New Approximation to Distribution of Positive RVs Applied to Gaussian Quadratic Forms (original) (raw)
Related papers
2015
• This paper provides an accessible methodology for approximating the distribution of a general linear combination of non-central chi-square random variables. Attention is focused on the main application of the results, namely the distribution of positive definite and indefinite quadratic forms in normal random variables. After explaining that the moments of a quadratic form can be determined from its cumulants by means of a recursive formula, we propose a moment-based approximation of the density function of a positive definite quadratic form, which consists of a gamma density function that is adjusted by a linear combination of Laguerre polynomials or, equivalently, by a single polynomial. On expressing an indefinite quadratic form as the difference of two positive definite quadratic forms, explicit representations of approximations to its density and distribution functions are obtained in terms of confluent hypergeometric functions. The proposed closed form expressions converge rapidly and provide accurate approximations over the entire support of the distribution. Additionally, bounds are derived for the integrated squared and absolute truncation errors. An easily implementable algorithm is provided and several illustrative numerical examples are presented. In particular, the methodology is applied to the Durbin-Watson statistic. Finally, relevant computational considerations are discussed. Linear combinations of chi-square random variables and quadratic forms in normal variables being ubiquitous in statistics, the distribution approximation technique introduced herewith should prove widely applicable.
The distribution of positive definite quadratic forms in normal random vectors is first approximated by generalized gamma and Pearson-type density functions. The distribution of indefinite quadratic forms is then obtained from their representation in terms of the difference of two positive definite quadratic forms. In the case of the Pearson-type approximant, explicit representations are obtained for the density and distribution functions of an indefinite quadratic form. A moment-based technique whereby the initial approximations are adjusted by means of polynomials is being introduced. A detailed algorithm describing the steps involved in the methodology advocated herein is provided as well. It is also explained that the distributional results apply to the ratios of certain quadratic forms. Two numerical examples are presented: the first involves an indefinite quadratic form while the second approximates the distribution of the Durbin-Watson statistic, which is shown to be expressible as a ratio of quadratic forms.
Asymptotic expansions for the distribution of quadratic forms in normal variables
Annals of the Institute of Statistical Mathematics, 1988
Higher order asymptotic expansions for the distribution of quadratic forms in normal variables are obtained. The Cornish-Fisher inverse expansions for the percentiles of the distribution are also given. The tesulting formula for a definite quadratic form guarantees accuracy almost up to fourth decimal place if the distribution is not very skew. The normalizing transformation investigated by Jensen and Solomon (1972, J. Amer. Statist. Assoc., 67, 898-902) is reconsidered based on the rate of convergence to the normal distribution.
Title Two Simple Approximations to the Distributions of Quadratic Forms Permalink
Many test statistics are asymptotically equivalent to quadratic forms of normal variables, which are further equivalent to T = ∑d i=1 λiz 2 i with zi being independent and following N(0, 1). Two approximations to the distribution of T have been implemented in popular software and are widely used in evaluating various models. It is important to know how accurate these approximations are when compared to each other and to the exact distribution of T . The paper systematically studies the quality of the two approximations and examines the effect of λi’s and the degrees of freedom d by analysis and Monte Carlo. The results imply that one approximation can be as good as the exact distribution when d is large. When the coefficient of variation of the λi’s is small, another approximation is also adequate for practical model inference. The results are applied to a study of alcoholism and psychological symptoms.
Approximating the Distribution of Quadratic Forms Using Orthogonal Polynomials
1999
In this paper we propose some techniques to approximate the distribution of quadratic forms in normal random variables. The first two techniques will be based on orthogonal polynomials, namely; the Hermitte and Laguerre polynomials. The third technique will be based on a mixture of normal and truncated gamma distributions. The performance of these approximations will be studied and discussed for various quadratic forms.
IEEE Transactions on Vehicular Technology, 2019
This paper proposes a novel approach to the statistical characterization of non-central complex Gaussian quadratic forms (CGQFs). Its key strategy is the generation of an auxiliary random variable (RV) that replaces the original CGQF and converges in distribution to it. The technique is valid for both definite and indefinite CGQFs and yields simple expressions of the probability density function (PDF) and the cumulative distribution function (CDF) that only involve elementary functions. This overcomes a major limitation of previous approaches, where the complexity of the resulting PDF and CDF does
Two Simple Approximations to the Distributions of Quadratic Forms - eScholarship
2007
Many test statistics are asymptotically equivalent to quadratic forms of normal variables, which are further equivalent to T = d i=1 λ i z 2 i with z i being independent and following N (0, 1). Two approximations to the distribution of T have been implemented in popular software and are widely used in evaluating various models. It is important to know how accurate these approximations are when compared to each other and to the exact distribution of T. The paper systematically studies the quality of the two approximations and examines the effect of λ i 's and the degrees of freedom d by analysis and Monte Carlo. The results imply that one approximation can be as good as the exact distribution when d is large. When the coefficient of variation of the λ i 's is small, another approximation is also adequate for practical model inference. The results are applied to a study of alcoholism and psychological symptoms.