among the proliferating nonstationary design metrics.

Background

Hydraulic structures (dams, levees, storm drainage) are designed against hydrologic extremes using frequency analysis of annual maximum series (AMS), the largest flood or rainfall in each year. Under stationarity these annual maxima are treated as independent draws from a common distribution, usually the generalized extreme value (GEV) distribution, the asymptotic family for block maxima whose location, scale, and shape parameters determine mean level, spread, and tail heaviness. A design quantile $z_q$ is assigned a return period $T=1/p$ with $p=1-q$ the annual exceedance probability; the waiting time to the first exceedance then follows a geometric law with mean $T$, the number of exceedances in an $n$-year design life follows a binomial distribution, and the risk of at least one exceedance is $1-(1-p)^n$. Urbanization, land-use change, and climate change break the constant-$p$ assumption. A nonstationary GEV with covariate-dependent parameters still yields an annual exceedance probability of a fixed design quantile, but it varies from year to year, written $p_j$ for year $j$. For independent years with time-varying $p_j$, the number of exceedances $Y$ over the design life follows the Poisson binomial distribution (the law of a sum of independent, not identically distributed Bernoulli trials), whose exact probability mass function is a sum over subsets of years, and the waiting time to the first exceedance follows a nonstationary geometric law. Obeysekera and Salas (2016) assembled these pieces into a frequency framework for recurrent extremes — the chance that a threshold is exceeded several times, not just once — including a return-period analogue defined through the expected waiting time (EWT) and a design criterion based on the expected number of exceedances (ENE), which fixes a tolerable mean count $m$ over the design life. Related metrics in the same literature are the design life level (DLL) of Rootzén and Katz (2013), a quantile whose exceedance probability over a fixed design life equals a target, and equivalent reliability (ER), which requires the nonstationary design to reproduce the survival probability a stationary design would have had. The 2016 paper closed with five unresolved directions: k-th-exceedance order statistics and negative-binomial analogues under nonstationarity; combined propagation of parameter, model-structure, and scenario uncertainty; generalization to seasonal or monthly extremes, to peaks-over-threshold (POT) sampling — which models all exceedances above a threshold rather than only annual maxima — and to serially dependent (clustered) years; efficient exact evaluation of Poisson binomial probabilities; and consolidation of the alternative nonstationary design metrics into a coherent framework. Later work materially changed this list. Salas and Obeysekera (2019) derived, for independent years, the nonstationary negative-binomial distribution of the waiting time $X_k$ to the first occurrence of the k-th exceedance — for constant $p$ the classical Pascal (negative binomial) waiting-time law — together with recursive expectation formulas, Monte-Carlo validation, and a nonstationary design application. Serinaldi and Lombardo (2020) derived the waiting-time distribution of the k-th extreme event under serial dependence for constant $p$, modeling clustered exceedance counts with beta-binomial and beta-negative-binomial structures (families for exchangeable, positively correlated Bernoulli trials), and stated that the joint case — nonidentically distributed and serially dependent exceedances — still lacks tractable distributions. Efficient exact Poisson-binomial computation is available in the statistics literature and maintained software. What remains is the joint theory, its uncertainty treatment, and the metric consolidation that such a theory would make possible.

Problem Statement

For annual (and, as generalizations, seasonal or peaks-over-threshold) hydrologic extremes whose annual exceedance probabilities $p_j$ of a fixed design threshold vary in time according to a fitted nonstationary extreme-value model and whose year-to-year exceedance indicators may be serially dependent (clustered or persistent), develop a probabilistic theory of recurrent exceedances, with explicitly stated distributions, that must satisfy all three of the following requirements: (i) it gives the joint distributions of the waiting time $X_k$ to the first occurrence of the k-th exceedance and of the number of exceedances $Y$ over an n-year design life, generalizing both the independent-year nonstationary negative-binomial theory of Salas and Obeysekera (2019) and the constant-probability beta-negative-binomial theory of Serinaldi and Lombardo (2020) to the combined case of nonstationarity and serial dependence, for an explicitly stated parametric class of serial-dependence structures whose members reduce to the Salas–Obeysekera negative-binomial law as the dependence vanishes and reduce to the Serinaldi–Lombardo beta-negative-binomial law when the annual exceedance probabilities $p_j$ become constant; (ii) it defines from these distributions return-period, reliability, and risk metrics and design criteria for recurrent exceedances, extending the expected-waiting-time (EWT) and expected-number-of-exceedances (ENE) concepts introduced by Obeysekera and Salas (2016) to the combined nonstationary-dependent case, and states the relations among the alternative nonstationary design metrics (EWT, ENE, DLL, ER) that hold under the combined nonstationary-dependent case; and (iii) it propagates the uncertainty of the exceedance-probability trajectory — parameter-estimation uncertainty, model-structure (covariate-form) uncertainty, and scenario uncertainty — into these recurrent-exceedance metrics.

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