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

In this paper, four new Chi-Square type statistics are presented for testing the hypothesis of a uniform null versus specified trend alternatives. The powers of these test statistics are compared with the powers of the statistics considered by Steele and Chaseling [8]. The four test statistics are shown to have superior or equivalent powers to the powers of the test statistics considered by the authors for certain trend alternatives and for certain conditions placed on the ce...

Since its launch in 2008 the Fermi Large Area Telescope provides regular monitoring of a large sample of gamma-ray sources on time scales from hours to years. Together with observations at other wavelengths it is now possible to study variability and correlation properties in a much more systematic and detailed way than ever before. The paper describe some of the statistical methods and tools that have been, or can be, used to characterize variability and to study the relatio...

Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution -- the part of the distribution representing large but rare events -- and by the difficulty of identifying the range over which power-law behavior holds. Commonly used m...

The power density spectrum of a light curve is often calculated as the average of a number of spectra derived on individual time intervals the light curve is divided into. This procedure implicitly assumes that each time interval is a different sample function of the same stochastic ergodic process. While this assumption can be applied to many astrophysical sources, there remains a class of transient, highly nonstationary and short-lived events, such as gamma-ray bursts, for ...

Spectral estimation (SE) aims to identify how the energy of a signal (e.g., a time series) is distributed across different frequencies. This can become particularly challenging when only partial and noisy observations of the signal are available, where current methods fail to handle uncertainty appropriately. In this context, we propose a joint probabilistic model for signals, observations and spectra, where SE is addressed as an exact inference problem. Assuming a Gaussian p...

We discuss methods currently in use for determining the significance of peaks in the periodograms of time series. We discuss some general methods for constructing significance tests, false alarm probability functions, and the role played in these by independent random variables and by empirical and theoretical cumulative distribution functions. We also discuss the concept of "independent frequencies" in periodogram analysis. We propose a practical method for estimating the si...

The power-spectrum subband energy ratio (PSER) has been applied in a variety of fields, but reports on its statistical properties have been limited. As such, this study investigates these characteristics in the presence of additive Gaussian white noise for both pure noise and mixed signals. By analyzing the probability and independence of power spectrum bins, and the relationship between the F and beta distributions, we develop a probability distribution for the PSER. Results...

Histograms with weighted entries are used to estimate probability density functions. Computer simulation is the main application of this type of histograms. A review on chi-square tests for comparing weighted histograms is presented in this paper. Improvements to these tests that have a size closer to its nominal value are proposed. Numerical examples are presented for evaluation and demonstration of various applications of the tests.

Here, we present a new method to evaluate the expectation value of the power spectrum of a time series. A statistical approach is adopted to define the method. After its demonstration, it is validated showing that it leads to the known properties of the power spectrum when the time series contains a periodic signal. The approach is also validated in general with numerical simulations. The method puts into evidence the importance that is played by the probability density funct...

The analysis of time variability, whether fast variations on time scales well below the second or slow changes over years, is becoming more and more important in high-energy astronomy. Many sophisticated tools are available for data analysis and complex practical aspects are described in technical papers. Here, we present the basic concepts upon which all these techniques are based. It is intended as a condensed primer of Fourier analysis, dealing with fundamental aspects tha...