For a sample of independent and identically distributed observations, the parameters of the Generalized Pareto Distribution (GPD) can be estimated by the Maximum Likelihood (ML) method. In this paper, we drop the assumption of identically distributed random variables. We consider independent observations from GPD distributions having a common shape parameter but possibly an increasing trend in the scale parameter. Such a model, with increasing scale parameter, can be used to describe a trend in the observed extremes as time progresses. Estimating an increasing trend in a distribution parameter is common in the field of isotonic regression. We use ideas and tools from that area to compute ML estimates of the GPD
parameters. We also study these estimates in a simulation experiment. Moreover, we apply the approach to the Central England Temperature (CET) data.
M Roth, G Jongbloed, TA Buishand. Monotone trends in the GPD scale parameter
submitted, Journal of Applied Statistics, 2016