The exact distribution of indefinite quadratic forms in noncentral normal vectors

The exact density of the difference of two linear combinations of independent noncentral chi-square variables is obtained in terms of Whittaker's function and expressed in closed forms. Two distinct representations are required in order to cover all the possible cases. The corresponding expressions for the exact distribution function are also given.     

Sharp estimates for the CDF of quadratic forms of MPE random vectors

In this paper we develop an efficient analytical expansion of the cumulative distribution function (cdf) XBX^t where X=(X₁,…,X_n+1) with n≥2, follows a multivariate power exponential distribution (MPE). Our approach provides a sharp estimate of the cumulative distribution function of a quadratic form of MPE, together with explicit error estimates.     

Asymptotic expansions for the distribution of quadratic forms in normal variables

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 resulting 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.Faculty of Science, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812 Japan     

Random weighted projections,random quadratic forms and random eigenvectors

We present a concentration result concerning random weighted projections in high dimensional spaces. As applications, we prove (1) New concentration inequalities for random quadratic forms. (2) The infinity norm of most unit eigenvectors of a random ±1 matrix is of order . (3) An estimate on the threshold for the local semi‐circle law which is tight up to a factor. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 792–821, 2015     

Wishartness and independence of matrix quadratic forms in a normal random matrix

Let Y be an n×p multivariate normal random matrix with general covariance Σ_Y. The general covariance Σ_Y of Y means that the collection of all np elements in Y has an arbitrary np×np covariance matrix. A set of general, succinct and verifiable necessary and sufficient conditions is established for matrix quadratic forms Y^′W_iY's with the symmetric W_i's to be an independent family of random matrices distributed as Wishart distributions. Moreover, a set of general necessary and sufficient conditions is obtained for matrix quadratic forms Y^′W_iY's to be an independent family of random matrices distributed as noncentral Wishart distributions. Some usual versions of Cochran's theorem are presented as the special cases of these results.     

An invariance property of quadratic forms in random vectors with a selection distribution,with application to sample variogram and covariogram estimators

We study conditions under which an invariance property holds for the class of selection distributions. First, we consider selection distributions arising from two uncorrelated random vectors. In that setting, the invariance holds for the so-called C{\cal{C}} -class and for elliptical distributions. Second, we describe the invariance property for selection distributions arising from two correlated random vectors. The particular case of the distribution of quadratic forms and its invariance, under various selection distributions, is investigated in more details. We describe the application of our invariance results to sample variogram and covariogram estimators used in spatial statistics and provide a small simulation study for illustration. We end with a discussion about other applications, for example such as linear models and indices of temporal/spatial dependence.     

Computing the distribution function of the ratio of quadratic forms in normal variables

Formulas are given for computing the ratio of quadratic forms in normal variables. The doubly noncentral F-distribution function is computed.Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 123–128, 1986.     

The asymptotic behavior of quadratic forms in -mixing random variables

Let {Z_i,i≥1} be a linear process defined by with {d_j,j≥0} being a regular varying sequence of real numbers and {ξ_t,−∞<t<∞} being a sequence of -mixing random variables. The present paper studies the asymptotic behavior of the quadratic form under some mild assumptions on d_j and ξ_t. Meanwhile, the similar results of α-mixing random variables are presented.     

The limiting theorems of random quadratic forms and their application

The strong convergence and convergence rate of the random quadratic forms s1T(S1S1T)Ms1 and s1T(SST)ms1 are set up. The application of these results in wireless communication is given. Simulation results are presented.     

Systems of quadratic Diophantine inequalities and the value distribution of quadratic forms

Let Q ₁,…,Q _r be quadratic forms with real coefficients. We prove that the set is dense in , provided that the system Q ₁(x) = 0,…,Q _r (x) = 0 has a nonsingular real solution and all forms in the real pencil generated by Q ₁,…,Q _r are irrational and have rank larger than 8r. Moreover, we give a quantitative version of the above assertion. As an application we study higher correlation functions of the value distribution of a positive definite irrational quadratic form. Author’s address: Institut für Statistik, Technische Universit?t Graz, A-8010 Graz, Austria     

Estimation of the confidence limits for the quadratic forms in normal variables using a simple Gaussian distribution approximation

Summary  In this paper a simple Gaussian approximation of the distribution of the weighted sum of squared normal variables is proposed. The proposed approximation is computationally less complex compared to other known approximations. However, the convergence towards Gaussian distribution is guaranteed provided the weights comply with certain limit conditions. The suggested approximation is applied to the calculation of confidence limits of the quadratic forms in normal variables. These problems can be encountered in a number of statistical decision making tasks. The accuracy of the estimated confidence limit is investigated on several simulation examples.     

Systems of quadratic Diophantine inequalities and the value distribution of quadratic forms

Let Q ₁,…,Q _r be quadratic forms with real coefficients. We prove that the set {(Q₁(x),?,Q_r(x)) | x ? \Bbb Z^s}\{(Q_1(x),\ldots ,Q_r(x))\,\vert\, x\in{\Bbb Z}^s\} is dense in \Bbb R^r{\Bbb R}^r , provided that the system Q ₁(x) = 0,…,Q _r (x) = 0 has a nonsingular real solution and all forms in the real pencil generated by Q ₁,…,Q _r are irrational and have rank larger than 8r. Moreover, we give a quantitative version of the above assertion. As an application we study higher correlation functions of the value distribution of a positive definite irrational quadratic form.     

On the conditional probability density functions of multivariate uniform random vectors and multivariate normal random vectors

It is shown that the conditional probability density function of Y₁ given (1/n) Σ_i=1ⁿ Y_i=1Y_i^t = Σ, where Y₁, Y₂,…, Y_n are i.i.d, p-variate uniform random vectors with mean 0 equals to that of Y₁ given (1/n) Σ_i=1ⁿ Y_iY_i^t,…, Y_n are i.i.d, p-variate normal random vectors with mean 0 and covariance matrix Σ.     

Bounded laws of the iterated logarithm for quadratic forms in Gaussian random variables

In this paper we prove bounded laws of the iterated logarithm for Gaussian quadratic forms. The underlying sequence of Gaussian variables is assumed to satisfy quite general conditions on its covariance structure. Basic tools are maximal inequalities of exponential type for sums of dependent random variables which may be of own interest. Several examples illustrate the sharpness of the results. In a particular section the bounded law of the iterated logarithm is shown for quadratic variation of Brownian motion.     

The convergence rate of distribution of the maximum of random vectors

Technology University, Studentu 50, 3028 Kaunas, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 32, No. 2, pp. 229–236, April–June, 1992.     

On the Kolmogorov inequalities for quadratic forms of dependent uniformly bounded random variables

Some Kolmogorov probability inequalities for quadratic forms and weighted quadratic forms of negative superadditive dependent (NSD) uniformly bounded random variables are provided. Using these inequalities, some complete convergence of randomized quadratic forms under some suitable conditions are evaluated. Moreover, various examples are presented in which the given conditions of our results are satisfied.     

Random vectors satisfying Khinchine–Kahane type inequalities for linear and quadratic forms

We study the behaviour of moments of order p (1 < p < ∞) of affine and quadratic forms with respect to non log‐concave measures and we obtain an extension of Khinchine–Kahane inequality for new families of random vectors by using Pisier's inequalities for martingales. As a consequence, we get some estimates for the moments of affine and quadratic forms with respect to a tail volume of the unit ball of lⁿ_q (0 < q < 1). (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)