Sum Of Chi-squared Random Variables Research Articles

Testing independence between discrete random variables

In this paper we develop a likelihood ratio test for independence between multivariate discrete random vectors whose components may be dependent or may have different marginal distributions. A special case that tests independence between two discrete random variables is also discussed. The model under the null hypothesis is constructed based on the marginal distributions of the random variables, while the model under the alternative is built by adding a common random effect which is not restricted to follow the same distribution as those of the other variables. The test statistic is shown to admit an asymptotic distribution of a weighted sum of chi-square random variables, each with one degree of freedom. In order to avoid the high computational cost of finding the parameters of the asymptotic distribution, a permutation-based procedure is introduced. It is shown in simulations that the permutation-based realization yields empirical level close to the nominal level, while achieving power comparable to or higher than that of existing competitors in the literature. The application of the test is illustrated with a real data set.

On the Null Distribution of Bayes Factors in Linear Regression

ABSTRACTWe show that under the null, the is asymptotically distributed as a weighted sum of chi-squared random variables with a shifted mean. This claim holds for Bayesian multi-linear regression with a family of conjugate priors, namely, the normal-inverse-gamma prior, the g-prior, and the normal prior. Our results have three immediate impacts. First, we can compute analytically a p-value associated with a Bayes factor without the need of permutation. We provide a software package that can evaluate the p-value associated with Bayes factor efficiently and accurately. Second, the null distribution is illuminating to some intrinsic properties of Bayes factor, namely, how Bayes factor quantitatively depends on prior and the genesis of Bartlett’s paradox. Third, enlightened by the null distribution of Bayes factor, we formulate a novel scaled Bayes factor that depends less on the prior and is immune to Bartlett’s paradox. When two tests have an identical p-value, the test with a larger power tends to have a larger scaled Bayes factor, a desirable property that is missing for the (unscaled) Bayes factor. Supplementary materials for this article are available online.

A comparison of efficient approximations for a weighted sum of chi-squared random variables

In many applications, the cumulative distribution function (cdf) $$F_{Q_N}$$ of a positively weighted sum of N i.i.d. chi-squared random variables $$Q_N$$ is required. Although there is no known closed-form solution for $$F_{Q_N}$$ , there are many good approximations. When computational efficiency is not an issue, Imhof’s method provides a good solution. However, when both the accuracy of the approximation and the speed of its computation are a concern, there is no clear preferred choice. Previous comparisons between approximate methods could be considered insufficient. Furthermore, in streaming data applications where the computation needs to be both sequential and efficient, only a few of the available methods may be suitable. Streaming data problems are becoming ubiquitous and provide the motivation for this paper. We develop a framework to enable a much more extensive comparison between approximate methods for computing the cdf of weighted sums of an arbitrary random variable. Utilising this framework, a new and comprehensive analysis of four efficient approximate methods for computing $$F_{Q_N}$$ is performed. This analysis procedure is much more thorough and statistically valid than previous approaches described in the literature. A surprising result of this analysis is that the accuracy of these approximate methods increases with N.

Computational aspects of Cui-Freeden statistics for equidistribution on the sphere

Abstract. In this paper, we derive the asymptotic statistical properties of a class of generalized discrepancies introduced by Cui and Freeden (SIAM J. Sci. Comput., 1997) to test equidistribution on the sphere. We show that they have highly desirable properties and encompass several statistics already proposed in the literature. In particular, it turns out that the limiting distribution is an (infinite) weighted sum of chi-squared random variables. Issues concerning the approximation of this distribution are considered in detail and explicit bounds for the approximation error are given. The statistics are then applied to assess the equidistribution of Hammersley low discrepancy sequences on the sphere and the uniformity of a dataset concerning magnetic orientations.

Coexistence of Dynamic Spectrum Access Based Heterogeneous Networks

This paper addresses the coexistence issue of distributed heterogeneous networks where the network nodes are cognitive radio terminals. These nodes, operating as secondary users (SUs), might interfere with primary users (PUs) who are licensed to use a given frequency band. Further, due to the lack of coordination and the dissimilarity of the radio access technologies (RATs) among these wireless nodes, they might interfere with each other. To solve this coexistence problem, we propose an architecture that enables coordination among the distributed nodes. The architecture provides coexistence solutions and sends reconfiguration commands to SU networks. As an example, time sharing is considered as a solution. Further, the time slot allocation ratios and transmit powers are parameters encapsulated in the reconfiguration commands. The performance of the proposed scheme is evaluated in terms of the coexistence between PUs and SUs, as well as the coexistence among SUs. The former addresses the interference from SUs to PUs, whereas the latter addresses the sharing of an identified spectrum opportunity among heterogeneous SU networks for achieving an efficient spectrum usage. In this study, we first introduce a new parameter named as quality of coexistence (QoC), which is defined as the ratio between the quality of SU transmissions and the negative interference to PUs. In this study we assume that the SUs have multiple antennas and employ fixed transmit power control (fixed-TPC). By using the approximation to the distribution of a weighted sum of chi-square random variables (RVs), we develop an analytical model for the time slot allocation among SU networks. Using this analytical model, we obtain the optimal time slot allocation ratios as well as transmit powers of the SU networks by maximizing the QoC. This leads to an efficient spectrum usage among SUs and a minimized negative influence to the PUs. Results show that in a particular scenario the QoC can be increased by 30%.

EDGEWORTH SERIES APPROXIMATION FOR CHI-SQUARE TYPE CHANCE CONSTRAINTS

We introduce two methods for approximation to distribution ofweighted sum of chi-square random variables. These methods can be more useful than the known methods in literature to transform chi-square type chanceconstrained programming (CCP) problem into deterministic problem. Therefore, these are compared with Sengupta (1970)’s method. Some examples areillustrated for the purpose of comparing the solutions of these methods

On the Asymptotic Distributions of Bandwidth Estimates

The problem of automatic bandwidth selection for a kernel regression estimate is studied. Since the bandwidth considerably affects the features of the estimated curve, it is important to understand the behavior of bandwidth selection procedures. The bandwidth estimate considered here is the minimizer of Mallows' criterion. Though it was established that the bandwidth estimate is asymptotically normal, it is well recognized that the rate of convergence is extremely slow. In simulation studies, it is often observed that the normal distribution does not provide a satisfactory approximation. In this paper, the bandwidth estimate is shown to be approximately equal to a constant plus a linear combination of independent exponential random variables. In practice, the distribution of a weighted sum of chi-squared random variables can be approximated by a multiple of a chi-squared distribution. Simulation results indicate that this provides a very good approximation even for a modest sample size. It is shown that the degrees of freedom of the chi-squared distribution goes to infinity rather slowly (at the rate $T^{1/5}$ for a nonnegative kernel). This explains why the distributions of the bandwidth estimates converge to the normal distribution so slowly.

A General formula for the upper tail significance levels of empirical distribution function test statistics

In this paper ve obtain an asymptotic expression for the upper tail area of the distribution of an infinite weighted sum of chi-square random variables and show how this can be applied to distributions of various goodness of fit test statistics. Results obtained by this general approach are comparable with those reported previously in the literature. In the case of the Cramer-von Mises statistic an empirical adjustment is given vhich significantly improves on previous approximations. For the Kuiper statistic the corresponding empirical adjustment leads to an existing highly accurate approximation.