arXiv subjects

Narayanaswamy Balakrishnan

Publications and source records attributed to Narayanaswamy Balakrishnan.

41 records · Page 3Linked to original sources

arXiv2018

Orthogonal polynomials in the Cumulative Ord family and its application to variance bounds

This article presents and reviews several basic properties of the Cumulative Ord family of distributions; this family contains all the commonly used discrete distributions. A complete classification of the Ord family of probability mass functions is related to the orthogonality of the corresponding Rodrigues polynomials. Also, for any random variable $X$ of this family and for any suitable function $g$ in $L^2(\mathbb{R},X)$, the article provides useful relationships between the Fourier coefficients of $g$ (with respect to the orthonormal polynomial system associated to $X$) and the Fourier coefficients of the forward difference of $g$ (with respect to another system of polynomials, orthonormal with respect to another distribution of the system). Finally, using these properties, a class of bounds for the variance of $g(X)$ is obtained, in terms of the forward differences of $g$. These bounds unify and improve several existing results.

math.PR↗

arXiv2017

An Algorithm for Computing the Distribution Function of the Generalized Poisson-Binomial Distribution

The Poisson-binomial distribution is useful in many applied problems in engineering, actuarial science, and data mining. The Poisson-binomial distribution models the distribution of the sum of independent but not identically distributed Bernoulli random variables whose success probabilities vary. In this paper, we extend the Poisson-binomial distribution to the generalized Poisson-binomial (GPB) distribution. The GPB distribution is defined in cases where the Bernoulli variables can take any two arbitrary values instead of 0 and~1. The GPB distribution is useful in many areas such as voting theory, actuarial science, warranty prediction, and probability theory. With few previous works studying the GPB distribution, we derive the probability distribution via the discrete Fourier transform of the characteristic function of the distribution. We develop an efficient algorithm for computing the distribution function, which uses the fast Fourier transform. We test the accuracy of the developed algorithm upon comparing it with enumeration-based exact method and the results from the binomial distribution. We also study the computational time of the algorithm in various parameter settings. Finally, we discus the factors affecting the computational efficiency of this algorithm, and illustrate the use of the software package.

stat.CO↗

arXiv2016

Empirical phi-divergence test-statistics for the equalityof means of two populations

Empirical phi-divergence test-statistics have demostrated to be a useful technique for the simple null hypothesis to improve the finite sample behaviour of the classical likelihood ratio test-statistic, as well asfor model misspecification problems, in both cases for the one population problem. This paper introduces this methodology for two sample problems. A simulation study illustrates situations in which the new test-statistics become a competitive tool with respect to the classical z-test and the likelihood ratio test-statistic.

stat.ME↗

arXiv2014

Empirical phi-divergence test statistics for testing simple and composite null hypotheses

The main purpose of this paper is to introduce first a new family of empirical test statistics for testing a simple null hypothesis when the vector of parameters of interest are defined through a specific set of unbiased estimating functions. This family of test statistics is based on a distance between two probability vectors, with the first probability vector obtained by maximizing the empirical likelihood on the vector of parameters, and the second vector defined from the fixed vector of parameters under the simple null hypothesis. The distance considered for this purpose is the phi-divergence measure. The asymptotic distribution is then derived for this family of test statistics. The proposed methodology is illustrated through the well-known data of Newcomb's measurements on the passage time for light. A simulation study is carried out to compare its performance with that of the empirical likelihood ratio test when confidence intervals are constructed based on the respective statistics for small sample sizes. The results suggest that the "empirical modified likelihood ratio test statistic" provides a competitive alternative to the empirical likelihood ratio test statistic, and is also more robust than the empirical likelihood ratio test statistic in the presence of contamination in the data. Finally, we propose empirical phi-divergence test statistics for testing a composite null hypothesis and present some asymptotic as well as simulation results for evaluating the performance of these test procedures.

math.ST↗

arXiv2014

On the method of pivoting the CDF for exact confidence intervals with illustration for exponential mean under life-test with time constraints

Two requirements for pivoting a cumulative distribution function (CDF) in order to construct exact confidence intervals or bounds for a real-valued parameter $θ$ are the monotonicity of this CDF with respect to $θ$ and the existence of solutions of some pertinent equations for $θ$. The second requirement is not fulfilled by the CDF of the maximum likelihood estimator of the exponential scale parameter when the data come from some life-testing scenarios such as type-I censoring, hybrid type-I censoring, and progressive type-I censoring that are subject to time constraints. However, the method has been used in these cases probably because the non-existence of the solution usually happens only with small probability. Here, we illustrate the problem by giving formal details in the case of type-I censoring and by providing some further examples. We also present a suitable extension of the basic pivoting method which is applicable in situations wherein the considered equations have no solution.

math.ST↗