False-alarm probability in relation to oversampled power spectra, with applicatio to Super-Kamiokande solar neutrino data

Abstract

The term "false-alarm probability" denotes the probability that at least one out of M independent power values in a prescribed search band of a power spectrum computed from a white-noise time series is expected to be as large as or larger than a given value. The usual formula is based on the assumption that powers are distributed exponentially, as one expects for power measurements of normally distributed random noise. However, in practice, one typically examines peaks in an oversampled power spectrum. It is therefore more appropriate to compare the strength of a particular peak with the distribution of peaks in oversampled power spectra derived from normally distributed random noise. We show that this leads to a formula for the false-alarm probability that is rather more conservative than the familiar formula. We also show how to combine these results with a Bayesian method for estimating the probability of the null hypothesis (that there is no oscillation in the time series), and we discuss as an example the application of these procedures to Super-Kamiokande solar neutrino data.