Estimating peaks of stationary random processes: a peaks-over-threshold approach

Estimating properties of the distribution of the peak of a stationary process from a single finite realization is a problem that arises in a variety of science and engineering applications. Further, it is often the case that the realization is of length T while the distribution of the peak is sought for a different length of time, T ₁ > T. Current methods for estimating peaks of time series have drawbacks that motivated the development of a new procedure, based on the peaks-over-threshold method, an advantage of which is that it often results in an increased size of the relevant extreme value data set compared with epochal procedures. 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 epochal procedure for estimating peaks combined with Best Linear Unbiased Estimates (BLUE) of the Gumbel parameters was found to depend, in some cases very significantly, upon the number of partitions being used. The proposed procedure is based on a Poisson process model for the thresholded pressure coefficient y, with threshold u. The estimated peak depends upon the choice of the threshold. However, unlike for the choice of the number n of partitions for the epochal procedure, a criterion is available that allows the analyst to make an optimal choice (according to a chosen metric) of the threshold value. Two versions of the proposed new procedure have been developed. One version, denoted by FpotMax, includes estimation of a tail length parameter with a similar interpretation to the generalized extreme value distribution tail length parameter. The second version, denoted by GpotMax, assumes that the tail length parameter vanishes, resulting in a tail of the Gumbel distribution type.