Bayesian prediction in doubly stochastic Poisson process
A stochastic marked point process model based on doubly stochastic Poisson process is considered in the problem of prediction for the total size of future marks in a given period, given the history of the process. The underlying marked point process <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$(T_{i},Y_{i})_{i\ge 1}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>i</mi> <mo>≥</mo> <mn>1</mn> </mrow> </msub> </math> </EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$T_{i}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>T</mi> <mi>i</mi> </msub> </math> </EquationSource> </InlineEquation> is the time of occurrence of the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$i$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>i</mi> </math> </EquationSource> </InlineEquation>th event and the mark <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$Y_{i}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>Y</mi> <mi>i</mi> </msub> </math> </EquationSource> </InlineEquation> is its characteristic (size), is supposed to be a non-homogeneous Poisson process on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$\mathbb {R}_{+}^{2}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msubsup> <mi mathvariant="double-struck">R</mi> <mrow> <mo>+</mo> </mrow> <mn>2</mn> </msubsup> </math> </EquationSource> </InlineEquation> with intensity measure <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$P\times \varTheta $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>P</mi> <mo>×</mo> <mi mathvariant="italic">Θ</mi> </mrow> </math> </EquationSource> </InlineEquation>, where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$P$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>P</mi> </math> </EquationSource> </InlineEquation> is known, whereas <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$\varTheta $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">Θ</mi> </math> </EquationSource> </InlineEquation> is treated as an unknown measure of the total size of future marks in a given period. In the problem of prediction considered, a Bayesian approach is used assuming that <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$\varTheta $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">Θ</mi> </math> </EquationSource> </InlineEquation> is random with prior distribution presented by a gamma process. The best predictor with respect to this prior distribution is constructed under a precautionary loss function. A simulation study for comparing the behavior of the predictors under various criteria is provided. Copyright The Author(s) 2014