Return level plot

A return level plot graphs $T$ horizontally and $z_T$ vertically, often using a logarithmic scale for $T$. To read it, choose a return period, move up to the curve, and read the corresponding return level. Conversely, a threshold's horizontal coordinate gives the reciprocal of its annual exceedance probability. The usual plot of fitted quantiles against return periods, and optional pointwise confidence intervals, are documented in Mathieu Ribatet, POT package: return level plot.

The near-straight shape has a mathematical explanation: since $-\log (1-1/T)\sim 1/T$ as $T\to \infty$, the Gumbel distribution gives $z_T=\mu +b\log T+o(1)$. Multiplying a large return period by ten therefore increases the return level by approximately $b\log 10$, about $4.605$ units for this toy curve. A curve indexed by rarity puts several thresholds on one common scale while retaining their measurement units.

For a fitted curve, a pointwise confidence interval concerns uncertainty in the estimated quantile at a specified $T$; it is distinct from a prediction interval for a future annual sample maximum, and pointwise intervals need not form a simultaneous confidence band. A far-right curve value can also be an extrapolation beyond the observation record: extending the line adds no observations to support the chosen tail model. The diagram above uses fixed toy parameters, so it shows neither estimation uncertainty nor an observational record. Its interpretation throughout assumes the stated annual block and stationarity.