Inhomogeneous Poisson process 6321 Model

Consider the time points {Tout(j)}/= when an extreme has occurred. The points may have resulted from POT data or block extremes, or those event times may constitute the only information left about the occurrence of an extreme or outlier. Consider the number of extreme events to be described by a discrete random variable, M. Let the number of extremes at continuous time T be described by the random process M(T). Its realization consists of step functions with "unit jumps" at Tout(j). The process M(T) is called a point process (see Karr (1986) for a complete definition).

Consider the incremental process, dM(T) = M(T + ¿T) — M(T), which represents the number of events in the time interval [T; T + ¿T]. Let ¿T be arbitrarily small, so that not two or more events occur within the interval and dM(T) takes only two values, namely,

|1 with probability A ■ ¿T, dM (T) = < (6.28)

[0 with probability 1 — A ■ ¿T, where A > 0 is a constant. Then

Assume further that the events occur independently of each other,

The point process M(T) is then specified as a homogeneous Poisson process with occurrence rate parameter A.

The parameter of interest for the analysis of climate extremes is A. Its units are one over time units. It gives the probability per time interval that an extreme occurs. For studying nonstationarity and trends in climate risk, we now introduce time-dependence and denote the function A(T) as occurrence rate. The process is then denoted as inhomogeneous Poisson process (Cox and Lewis 1966). Nonparametric occurrence rate estimation

The kernel approach (Diggle 1985) estimates the occurrence rate as

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