A semi-parametric approach for comparing the estimated spectra of two stationary point processes

In this paper, we provide a semi-parametric test for the hypothesis that the spectra of two stationary point processes (SPPs) are the same. The estimates of the second-order spectral density functions of the SPPs are obtained by using two different approaches: (a) by smoothing the modified periodogram statistics using a moving average weighting scheme, (b) by employing the Welch's method on the modified periodogram statistics. The test is based on the likelihood ratio function under two alternative hypotheses. The first alternative hypothesis suggests that the ratio of the spectra is a constant (there is a shift to the power) while the second one adopts a quadratic model for the logarithmic ratio of the spectra. A comparison with a log-linear model indicates coincidence in the results. This is explained by using an illustrative example from the field of neurophysiology. It is shown that the information transferred to the spinal cord by the sensory axons, closely related with the complex physiological system called muscle spindle, under the influence of two different stimuli can be separated in two parts. The first part corresponds to the range of frequencies 0-19 Hz while the second to the range of frequencies 19-100 Hz. However, the dependence on the frequency in both parts is of a quadratic form.