Journal Article

The distribution of empirical periodograms: Lomb-Scargle and PDM spectra

A. Schwarzenberg-Czerny

in Monthly Notices of the Royal Astronomical Society

Published on behalf of The Royal Astronomical Society

Volume 301, issue 3, pages 831-840
Published in print December 1998 | ISSN: 0035-8711
Published online December 1998 | e-ISSN: 1365-2966 | DOI: http://dx.doi.org/10.1111/j.1365-8711.1998.02086.x
The distribution of empirical periodograms: Lomb-Scargle and PDM spectra

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Abstract

The theoretical probability distributions of periodograms are derived for the assumed variance of noise. In practice, however, the variance is estimated from data and hence it is a random variable itself. The empirical periodograms, i.e. the periodograms normalized using the estimated variance, therefore follow a distribution different from that predicted by theory. We demonstrate that in general many empirical periodograms follow the beta distribution. In particular, as an example we consider a Lomb &38; Scargle (L-S) modified power spectrum with an exponential theoretical distribution. We derive its easy-to-use analytical empirical distribution. We demonstrate that the difference between the tails of the empirical and theoretical distributions is large enough to have a profound effect on the statistical significance of signal detections. The difference persists despite generally good asymptotic convergence of the distributions near their centres. Hence we argue that even for well-behaved statistics (e.g. L-S) one has to use our new empirical beta distributions rather than the theoretical ones. Our conclusions are illustrated by a realistic example. In the example we demonstrate a significant difference between the theoretical and empirical distributions. Additionally, we provide an example of conversion between analysis of variance (AOV), power-spectrum, PDM and χ2 periodograms.

Keywords: methods: data analysis; methods: statistical; binaries: eclipsing; stars: oscillations; pulsars: general; X-rays: stars

Journal Article.  0 words. 

Subjects: Astronomy and Astrophysics

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