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We take a wavelet based approach to the analysis of point processes and theestimation of the first order intensity under a continuous time setting. Amultiresolution analysis of a point process is formulated which motivates thedefinition of homogeneity at different scales of resolution, termed $J$-thlevel homogeneity. Further to this, the activity in a point processes' firstorder behavior at different scales of resolution is also defined and termed$L$-th level innovation. Likelihood ratio tests for both these properties areproposed with asymptotic distributions provided, even when only a singlerealization of the point process is observed. The test for $L$-th levelinnovation forms the basis for a collection of statistical strategies forthresholding coefficients in a wavelet based estimator of the intensityfunction. These thresholding strategies are shown to outperform the existinglocal hard thresholding strategy on a range of simulation scenarios.

Multiresolution analysis of point processes and statistical thresholding for wavelet-based intensity estimation