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

A new, computationally- and statistically-efficient algorithm, the Fast $\chi^2$ algorithm, can find a periodic signal with harmonic content in irregularly-sampled data with non-uniform errors. The algorithm calculates the minimized $\chi^2$ as a function of frequency at the desired number of harmonics, using Fast Fourier Transforms to provide $O (N \log N)$ performance. The code for a reference implementation is provided.

The purpose of this article is to carry out a power-spectrum analysis (based on likelihood methods) of the Super-Kamiokande 5-day dataset that takes account of the asymmetry in the error estimates. Whereas the likelihood analysis involves a linear optimization procedure for symmetrical error estimates, it involves a nonlinear optimization procedure for asymmetrical error estimates. We find that for most frequencies there is little difference between the power spectra derive...

This paper presents new families of Rao-type test statistics based on the minimum density power divergence estimators which provide robust generalizations for testing simple and composite null hypotheses. The asymptotic null distributions of the proposed tests are obtained and their robustness properties are also theoretically studied. Numerical illustrations are provided to substantiate the theory developed. On the whole, the proposed tests are seen to be excellent alternati...

We present the statistical approach to the combining of signal significances.

The statistical properties of the multivariate Gamma-Gamma ($\Gamma \Gamma$) distribution with arbitrary correlation have remained unknown. In this paper, we provide analytical expressions for the joint probability density function (PDF), cumulative distribution function (CDF) and moment generation function of the multivariate $\Gamma \Gamma$ distribution with arbitrary correlation. Furthermore, we present novel approximating expressions for the PDF and CDF of the sum of $\Ga...

Exponential distributions appear in a wide range of applications including chemistry, nuclear physics, time series analyses, and stock market trends. There are conceivable circumstances in which one would be interested in the cumulative probability distribution of the sum of some number of exponential variables, with potentially differing constants in their exponents. In this article we present a pedagogical derivation of the cumulative distribution, which reproduces the know...

Spectral analysis in conjunction with discrete data in one and more dimensions can become a challenging task, because the methods are sometimes difficult to understand. This paper intends to provide an overview about the usage of the Fourier transform, its related methods and focuses on the subtleties to which the users must pay attention. Typical questions, which are often addressed to the data, will be discussed. Such a problem can be the issue of frequency or band limitati...

Weighted histograms are used for the estimation of probability density functions. Computer simulation is the main domain of application of this type of histogram. A review of chi-square goodness of fit tests for weighted histograms is presented in this paper. Improvements are proposed to these tests that have size more close to its nominal value. Numerical examples are presented in this paper for evaluation of tests and to demonstrate various applications of tests.

This paper introduces a comprehensive framework to adjust a discrete test statistic for improving its hypothesis testing procedure. The adjustment minimizes the Wasserstein distance to a null-approximating continuous distribution, tackling some fundamental challenges inherent in combining statistical significances derived from discrete distributions. The related theory justifies Lancaster's mid-p and mean-value chi-squared statistics for Fisher's combination as special cases....

Analysing data that consists of one or more truncated exponential functions is of great interest in a wide range of fields. Data consisting of one or more exponential functions are measured by a wide range of instruments, examples include: nuclear magnetic resonance, (NMR); magnetic resonance imaging (MRI); the analysis of radioactive decay data; chromatography; lifetime fluorescence imaging; spin resonance, (ESR). Many of the algorithms that are currently in used for 'charac...