Inspired by Stein's lemma, we derive two expressions for the joint moments of elliptical distributions. We use two different methods to derive $E[X_{1}^{2}f(\mathbf{X})]$ for any measurable function $f$ satisfying some regularity conditions. Then, by applying this result, we obtain new formulae for expectations of product of normally distributed random variables, and also present simplified expressions of $E[X_{1}^{2}f(\mathbf{X})]$ for multivariate Student-$t$, logistic and Laplace distributions.
In this paper, we define doubly truncated moment (DTM), doubly truncated skewness (DTS) and kurtosis (DTK). We derive DTM formulae for elliptical family, with emphasis on normal, student-$t$, logistic, Laplace and Pearson type VII distributions. We also present explicit formulas of the DTE (doubly truncated expectation), DTV (doubly truncated variance), DTS and DTK for those distributions. As illustrative example, DTEs, DTVs, DTSs and DTKs of three industry segments' (Banks, Insurance, Financial and Credit Service) stock return in London stock exchange are discussed.
In this paper, we focus on multivariate doubly truncated first two moments of generalized skew-elliptical (GSE) distributions and derive explicit expressions for them. It includes many useful distributions, for examples, generalized skew-normal (GSN), generalized skew-Laplace (GSLa), generalized skew-logistic (GSLo) and generalized skew student-$t$ (GSSt) distributions, all as special cases. We also give formulas of multivariate doubly truncated expectation and covariance for GSE distributions. As applications, we show the results of multivariate tail conditional expectation (MTCE) and multiva…
Let (X,Y) be a bivariate elliptical random vector with associated random radius in the Gumbel max-domain of attraction. In this paper we obtain a second order asymptotic expansion of the joint survival probability P(X > x, Y> y) for x,y large. Further, based on the asymptotic bounds we discuss some aspects of the statistical modelling of joint survival probabilities and the survival conditional excess probability.
In this paper, we compute doubly truncated moments for the selection elliptical (SE) class of distributions, which includes some multivariate asymmetric versions of well-known elliptical distributions, such as, the normal, Student's t, slash, among others. We address the moments for doubly truncated members of this family, establishing neat formulation for high order moments as well as for its first two moments. We establish sufficient and necessary conditions for the existence of these truncated moments. Further, we propose optimized methods able to deal with extreme setting of the parameters…
In this study, we derive the exact distributions of eigenvalues of a singular Wishart matrix under an elliptical model. We define generalized heterogeneous hypergeometric functions with two matrix arguments and provide convergence conditions for these functions. The joint density of eigenvalues and the distribution function of the largest eigenvalue for a singular elliptical Wishart matrix are represented by these functions. Numerical computations for the distribution of the largest eigenvalue were conducted under the matrix-variate $t$ and Kotz-type models.
We derive the form of the variance-covariance matrix for any affine equivariant matrix-valued statistics when sampling from complex elliptical distributions. We then use this result to derive the variance-covariance matrix of the sample covariance matrix (SCM) as well as its theoretical mean squared error (MSE) when finite fourth-order moments exist. Finally, illustrative examples of the formulas are presented.
This paper describes the Elliptical Quartic Exponential distribution in $\mathbb{R}^D$, obtained via a maximum entropy construction by imposing second and fourth moment constraints. I discuss relationships to related work, analytical expressions for the normalization constant and the entropy, and the conditional and marginal distributions.
Some ideas studied by Lele (1993), under a Gaussian perturbation model, are generalised in the setting of matrix multivariate elliptical distributions. In particular, several inaccuracies in the published statistical perturbation model are revised. In addition, a number of aspects about identifiability and estimability are also considered. Instead of using the Euclidean distance matrix for proposing consistent estimates, this paper determines exact formulae for the moments of matrix $\mathbf{B} = \mathbf{X}^{c}\left(\mathbf{X}^{c}\right)^{T}$, where $\mathbf{X}^{c}$ is the centered landmarks m…
We address structured covariance estimation in Elliptical distribution. We assume it is a priori known that the covariance belongs to a given convex set, e.g., the set of Toeplitz or banded matrices. We consider the General Method of Moments (GMM) optimization subject to these convex constraints. Unfortunately, GMM is still non-convex due to objective. Instead, we propose COCA - a convex relaxation which can be efficiently solved. We prove that the relaxation is tight in the unconstrained case for a finite number of samples, and in the constrained case asymptotically. We then illustrate the ad…
Although the assumption of elliptical symmetry is quite common in multivariate analysis and widespread in a number of applications, the problem of testing the null hypothesis of ellipticity so far has not been addressed in a fully satisfactory way. Most of the literature in the area indeed addresses the null hypothesis of elliptical symmetry with specified location and actually addresses location rather than non-elliptical alternatives. In this paper, we are proposing new classes of testing procedures, both for specified and unspecified location. The backbone of our construction is Le Cam's as…
Let $X$ be an $M\times N$ random matrix consisting of independent $M$-variate elliptically distributed column vectors $\mathbf{x}_{1},\dots,\mathbf{x}_{N}$ with general population covariance matrix $Σ$. In the literature, the quantity $XX^{*}$ is referred to as the sample covariance matrix after scaling, where $X^{*}$ is the transpose of $X$. In this article, we prove that the limiting behavior of the scaled largest eigenvalue of $XX^{*}$ is universal for a wide class of elliptical distributions, namely, the scaled largest eigenvalue converges weakly to the same limit regardless of the distrib…
Truncated multivariate distributions arise extensively in econometric modelling when non-negative random variables are intrinsic to the data-generation process. More broadly, truncated multivariate distributions have appeared in censored and truncated regression models, simultaneous equations modelling, multivariate regression, and applications going back to the now-classic papers of Amemiya (1974) and Heckman (1976). In some applications of truncated multivariate distributions, there arises the problem of characterizing the distribution through correlation and independence properties of sub-v…
According to a well-known theorem of Cramér and Wold, if $P$ and $Q$ are two Borel probability measures on $\mathbb{R}^d$ whose projections $P_L,Q_L$ onto each line $L$ in $\mathbb{R}^d$ satisfy $P_L=Q_L$, then $P=Q$. Our main result is that, if $P$ and $Q$ are both elliptical distributions, then, to show that $P=Q$, it suffices merely to check that $P_L=Q_L$ for a certain set of $(d^2+d)/2$ lines $L$. Moreover $(d^2+d)/2$ is optimal. The class of elliptical distributions contains the Gaussian distributions as well as many other multivariate distributions of interest. Our theorem contrasts wit…
The sub-Gaussian stable distribution is a heavy-tailed elliptically contoured law which has interesting applications in signal processing and financial mathematics. This work addresses the problem of feasible estimation of distributions. We present a method based on application of the method of moments to the empirical characteristic function. Further, we show almost sure convergence of our estimators, discover their limiting distribution and demonstrate their finite-sample performance.
Multivariate rapid variation describes decay rates of joint light tails of a multivariate distribution. We impose a local uniformity condition to control decay variation of distribution tails along different directions, and using higher-order tail dependence of copulas, we prove that a rapidly varying multivariate density implies rapid variation of the joint distribution tails. As a corollary, rapid variation of skew-elliptical distributions is established under the assumption that the underlying density generators belong to the max-domain of attraction of the Gumbel distribution.
