Estimating Peaks of Stationary Random Processes: A Peaks-over-Threshold Approach

Estimating properties of the distribution of the peak of a stochastic process from a single finite realization is a problem that arises in a variety of science and engineering applications. Furthermore, it is often the case that the realization is of length $T$ whereas the distribution of the peak is sought for a different length of time, $T_1>T$. The procedure proposed here is based on a peaks-over-threshold extreme value model, which has an advantage over classical models used in epochal procedures because it often results in an increased size of the relevant extreme value data set. For further comparison, the translation approach depends upon the estimate of the marginal distribution of a non-Gaussian time series, which is typically difficult to perform reliably. The proposed procedure is based on a two-dimensional Poisson process model for the pressure coefficients $y$, above the threshold $B$. The estimated distribution of the peak value depends upon the choice of the threshold. The threshold choice is automated by selecting the threshold that minimizes a metric that captures the trade-off between bias and variance in estimation. Two versions of the proposed new procedure are developed. One version, denoted FpotMax, includes estimation of a tail length parameter with a similar interpretation of the generalized extreme value distribution tail length parameter. The second version, denoted GpotMax, assumes that the tail length parameter vanishes.