On the estimation of poles in intensity functions

Research Report SRR97-006

On the estimation of poles in intensity functions

Edwin Choi and Peter Hall

Abstract: Motivated by spatial data on earthquakes, we consider the problem of estimating properties of poles in point-process intensity (and probability density) functions. Our methods are semiparametric, requiring only "asymptotic" models for the intensity. They produce estimates of the location and strength of poles. Strength is expressed in terms of an exponent of regular variation, and is simply related to the correlation dimension of the underlying point process. It is argued that existing methods for estimating pole strength are restrictive in terms of the range of strengths that they allow. They are statistically consistent only in that half of the range which corresponds to relatively strong poles, and perform poorly for poles whose strengths lie in the middle of the range. Application of our alternative methods to real data sets suggests that in practice, pole strengths often lie the middle range. Our methods may also be used to estimate pole lines, and to track the migration of pole location with time.

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