Asymptotic expansions for the distribution of quadratic forms in normal variables (original) (raw)
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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.
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.
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.
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...
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.
Expectation of quadratic forms in normal and nonnormal variables with applications
Journal of Statistical Planning and Inference, 2010
We derive some new results on the expectation of quadratic forms in normal and nonnormal variables. Using a nonstochastic operator, we show that the expectation of the product of an arbitrary number of quadratic forms in noncentral normal variables follows a recurrence formula. This formula includes the existing result for central normal variables as a special case. For nonnormal variables, while the existing results are available only for quadratic forms of limited order (up to 3), we derive analytical results to a higher order 4. We use the nonnormal results to study the e¤ects of nonnormality on the …nite sample mean squared error of the OLS estimator in an AR(1) model and the QMLE in an MA(1) model.
Quadratic forms in skew normal variates
Journal of Mathematical Analysis and Applications, 2002
In this paper first a characterization of the multivariate skew normal distribution is given. Then the joint moment generating functions of two quadratic forms, and a linear compound and a quadratic form in skew normal variates, have been derived and conditions for their independence are given. Distribution of the ratios of quadratic forms in skew normal variates has also been studied. 2002 Elsevier Science (USA). All rights reserved.