Statistics of the Chi-Square Type, with Application to the Analysis of Multiple Time-Series Power Spectra

The interpretation of Fourier spectra in the time domain is critically examined. Power density spectra defined and calculated in the time domain are compared with Fourier spectra in the frequency domain for three different types of variability: periodic signals, Markov processes and random shots. The power density spectra for a sample of neutron stars and black hole binaries are analyzed in both the time and the frequency domains. For broadband noise, the two kinds of power s...

The least-squares (or Lomb-Scargle) periodogram is a powerful tool which is used routinely in many branches of astronomy to search for periodicities in observational data. The problem of assessing statistical significance of candidate periodicities for different periodograms is considered. Based on results in extreme value theory, improved analytic estimations of false alarm probabilities are given. They include an upper limit to the false alarm probability (or a lower limit ...

Power-law type distributions are extensively found when studying the behaviour of many complex systems. However, due to limitations in data acquisition, empirical datasets often only cover a narrow range of observation, making it difficult to establish power-law behaviour unambiguously. In this work we present a statistical procedure to merge different datasets with the aim of obtaining a broader fitting range for the statistics of different experiments or observations of the...

We apply Fourier analysis to 214 light curves of long gamma-ray bursts and study the statistical properties of their power density spectra (PDSs). The averaged PDS is found to follow a power law of index -5/3 over almost 2 decades of frequency with a break at about 2 Hz. Individual PDSs are exponentially distributed around the power law. It provides evidence that the diversity of the bursts is due to random realizations of the same process which is self-similar over the full ...

Period searches in event data have traditionally used the Rayleigh statistic, $R^2$. For X-ray pulsars, the standard has been the $Z^2$ statistic, which sums over more than one harmonic. For $\gamma$-rays, the $H$-test, which optimizes the number of harmonics to sum, is often used. These periodograms all suffer from the same problem, namely artefacts caused by correlations in the Fourier components that arise from testing frequencies with a non-integer number of cycles. This ...

This paper demonstrates that basic statistics (mean, variance) of the logarithm of the variate itself can be used in the calculation of differential entropy among random variables known to be multiples and powers of a common underlying variate. For the same set of distributions, the variance of the differential self-information is shown also to be a function of statistics of the logarithmic variate. Then entropy and its "variance" can be estimated using only statistics of the...

We develop an approach to spectral estimation that has been advocated by Ferrante, Masiero and Pavon and, in the context of the scalar-valued covariance extension problem, by Enqvist and Karlsson. The aim is to determine the power spectrum that is consistent with given moments and minimizes the relative entropy between the probability law of the underlying Gaussian stochastic process to that of a prior. The approach is analogous to the framework of earlier work by Byrnes, Geo...

This paper and its companion form an extended version of notes provided to participants in the Valencia September 2004 summer school on Data Analysis in Cosmology. The papers offer a pedagogical introduction to the problem of estimating the power spectrum from galaxy surveys. The intention is to focus on concepts rather than on technical detail, but enough mathematics is provided to point the student in the right direction. This first paper presents background material. It ...

A finite-energy signal is represented by a square-integrable, complex-valued function $t\mapsto s(t)$ of a real variable $t$, interpreted as time. Similarly, a noisy signal is represented by a random process. Time-frequency analysis, a subfield of signal processing, amounts to describing the temporal evolution of the frequency content of a signal. Loosely speaking, if $s$ is the audio recording of a musical piece, time-frequency analysis somehow consists in writing the musica...

Unbinned maximum likelihood is a common procedure for parameter estimation. After parameters have been estimated, it is crucial to know whether the fit model adequately describes the experimental data. Univariate Goodness of Fit procedures have been thoroughly analyzed. In multi-dimensions, Goodness of Fit test powers have rarely been studied on realistic problems. There is no definitive answer to regarding which method is better. Test performance is strictly related to speci...