On Approximating the Distributions of Ratios and Differences of Noncentral Quadratic Forms in Normal Vectors (original) (raw)
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Approximating the Distributions of Singular Quadratic Expressions and their Ratios
Noncentral indefinite quadratic expressions in possibly non-singular normal vectors are represented in terms of the difference of two positive definite quadratic forms and an independently distributed linear combination of standard normal random variables. This result also ap-plies to quadratic forms in singular normal vectors for which no general representation is currently available. The distribution of the positive definite quadratic forms involved in the representations is approximated by means of gamma-type distributions. We are also considering general ratios of quadratic forms, as well as ratios whose denominator involves an idempotent matrix and ratios for which the quadratic form in the denominator is positive definite. Additionally, an approximation to the density of ratios of quadratic expressions in singular normal vectors is being proposed. The results are applied to the Durbin-Watson statistic and Burg's estimator, both of which are expressible as ratios of quadrat...
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.
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• 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.
New Approximation to Distribution of Positive RVs Applied to Gaussian Quadratic Forms
IEEE Signal Processing Letters, 2019
This letter introduces a new approach to the problem of approximating the probability density function (PDF) and the cumulative distribution function (CDF) of a positive random variable. The novel approximation strategy is based on the analysis of a suitably defined sequence of auxiliary variables which converges in distribution to the target variable. By leveraging such convergence, simple approximations for both the CDF and PDF of the target variable are given in terms of the derivatives of its moment generating function (MGF). In contrast to classical approximation methods based on truncated series of moments or cumulants, our approximations always represent a valid distribution and the relative error between variables is independent of the variable under analysis. The derived results are then used to approximate the statistics of positive-definite real Gaussian quadratic forms, comparing our proposed approach with other existing approximations in the literature.
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.
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.
2007
From a suitable integral representation of the Laplace transform of a positive semi-definite quadratic form of independent real random variables with not necessarily identical densities a univariate integral representation is derived for the cumulative distribution function of the sample variance of i.i.d. random variables with a gamma density, supplementing former formulas of the author. Furthermore, from the above Laplace transform Fourier series are obtained for the density and the distribution function of the sample variance of i.i.d. random variables with a uniform distribution. This distribution can be applied e.g. to a statistical test for a scale parameter.