On positive definite quadratic forms in correlatedt variables

Summary In this paper we extend Ruben's [4] result for quadratic forms in normal variables. He represented the distribution function of the quadratic form in normal variables as an infinite mixture of chi-square distribution functions. In the central case, we show that the distribution function of a quadratic form int-variables can be represented as a mixture of beta distribution functions. In the noncentral case, the distribution function presented is an infinite series in beta distribution functions. An application to quadratic discrimination is given.     

Inferences in linear mixed models with skew-normal random effects

For the linear mixed model with skew-normal random effects, this paper gives the density function, moment generating function and independence conditions. The noncentral skew chi-square distribution is defined and its density function is shown. The necessary and sufficient conditions under which a quadratic form is distributed as noncentral skew chi-square distribution are obtained. Also, a version of Cochran's theorem is given, which modifies the result of Wang et al. (2009) and is used to set up exact tests for fixed effects and variance components of the proposed model. For illustration, our main results are applied to a real data problem.     

The bivariate noncentral chi-square distribution - A compound distribution approach

This paper proposes the bivariate noncentral chi-square (BNC) distribution by compounding the Poisson probabilities with the bivariate central chi-square distribution. The probability density and cumulative distribution functions of the joint distribution of the two noncentral chi-square variables are derived for arbitrary values of the correlation coefficient, degrees of freedom(s), and noncentrality parameters. Computational procedures to calculate the upper tail probabilities as well as the percentile points for selected values of the parameters, for both equal and unequal degrees of freedom, are discussed. The graphical representation of the distribution for different values of the parameters are provided. Some applications of the distribution are outlined.     

Asymptotic Approximations for the Distributions of the Multinomial Goodness-of-Fit Statistics under Local Alternatives

N. Cressie and T. R. C. Read (1984, J. Roy. Statist. Soc. B46, 440–464) introduced a class of multinomial goodness-of-fit statistics R^a based on power divergence. All R^a have the same chi-square limiting distribution under null hypothesis and have the same noncentral chi-square limiting distribution under local alternatives. In this paper, we investigate asymptotic approximations for the distributions of R^a under local alternatives. We obtain an expression of approximation for the distribution of R^a under local alternatives. The expression consists of continuous and discontinuous terms. Using the continuous term of the expression, we propose a new approximation of the power of R^a. We call the approximation AE approximation. By numerical investigation of the accuracy of the AE approximation, we present a range of sample size n that the omission of the discontinuous term exercises only slight influence on power approximation of R^a. We find that the AE approximation is effective for a much wider range of the value of a than the other power approximations, except for an approximation method which requires high computer performance.     

New asymptotic representations of the noncentral t-distribution

New asymptotic approximations of the noncentral t distribution are given a generalization of the Student's t distribution. Using new integral representations, we give new asymptotic expansions not only for large values of the noncentrality parameter but also for large values of the degrees of freedom parameter. In some cases, we accept more than one large parameter. These results are not only in terms of elementary functions, but also in terms of the complementary error function and the incomplete gamma function. A number of numerical tests demonstrate the performance of the asymptotic approximations.     

Distribution of quadratic forms under skew normal settings

For a class of multivariate skew normal distributions, the noncentral skew chi-square distribution is studied. The necessary and sufficient conditions under which a sequence of quadratic forms is generalized noncentral skew chi-square distributed random variables are obtained. Several examples are given to illustrate the results.     

Noncentral quadratic forms of the skew elliptical variables

In this paper the quadratic forms in the skew elliptical variables are studied. A family of the noncentral generalized Dirichlet distributions is introduced and their distribution functions and probability density functions are obtained. The moment generating functions of the quadratic forms in the skew normal variables are obtained. Sufficient and necessary conditions for the quadratic forms in the skew normal variables to have the noncentral generalized Dirichlet distributions are obtained. This leads to the noncentral Cochran's Theorem for the skew normal distribution.     

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.     

Moment-Based Approximations of Distributions Using Mixtures: Theory and Applications

There are a number of cases where the moments of a distribution are easily obtained, but theoretical distributions are not available in closed form. This paper shows how to use moment methods to approximate a theoretical univariate distribution with mixtures of known distributions. The methods are illustrated with gamma mixtures. It is shown that for a certain class of mixture distributions, which include the normal and gamma mixture families, one can solve for a p-point mixing distribution such that the corresponding mixture has exactly the same first 2p moments as the targeted univariate distribution. The gamma mixture approximation to the distribution of a positive weighted sums of independent central ² variables is demonstrated and compared with a number of existing approximations. The numerical results show that the new approximation is generally superior to these alternatives.     

Exact Cumulant Calculations for Pearson X2 and Zelterman Statistics for r-Way Contingency Tables

Abstract

We present a computer algebra procedure that calculates exact cumulants for Pearson X ² and Zelterman statistics for r-way contingency tables. The algorithm is an example of how an overwhelming algebraic problem can be solved neatly through computer implementation by emulating tactics that one uses by hand. For inference purposes the cumulants may be used to assess chi-square approximations or to improve this approximation via Edgeworth expansions. Edgeworth approximations are compared to the computerintensive techniques of Mehta and Patel that provide exact and arbitrarily close results. Comparisons to approximations that utilize the gamma distribution (Mielke and Berry) are also made.     

Splitting Patterns and Invariants of Quadratic Forms

Let φ be an anisotropic quadratic form over a field F of characteristic not 2. The splitting pattern of φ is defined to be the increasing sequence of nonnegative integers obtained by considering the Witt indices i_W(φ_k) of φ over K where K ranges over all field extensions of F. Restating earlier results by HURRELBRINK and REHMANN , we show how the index of the Clifford algebra of φ influences the splitting pattern. In the case where F is formally real, we investigate how the signatures of φ influence the splitting behaviour. This enables us to construct certain splitting patterns which have been known to exist, but now over much “simpler” fields like formally real global fields or ?(t). We also give a full classification of splitting patterns of forms of dimension less than or equal to 9 in terms of properties of the determinant and Clifford invariant. Partial results for splitting patterns in dimensions 10 and 11 are also provided. Finally, we consider two anisotropic forms φ and φ of the same dimension m with φ ? ? φ ∈ Iⁿ F and give some bounds on m depending on n which assure that they have the same splitting pattern.     

Statistically and Computationally Efficient Estimating Equations for Large Spatial Datasets

For Gaussian process models, likelihood-based methods are often difficult to use with large irregularly spaced spatial datasets, because exact calculations of the likelihood for n observations require O(n³) operations and O(n²) memory. Various approximation methods have been developed to address the computational difficulties. In this article, we propose new, unbiased estimating equations (EE) based on score equation approximations that are both computationally and statistically efficient. We replace the inverse covariance matrix that appears in the score equations by a sparse matrix to approximate the quadratic forms, then set the resulting quadratic forms equal to their expected values to obtain unbiased EE. The sparse matrix is constructed by a sparse inverse Cholesky approach to approximate the inverse covariance matrix. The statistical efficiency of the resulting unbiased EE is evaluated both in theory and by numerical studies. Our methods are applied to nearly 90,000 satellite-based measurements of water vapor levels over a region in the Southeast Pacific Ocean.     

Multiple comparisons of several homoscedastic multivariate populations

The limiting joint distribution of correlated Hotelling’s T ² statistics associated with multiple comparisons with a control in multivariate one-way layout model is a multivariate central nonsingular chi-square distribution with one-factorial correlation matrix, which has the distribution function expressed in a closed form as an integral of a product of noncentral chi-square distribution functions with respect to a central chi-square density function. For pairwise comparisons, it is a multivariate central singular chi-square distribution whose distribution function is generally intricate. To overcome the complexity of the (exact or asymptotic) distribution theory of -type statistics appeared in simultaneous confidence intervals of mean vectors, improved Bonferroni-type inequalities are applied to construct asymptotically conservative simultaneous confidence intervals for pairwise comparisons as well as comparisons with a control.     

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.     

Approximation of Singularities by Quantized-Tensor FEM

In d dimensions, first-order tensor-product finite-element (FE) approximations of the solutions of second-order elliptic problems are well known to converge algebraically, with rate at most 1/d in the energy norm and with respect to the number of degrees of freedom. On the other hand, FE methods of higher regularity may achieve exponential convergence, e.g. global spectral methods for analytic solutions and hp methods for solutions from certain countably normed spaces, which may exhibit singularities. In this note, we revisit, in one dimension, the tensor-structured approach to the h-FE approximation of singular functions. We outline a proof of the exponential convergence of such approximations represented in the quantized-tensor-train (QTT) format. Compared to special approximation techniques, such as hp, that approach is fully adaptive in the sense that it finds suitable approximation spaces algorithmically. The convergence is measured with respect to the number of parameters used to represent the solution, which is not the dimension of the first-order FE space, but depends only polylogarithmically on that. We demonstrate the convergence numerically for a simple model problem and find the rate to be approximately the same as for hp approximations. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Equivalent conditions for noncentral generalized Laplacianness and independence of matrix quadratic forms

Let Y be an n×p multivariate normal random matrix with general covariance Σ_Y and W be a symmetric matrix. In the present article, the property that a matrix quadratic form Y^′WY is distributed as a difference of two independent (noncentral) Wishart random matrices is called the (noncentral) generalized Laplacianness (GL). Then a set of algebraic results are obtained which will give the necessary and sufficient conditions for the (noncentral) GL of a matrix quadratic form. Further, two extensions of Cochran’s theorem concerning the (noncentral) GL and independence of a family of matrix quadratic forms are developed.     

Numerical computation of forced oscillations in coupled duffing equations

Oscillating phenomena in non-linear mechanical systems with two degrees of freedom described by coupled Duffing equations are studied from the computational view point. Galerkin approximations of order 7 are computed with a very high precision on an electronic computer by applying a numerical approximation method of Urabe for the Galerkin method. The existence of an exact isolated periodic solution in a small neighborhood of these Galerkin approximations is proved and the error bound of these Galerkin approximations is given. The stability of Stierel's integration method in combination with Galerkin approximations is shown.     

Bayesian Inference in the Noncentral Student-t Model

This article takes up Bayesian inference in linear models with disturbances from a noncentral Student-t distribution. The distribution is useful when both long tails and asymmetry are features of the data. The distribution can be expressed as a location-scale mixture of normals with inverse weights distributed according to a chi-square distribution. The computations are performed using Gibbs sampling with data augmentation. An empirical application to Standard and Poor's stock returns indicates that posterior odds strongly favor a noncentral Student-t specification over its symmetric counterpart.     

Forms of pointed Hopf algebras

Using descent theory, we study Hopf algebra forms of pointed Hopf algebras. It turns out that the set of isomorphism classes of such forms are in one-to-one correspondence to other known invariants, for example the set of isomorphism classes of Galois extensions with a certain group F, or the set of isometry classes of m-ary quadratic forms. Our theory leads to a classification of all Hopf algebras over a field of characteristic zero that become pointed after a base extension, in dimension p, p ² and p ³, with p odd. Received: 22 November 1998     

Weighted Polynomial Approximation in the Complex Plane

Given a pair (G,W) of an open bounded set G in the complex plane and a weight function W(z) which is analytic and different from zero in G , we consider the problem of the locally uniform approximation of any function f(z) , which is analytic in G , by weighted polynomials of the form {W ⁿ (z)P _n (z) } ^$\infinity$ _n=0 , where deg P_n n. The main result of this paper is a necessary and sufficient condition for such an approximation to be valid. We also consider a number of applications of this result to various classical weights, which give explicit criteria for these weighted approximations. May 1, 1996. Date revised: October 8, 1996.