Characterizing parameters of multivariate elliptical distributions*

This paper defines new parameters characterizing multivariate elliptical distributions. Mardia's coefficient of multivariate kurtosis is shown to be essentially one of these parameters. A simple relation is established between centered multivariate product moments and second moments of the variables. The general results are verified on the contaminated normal distribution as an example.     

Higher order moments of the estimated tangency portfolio weights

In this paper, we consider the estimated weights of the tangency portfolio. We derive analytical expressions for the higher order non-central and central moments of these weights when the returns are assumed to be independently and multivariate normally distributed. Moreover, the expressions for mean, variance, skewness and kurtosis of the estimated weights are obtained in closed forms. Later, we complement our results with a simulation study where data from the multivariate normal and t-distributions are simulated, and the first four moments of estimated weights are computed by using the Monte Carlo experiment. It is noteworthy to mention that the distributional assumption of returns is found to be important, especially for the first two moments. Finally, through an empirical illustration utilizing returns of four financial indices listed in NASDAQ stock exchange, we observe the presence of time dynamics in higher moments.     

Population moments and moments of the sample mean

This paper presents new formulae which simultaneously express and estimate moments of the sample mean and estimate population moments, from a simple random sample drawn without replcement from a finite population. By avoiding the generality of the multivariate case, these two problems are not only unified but are made significantly more tractable. Explicit solution are given up to eighth moments. Asymptotic results for infinite populations are also given.     

Multivariate lagrange distributions and a subfamiliy

Second order moments about its means, i.e. the variances and covari-ances for multivariate Lagrange distributions are derived in a matrix form. A subfamily of multivariate Lagrange distributions which can be characterized as the distributions of customers served in a busy period in queues with some conditions are considered. Theorems about their probability functions, one of which is a multivariate generalization of a formula by Takà cs(1989). are given and the means and second order moments about its means are considered. As an example, a multivariate Borel-Tanner distribution is derived.     

A class of maximum-entropy multivariate distributions

This paper characterizes a class of multivariate distributions that includes the multinormal and is contained in the exponential family. The wide range of possible applications of these distributions is suggested by some of hte characteristics germane to them: First, they maximize Shannon's entropy among all distributions that have finite moments of given orders. As such, they constitute a class of distributions that includes the multinormal and some likely alternatives. Second, they can exhibit several modes, and, further-more, they do so with a relatively small number of parameters (compared to mixtures of multinormals). Third, they are the stationary distributions of certain diffusion processes. Fourth, they approximate, near the multinormal, the multivariate Pearson family. And fifth, the maximum likelihood estimators of their population moments are the sample moments. Two possible methods of estimating the distributions are studied in this paper: maximum likelihood estimation, and a fast procedure that can be used to find consistent estimators of the parameters via sample moments. A FORTTAN subroutine that implements the latter method is also provided.     

Inverse moments of discrete distributions

In this paper an expression for the inverse moment of order r is given for the truncated binomial and Poisson distributions. This enables one to obtain inverse moments in a finite series. Some applications and multivariate generalizations are also given. The method also enables one to obtain relations between inverse moments and factorial moments and distributions of sums of variables.     

Further identities for the Wishart distribution with applications in regression

Matrix analogues are given for a known scalar identity which relates certain expectations with respect to the Wishart distribution. (The scalar identity was independently derived by C. Stein and L. Haff.) The matrix analogues are more aptly called “matrix extensions.” They can be derived by using the scalar identity; nevertheless, they are seen (in quite elementary terms) to be more general than the latter. A method of doing multivariate calculations is developed from the identities, and several examples are worked in detail. We compute the first two moments of the regression coefficients and another matrix arising in regression analysis. Also, we give a new result for the matrix analogue of squared multiple correlation: the bias correction of Ezekiel (1930), a result often used in model building, is extended to the case of two or more dependent variables.     

Relations and identities for the moments of order statistics from a sample containing a single outlier

In this paper, we derive several new recurrence relations and indentities satisfied by the single and the product moments of order statistics from a sample of size n in the presence of an outlier. These recurrence relations involve the first two single moments and the product moments in samples of sized n?1 and less. By making use of these recurrence relations we show that it is sufficient to evaluate at most two single rerents and (n?2)/2 product moment when n is even and two single moments and (n?2)/2 product moments when n is odd, in order to evaluate the first and second single moments and product moments of all order statistics in a sample of size n comprising an outlier, given these moments for the all sample of size less than n. These generalize the results of Govindarajulu (1963), Joshi (1971), and Joshi and Balakrishnan (1982) to the case when the sample includes a single outlier. We also establish some simple identitites involving linear combination of convariances of order statistics. These results can be effectively used to reduce the amount of numerical computation considerably and also to check the accuracy of the computations while evaluating means, variances and covariances of order statistics from an outlier model.     

Evaluating Multivariate GARCH Models in the Nordic Electricity Markets

ABSTRACT

This article considers a variety of specification tests for multivariate GARCH models that are used for dynamic hedging in electricity markets. The test statistics include the robust conditional moments tests for sign-size bias along with the recently introduced copula tests for an appropriate dependence structure. We consider this effort worthwhile, since quite often the tests of multivariate GARCH models are omitted and the models become selected ad hoc depending on the results they generate. Hedging performance comparisons, in terms of unconditional and conditional ex-post variance portfolio reduction, are conducted.     

On approximating the central and noncentral multivariate gamma distributions

In this paper a finite series approximation involving Laguerre polynomials is derived for central and noncentral multivariate gamma distributions. It is shown that if one approximates the density of any k nonnegative continuous random variables by a finite series of Laguerre polynomials up to the (n₁, …, n_k)th degree, then all the mixed moments up to the order (n₁, …, n_k) of the approximated distribution equal to the mixed moments up to the same order of the random variables. Some numerical results are given for the bivariate central and noncentral multivariate gamma distributions to indicate the usefulness of the approximations.     

Some results for multidimensional stationary independent increment processes

The characteristic function, cumulants and moments of vector-valued multidimensional processes, satisfying properties similar to stationary independent increments, are derived. By considering a set of additional postulates for such processes, it is shown that the marginal distribution of such processes is multivariate Poisson. Some of the results in this paper are extensions of the properties of the first two moments of a univariate one-dimensional process with stationary independent increments.     

On the Parameter Estimation of the Asymmetric Multivariate Laplace Distribution

This article examines a family of three-parameter multivariate Laplace distributions ML _p (a, μ, Σ) which is closed under constant shifts. Parameter vectors a and μ are called shift and shape parameter, respectively, positive definite p × p-matrix Σ is a scale parameter. The first three moments are derived and used for estimating the parameters. The behavior of the obtained estimates is explored in a simulation experiment.     

On fractional moments of quadratic expressions in normal variables1

Fractional moments, product cumulants and product moments of general quadratic expressions in singular and nonsingular normal variables are explicitly evaluated. A general method of deriving such moments is also indicated. Particular cases art; shown to agree with known results.     

Parameter estimation for multivariate diffusion processes with the time inhomogeneously positive semidefinite diffusion matrix

Statistical inference for the diffusion coefficients of multivariate diffusion processes has been well established in recent years; however, it is not the case for the drift coefficients. Furthermore, most existing estimation methods for the drift coefficients are proposed under the assumption that the diffusion matrix is positive definite and time homogeneous. In this article, we put forward two estimation approaches for estimating the drift coefficients of the multivariate diffusion models with the time inhomogeneously positive semidefinite diffusion matrix. They are maximum likelihood estimation methods based on both the martingale representation theorem and conditional characteristic functions and the generalized method of moments based on conditional characteristic functions, respectively. Consistency and asymptotic normality of the generalized method of moments estimation are also proved in this article. Simulation results demonstrate that these methods work well.     

Expectation identity for the binomial distribution and its application in the calculations of high-order binomial moments

ABSTRACT

For the exponential families normal, gamma, beta, Poisson, and negative binomial, there exists an expectation identity for each of the family. For the binomial family, we discover an expectation identity, which is useful in analytical calculations of its high-order moments.     

CONVERGENT SERIES EXPRESSIONS FOR INVERSE MOMENTS OF QUADRATIC FORMS IN NORMAL VARIABLES

Using relatively recent results from multivariate distribution theory, a direct approach to evaluating the inverse moments of a quadratic form in normal variables is proposed. Convergent infinite series expressions involving the invariant polynomials of matrix argument are obtained. The solution also depends upon a positive scalar which is arbitrarily chosen. For the solution to converge an upper bound upon this scalar is derived.     

Recurrence relations for the moments of truncated multinormal distribution

In this paper some recurrence relations of moments of doubly truncated multivariate normal distribution are obtained. The bivariate case is given as an example and some applications are indicated.     

Modified slash distribution

In this paper we introduce a modified slash distribution obtained by modifying the usual slash distribution. This new distribution is based on the quotient of two independent random variables, whose distributions are the normal and the power of an exponential distribution of scale parameter equals to two, respectively. In this way, the result is a new distribution whose kurtosis values are greater when compared with that of the slash distribution. We study the density, some properties, moments, kurtosis and make inferences by the method of moments and maximum likelihood. We introduce a multivariate version of this new distribution. Moreover, we provide two illustrations with real data showing that the new distribution fits better the data than the ordinary slash distribution.     

Moments of sample moments

The problem offinding expressions for sampling moments of sample moments has been ahistorically old one. This problem is treated here, with the use of partitions and multi partitions , for the univariate as well as the multivariate case. The systematic combinatorial approach minimizes the chance of omitting any

contributions and making errors in their computation. Componentwise identification is made possible , soerrors can be located. From the complete set of general moment formulae, s pecial cases may be obtained by identifying identical variables.     

EXPRESSIONS FOR INVERSE MOMENTS OF POSITIVE QUADRATIC FORMS IN NORMAL VARIABLES

A unified approach to the provision of exact expressions for inverse moments of positive quadratic forms in normal variables is described, using the method of (essentially) integrating the moment-generating-function (Cressie, Davis, Folks & Policello (1981). A number of special cases, many familiar from the literature, are reviewed; some allow closed form representations, others are expressed in terms of well-known mathematical functions. In the general case, our approach affords representations suitable for numerical computations.