Approximating by the Wishart distribution

Approximations of density functions are considered in the multivariate case. The results are presented with the help of matrix derivatives, powers of Kronecker products and Taylor expansions of functions with matrix argument. In particular, an approximation by the Wishart distribution is discussed. It is shown that in many situations the distributions should be centred. The results are applied to the approximation of the distribution of the sample covariance matrix and to the distribution of the non-central Wishart distribution.     

On the structure of the Wishart distribution

In this paper it is shown that every nonnegative definite symmetric random matrix with independent diagonal elements and at least one nondegenerate nondiagonal element has a noninfinitely divisible distribution. Using this result it is established that every Wishart distribution W_p(k, Σ, M) with both p and rank (Σ) ≥ 2 is noninfinitely divisible. The paper also establishes that any Wishart matrix having distribution W_p(k, Σ, 0) has the joint distribution of its elements in the rth row and rth column to be infinitely divisible for every r = 1,2,…,p.     

A characterization of Wishart processes and Wishart distributions

A characterization of the existence of non-central Wishart distributions (with shape and non-centrality parameter) as well as the existence of solutions to Wishart stochastic differential equations (with initial data and drift parameter) in terms of their exact parameter domains is given. These two families are the natural extensions of the non-central chi-square distributions and the squared Bessel processes to the positive semidefinite matrices.     

Eigenfunctions of expected value operators in the Wishart distribution,II

Let V = (v_ij) denote the k × k symmetric scatter matrix following the Wishart distribution W(k, n, Σ). The problem posed is to characterize the eigenfunctions of the expectation operators of the Wishart distribution, i.e., those scalar-valued functions f(V) such that (E_nf)(V) = λ_n,kf(V). A finite sequence of polynomial eigenspaces, EP spaces, exists whose direct sum is the space of all homogeneous polynomials. These EP subspaces are invariant and irreducible under the action of the congruence transformation V → T′VT. Each of these EP subspaces contains an orthogonally invariant subspace of dimension one. The number of EP subspaces is determined and eigenvalues are computed. Bi-linear expansions of $\text{|I + VA|}{}^{\mathrm{<mtext>?n</mtext><mtext>2</mtext>}}$ and (tr VA)^r into eigenfunctions are given. When f(V) is an EP polynomial, then f(V^?1) is an EP function. These EP subspaces are identical to the more abstractly defined polynomial subspaces studied by James.     

Nonparametric Empirical Bayes Estimation of the Matrix Parameter of the Wishart Distribution

We consider independent pairs (X₁, Σ₁), (X₂, Σ₂), …, (X_n, Σ_n), where eachΣ_iis distributed according to some unknown density functiong(Σ) and, givenΣ_i=Σ,X_ihas conditional density functionq(xΣ) of the Wishart type. In each pair the first component is observable but the second is not. After the (n+1)th observationX_n+1is obtained, the objective is to estimateΣ_n+1corresponding toX_n+1. This estimator is called the empirical Bayes (EB) estimator ofΣ. An EB estimator ofΣis constructed without any parametric assumptions ong(Σ). Its posterior mean square risk is examined, and the estimator is demonstrated to be pointwise asymptotically optimal.     

The inverted complex Wishart distribution and its application to spectral estimation

The inverted complex Wishart distribution and its use for the construction of spectral estimates are studied. The density, some marginals of the distribution, and the first- and second-order moments are given. For a vector-valued time series, estimation of the spectral density at a collection of frequencies and estimation of the increments of the spectral distribution function in each of a set of frequency bands are considered. A formal procedure applies Bayes theorem, where the complex Wishart is used to represent the distribution of an average of adjacent periodogram values. A conjugate prior distribution for each parameter is an inverted complex Wishart distribution. Use of the procedure for estimation of a 2 × 2 spectral density matrix is discussed.     

Representations for eigenfunctions of expected value operators in the Wishart distribution

Recent articles by Kushner and Meisner (1980) and Kushner, Lebow and Meisner (1981) have posed the problem of characterising the ‘EP’ functions f(S) for which Ef(S) for which E(f(S)) = λ_nf(Σ) for some λ_n ? $\text{R}$, whenever the m × m matrix S has the Wishart distribution W(m, n, Σ). In this article we obtain integral representations for all nonnegative EP functions. It is also shown that any bounded EP function is harmonic, and that EP polynomials may be used to approximate the functions in certain L^p spaces.     

Guaranteed error bounds for the initial value problem using polytope arithmetic

In this paper we propose and compare modifications of the method of co-ordinate transformations for finding guaranteed bounds for the numerical solution of the initial value problem. These modifications are judged on their success in overcoming exponentially too large growth of the computed error bound.     

An identity for the Wishart distribution with applications

Let S_p×p have a Wishart distribution with unknown matrix Σ and k degrees of freedom. For a matrix T(S) and a scalar h(S), an identity is obtained for E_∑tr[h(S)T∑⁻¹]. Two applications are given. The first provides product moments and related formulae for the Wishart distribution. Higher moments involving S can be generated recursively. The second application concerns good estimators of ∑ and ∑⁻¹. In particular, identities for several risk functions are obtained, and estimators of ∑ (∑⁻¹) are described which dominate aS(bS⁻¹), a ≤ 1/k (b ≤ k − p − 1). [3] Ann. Statist. 7 No. 5; (1980) Ann. Statist. 8 used special cases of the identity to find unbiased risk estimators. These are unobtainable in closed form for certain natural loss functions. In this paper, we treat these case as well. The dominance results provide a unified theory for the estimation of ∑ and ∑⁻¹.     

The Hyperoctahedral Group,Symmetric Group Representations and the Moments of the Real Wishart Distribution

In this paper, we compute all the moments of the real Wishart distribution. To do so, we use the Gelfand pair (S_2k,H), where H is the hyperoctahedral group, the representation theory of H and some techniques based on graphs.     

Convergence on error correction methods for solving initial value problems

Higher-order semi-explicit one-step error correction methods(ECM) for solving initial value problems are developed. ECM provides the excellent convergence O(h^2p+2)$O\left(h^{2p+2}\right)$ one wants to get without any iteration processes required by most implicit type methods. This is possible if one constructs a local approximation having a residual error O(h^p)$O\left(h^p\right)$ on each time step. As a practical example, we construct a local quadratic approximation. Further, it is shown that special choices of parameters for the local quadratic polynomial lead to the known explicit second-order methods which can be improved into a semi-explicit type ECM of the order of accuracy 6$6$. The stability function is also derived and numerical evidences are presented to support theoretical results with several stiff and non-stiff problems. It should be remarked that the ECM approach developed here does not yield explicit methods, but semi-implicit methods of the Rosenbrock type. Both ECM and Rosenbrock’s methods require to solve a few linear systems at each integration step, but the ECM approach involves 2p+2$2p+2$ evaluations of the Jacobian matrix per integration step whereas the Rosenbrock method demands one evaluation only. However, it is much easier to get high order methods by using the ECM approach.     

Discretization error for the maximum of a Gaussian field

The paper considers the difference between (a) the true maximum of a Gaussian field on a square and (b) its maximum on a regular grid. This difference is called the discretization error. A kind of Slepian model is used to study the behavior of the field around the location of the maximum. We show that the normalized discretization error can be bounded by a quantity that converges to a uniform variable, depending on the Hessian matrix at the point of the maximum. The bound is applied to simulated and real data (satellite positioning data).     

Asymptotic expansions of the moments of extremes from general error distribution

In this paper, we derive the asymptotic expansions of the moments of normalized partial maxima for general error distribution. A byproduct is to deduce the convergence rates of the moments of normalized maxima to the moments of the corresponding extreme value distribution.     

The Expected loss in the discretization of multistage stochastic programming problems—estimation and convergence rate

In the present paper, the approximate computation of a multistage stochastic programming problem (MSSPP) is studied. First, the MSSPP and its discretization are defined. Second, the expected loss caused by the usage of the “approximate” solution instead of the “exact” one is studied. Third, new results concerning approximate computation of expectations are presented. Finally, the main results of the paper—an upper bound of the expected loss and an estimate of the convergence rate of the expected loss—are stated.     

Integrated squared error of kernel-type estimator of distribution function

Let X ₁ ,...,X _n be a random sample drawn from distribution function F(x) with density function f(x) and suppose we want to estimate X(x). It is already shown that kernel estimator of F(x) is better than usual empirical distribution function in the sense of mean integrated squared error. In this paper we derive integrated squared error of kernel estimator and compare the error with that of the empirical distribution function. It is shown that the superiority of kernel estimators is not necessarily true in the sense of integrated squared error.     

On the possibility of two-sided error bounds in the numerical solution of initial value problems

Summary Letu denote the approximation produced by a finite-difference method for solving an initial value problem for a given differential equation. Suppose the finite-difference equation is perturbed by a quantityw, e.g. due to round-off or truncation errors. Then, instead ofu, one obtains a solution which we denote by In this paper a condition is presented which is necessary and sufficient for the existence of a two-sided estimate of the error-u in terms of the perturbationw. The paper is concluded with applications in the fields of ordinary and partial parabolic differential equations.