Estimating the rate (first-order intensity) of a point process is a task of great interest in the understanding of its nature. In this work we first address the estimation of the rate of an orderly point process on the real line using a multiresolution wavelet expansion approach. Implementing Haar wavelets, we find that in the case of a Poisson process the piecewise constant wavelet estimator of the rate has a scaled Poisson distribution. We apply this result in the design of a likelihood ratio test for a multiresolution formulation of the homogeneity of a Poisson process. We demonstrate this method with simulations and provide Type 1 error and empirical power plots under specific models.