Some comments on zonal polynomials and their expected values with respect to elliptical distributions

In this paper, we give alternative proofs of some results in [15] (Li R.,1997) about the expected value of zonal polynomials.     

Some properties of invariant polynomials with matrix arguments and their applications in econometrics

Summary Further properties are derived for a class of invariant polynomials with several matrix arguments which extend the zonal polynomials. Generalized Laguerre polynomials are defined, and used to obtain expansions of the sum of independent noncentral Wishart matrices and an associated generalized regression coefficient matrix. The latter includes thek-class estimator in econometrics.     

The expected values of zonal polynomials of elliptically contoured distributions

Based on the work of Constantine, Waal, Shah, Khatri, and Muirhead, we have made further studies on the expected values of zonal polynomials and elementary symmetric functions of a Wishart matrix in this paper. Under certain conditions, we found the expected values of zonal polynomials and elementary symmetric functions of matrixXX, where the matrixX has matrix elliptically contoured distribution and obtain two interesting results. Three examples are given.     

A Rodrigues-type formula for Gegenbauer matrix polynomials

This paper centers on the derivation of a Rodrigues-type formula for the Gegenbauer matrix polynomial. A connection between Gegenbauer and Jacobi matrix polynomials is given.     

Multivariate tail conditional expectation for elliptical distributions

In this paper we introduce a novel type of a multivariate tail conditional expectation (MTCE) risk measure and explore its properties. We derive an explicit closed-form expression for this risk measure for the elliptical family of distributions taking into account its variance–covariance dependency structure. As a special case we consider the normal, Student-t and Laplace distributions, important and popular in actuarial science and finance. The motivation behind taking the multivariate TCE for the elliptical family comes from the fact that unlike the traditional tail conditional expectation, the MTCE measure takes into account the covariation between dependent risks, which is the case when we are dealing with real data of losses. We illustrate our results using numerical examples in the case of normal and Student-t distributions.     

The growth of laguerre matrix polynomials on bounded intervals

Let A be a matrix in ^r×r such that Re(z) > −1/2 for all the eigenvalues of A and let {π_n^(A,1/2) (x)} be the normalized sequence of Laguerre matrix polynomials associated with A. In this paper, it is proved that π_n^(A,1/2) (x) = O(n^(A)/2ln^r−1(n)) and π_n+1^(A,1/2) (x) − π_n^(A,1/2) (x) = O(n^((A)−1)/2ln^r−1(n)) uniformly on bounded intervals, where (A) = max{Re(z); z eigenvalue of A}.     

Differential coefficients of orthogonal matrix polynomials

We find explicit formulas for raising and lowering first order differential operators for orthogonal matrix polynomials. We derive recurrence relations for the coefficients in the raising and lowering operators. Some examples are given.     

On non-null distributions connected with testing that a real normal distribution is complex

The exact and the asymptotic non-null distribution of the maximal invariant corresponding to testing that the covariance matrix of a 2m-dimensional real normal distribution has complex structure is obtained.     

Fibonacci polynomials and Sylvester determinant of tridiagonal matrix

By means of left eigenvector method, we evaluate the determinant of a tridiagonal matrix, which extends the determinant due to Sylvester [5].     

ESTIMATION OF SCATTER MATRIX BASED ON i．i．d． SAMPLE FROM ELLIPTICAL DISTRIBUTIONS

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The efficient evaluation of the hypergeometric function of a matrix argument

We present new algorithms that efficiently approximate the hypergeometric function of a matrix argument through its expansion as a series of Jack functions. Our algorithms exploit the combinatorial properties of the Jack function, and have complexity that is only linear in the size of the matrix.

     

Noncentral elliptical configuration density

The noncentral configuration density, derived under an elliptical model, generalizes and corrects the Gaussian configuration and some Pearson results. Partition theory is then used to obtain explicit configuration densities associated with matrix variate symmetric Kotz type distributions (including the normal distribution), matrix variate Pearson type VII distributions (including t and Cauchy distributions), the matrix variate symmetric Bessel distribution (including the Laplace distribution) and the matrix variate symmetric Jensen-logistic distribution.     

A new extension of Hermite matrix polynomials and its applications

In this paper, an extension of the Hermite matrix polynomials is introduced. Some relevant matrix functions appear in terms of the two-variable Hermite matrix polynomials. Furthermore, in order to give qualitative properties of this family of matrix polynomials, the Chebyshev matrix polynomials of the second kind are introduced.     

Multi-variable Hermite matrix polynomials: Properties and applications

In this paper, the multi-variable Hermite matrix polynomials are introduced by algebraic decomposition of exponential operators. Their properties are established using operational methods. The matrix forms of the Chebyshev and truncated polynomials of two variable are also introduced, which are further used to derive certain operational representations and expansion formulae.     

A generalization of the inertia theorem for quadratic matrix polynomials

We show that the inertia of a quadratic matrix polynomial is determined in terms of the inertia of its coefficient matrices if the leading coefficient is Hermitian and nonsingular, the constant term is Hermitian, and the real part of the coefficient matrix of the first degree term is definite. In particular, we prove that the number of zero eigenvalues of such a matrix polynomial is the same as the number of zero eigenvalues of its constant term. We also give some new results for the case where the real part of the coefficient matrix of the first degree term is semidefinite.     

Orthogonal matrix polynomials, scalar-type Rodrigues’ formulas and Pearson equations

Some families of orthogonal matrix polynomials satisfying second-order differential equations with coefficients independent of n have recently been introduced (see [Internat. Math. Res. Notices 10 (2004) 461–484]). An important difference with the scalar classical families of Jacobi, Laguerre and Hermite, is that these matrix families do not satisfy scalar type Rodrigues’ formulas of the type (ΦⁿW)⁽ⁿ⁾W^-1, where Φ is a matrix polynomial of degree not bigger than 2. An example of a modified Rodrigues’ formula, well suited to the matrix case, appears in [Internat. Math. Res. Notices 10 (2004) 482].In this note, we discuss some of the reasons why a second order differential equation with coefficients independent of n does not imply, in the matrix case, a scalar type Rodrigues’ formula and show that scalar type Rodrigues’ formulas are most likely not going to play in the matrix valued case the important role they played in the scalar valued case. We also mention the roles of a scalar-type Pearson equation as well as that of a noncommutative version of it.     

Solvability of matrix equations in rings of quasi-diagonal matrices and similarity of matrix polynomials

A certain standard form is found for a complex matrix with respect to equivalent transformations by quasi-diagonal matrices. The solvability of certain matrix equations in the rings of quasi-diagonal matrices is examined using this standard form.     

Computations with half-range Chebyshev polynomials

An efficient construction of two non-classical families of orthogonal polynomials is presented in the paper. The so-called half-range Chebyshev polynomials of the first and second kinds were first introduced by Huybrechs in Huybrechs (2010) [5]. Some properties of these polynomials are also shown. Every integrable function can be represented as an infinite series of sines and cosines of these polynomials, the so-called half-range Chebyshev-Fourier (HCF) series. The second part of the paper is devoted to the efficient computation of derivatives and multiplication of the truncated HCF series, where two matrices are constructed for this purpose: the differentiation and the multiplication matrix.     

Rakhmanov's theorem for orthogonal matrix polynomials on the unit circle

Rakhmanov's theorem for orthogonal polynomials on the unit circle gives a sufficient condition on the orthogonality measure for orthogonal polynomials on the unit circle, in order that the reflection coefficients (the recurrence coefficients in the Szegő recurrence relation) converge to zero. In this paper we give the analog for orthogonal matrix polynomials on the unit circle.