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

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.