peakoverthreshold

Peak over threshold (POT) is a statistical approach within extreme value theory for modeling extreme events by focusing on data that exceed a high threshold. It is used to estimate the tail behavior and to infer extreme quantiles and return levels.

The theoretical justification comes from the Pickands–Balkema–de Haan theorem, which states that, for a wide class of distributions, the distribution of exceedances over a high threshold converges to a Generalized Pareto Distribution (GPD) as the threshold increases. In practice, one models the exceedances Y = X − u given X > u with a GPD.

Let X be a random variable or a time series and u a high threshold. The exceedances Y = X − u conditional on X > u are approximately distributed as a GPD with shape parameter xi and scale parameter sigma. The survival function is P(Y > y | X > u) = (1 + xi y / sigma)^(-1/xi) for y ≥ 0 (xi ≠ 0), while xi = 0 yields the exponential form P(Y > y | X > u) = exp(−y/sigma). The parameters xi and sigma, along with the threshold u, determine the tail behavior and the range of extrapolation.

Estimation typically proceeds by maximum likelihood or probability-weighted moments, applied to the exceedances. Practical issues include selecting u, ensuring approximate independence of exceedances (sometimes via declustering), and diagnostics such as mean residual life plots and threshold stability plots.

POT is widely used in hydrology, meteorology, finance, and environmental risk assessment. It is complementary to the block maxima method and can yield more efficient tail estimates by utilizing all exceedances above the threshold. Related topics include Generalized Pareto distribution, extreme value theory, the block maxima method, and declustering.

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