A note on the maximum correlation for Baker’s bivariate distributions with fixed marginals

We investigate Baker’s bivariate distributions with fixed marginals which are based on order statistics, and find conditions under which the correlation converges to the maximum for Fréchet-Hoeffding upper bound as the sample size tends to infinity. The convergence rate of the correlation is also investigated for some specific cases.     

Construction of multivariate distributions with given marginals

Summary We make some remarks on the problem how to construct probability measures with given marginals. Questions of this kind arise if one wants to build a stochastic model in a situation where one has some idea of the kind of dependence and knows exactly certain marginal distributions.     

Characterizations of multivariate life distributions

Characterizations of multivariate distributions has been a topic of great interest in applied statistics literature for the last three decades. In this paper, we develop characterizations of multivariate lifetime distributions by relationship between multivariate failure rates (reversed failure rates) and the left (right) truncated expectations of functions of random variables. We, then, discuss the application of the results to derive a multivariate Stein type identity.     

Some notes on multivariate generalized Pareto distributions

The investigation of multivariate generalized Pareto distributions (GPDs) has begun only recently and there are slightly varying definitions of GPDs available. In this article we investigate the one from Section 5.1 of Falk et al. [Laws of Small Numbers: Extremes and Rare Events, second ed., Birkhäuser, Basel, 2004], which does not differ in the area of interest from those of other authors. We first give an interpretation of the case of independence in terms of the peaks-over-threshold approach. This case is also used in dimension d=3 by Falk et al. [Laws of Small Numbers: Extremes and Rare Events, second ed., Birkhäuser, Basel, 2004] as a counterexample to show that GP functions are not necessarily distribution functions on their entire support. We generalize this counterexample to an arbitrary dimension d≥3 and demonstrate also that other GP functions show this behavior. Finally we show that different GPDs can lead to the same conditional probability measure in the area of interest.     

Variance function estimation in multivariate nonparametric regression with fixed design

Variance function estimation in multivariate nonparametric regression is considered and the minimax rate of convergence is established in the iid Gaussian case. Our work uses the approach that generalizes the one used in [A. Munk, Bissantz, T. Wagner, G. Freitag, On difference based variance estimation in nonparametric regression when the covariate is high dimensional, J. R. Stat. Soc. B 67 (Part 1) (2005) 19-41] for the constant variance case. As is the case when the number of dimensions d=1, and very much contrary to standard thinking, it is often not desirable to base the estimator of the variance function on the residuals from an optimal estimator of the mean. Instead it is desirable to use estimators of the mean with minimal bias. Another important conclusion is that the first order difference based estimator that achieves minimax rate of convergence in the one-dimensional case does not do the same in the high dimensional case. Instead, the optimal order of differences depends on the number of dimensions.     

Structural decompositions of multivariate distributions with applications in moment and cumulant

We provide lattice decompositions for multivariate distributions. The lattice decompositions reveal the structural relationship between the Lancaster/Bahadur model and the model of Streitberg (Ann. Statist. 18 (1990) 1878). For multivariate categorical data, the decompositions allows modeling strategy for marginal inference. The theory discussed in this paper illustrates the concept of reproducibility, which was discussed in Liang et al. (J. Roy. Statist. Soc. Ser. B 54 (1992) 3). For the purpose of delineating the relationship between the various types of decompositions of distributions, we develop a theory of polytypefication, the generality of which is exploited to prove results beyond interaction.     

A reliability model for multivariate exponential distributions

In this paper, we consider a counting process approach for characterizing a system having dependent component failure rates. We study the transient state probabilities and related reliability properties based on a series of Poisson shocks. We also show that the proposed infinitesimal generator representation can be used to characterize the bivariate exponential distributions of Freund, Marshall-Olkin, Block-Basu and Friday-Patil.     

Maximum entropy characterizations of the multivariate Liouville distributions

A random vector X=(X₁,X₂,…,X_n) with positive components has a Liouville distribution with parameter θ=(θ₁,θ₂,…,θ_n) if its joint probability density function is proportional to , θ_i>0 [R.D. Gupta, D.S.P. Richards, Multivariate Liouville distributions, J. Multivariate Anal. 23 (1987) 233-256]. Examples include correlated gamma variables, Dirichlet and inverted Dirichlet distributions. We derive appropriate constraints which establish the maximum entropy characterization of the Liouville distributions among all multivariate distributions. Matrix analogs of the Liouville distributions are considered. Some interesting results related to I-projection from a Liouville distribution are presented.     

Pseudo-inverse multivariate/matrix-variate distributions

The Moore-Penrose inverse of a singular or nonsquare matrix is not only existent but also unique. In this paper, we derive the Jacobian of the transformation from such a matrix to the transpose of its Moore-Penrose inverse. Using this Jacobian, we investigate the distribution of the Moore-Penrose inverse of a random matrix and propose the notion of pseudo-inverse multivariate/matrix-variate distributions. For arbitrary multivariate or matrix-variate distributions, we can develop the corresponding pseudo-inverse distributions. In particular, we present pseudo-inverse multivariate normal distributions, pseudo-inverse Dirichlet distributions, pseudo-inverse matrix-variate normal distributions and pseudo-inverse Wishart distributions.     

Orthant tail dependence of multivariate extreme value distributions

The orthant tail dependence describes the relative deviation of upper- (or lower-) orthant tail probabilities of a random vector from similar orthant tail probabilities of a subset of its components, and can be used in the study of dependence among extreme values. Using the conditional approach, this paper examines the extremal dependence properties of multivariate extreme value distributions and their scale mixtures, and derives the explicit expressions of orthant tail dependence parameters for these distributions. Properties of the tail dependence parameters, including their relations with other extremal dependence measures used in the literature, are discussed. Various examples involving multivariate exponential, multivariate logistic distributions and copulas of Archimedean type are presented to illustrate the results.     

Canonical representation of conditionally specified multivariate discrete distributions

Most work on conditionally specified distributions has focused on approaches that operate on the probability space, and the constraints on the probability space often make the study of their properties challenging. We propose decomposing both the joint and conditional discrete distributions into characterizing sets of canonical interactions, and we prove that certain interactions of a joint distribution are shared with its conditional distributions. This invariance opens the door for checking the compatibility between conditional distributions involving the same set of variables. We formulate necessary and sufficient conditions for the existence and uniqueness of discrete conditional models, and we show how a joint distribution can be easily computed from the pool of interactions collected from the conditional distributions. Hence, the methods can be used to calculate the exact distribution of a Gibbs sampler. Furthermore, issues such as how near compatibility can be reconciled are also discussed. Using mixed parametrization, we show that the proposed approach is based on the canonical parameters, while the conventional approaches are based on the mean parameters. Our advantage is partly due to the invariance that holds only for the canonical parameters.     

A large deviation limit theorem for multivariate distributions

A local limit theorem for large deviations of $\text{o(n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, where n is the sample size, is developed for multivariate statistics which are more general than standardised means, but which depend on n in much the same way. In particular, the cumulants of the statistic are of the same order in $\text{n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$ as those of a standardised mean. The theory is derived under conditions which correspond to those in earlier work by Richter on limit theorems for standardised means and by Chambers on the validity of Edgeworth expansions for multivariate statistics.     

Peaks-over-threshold stability of multivariate generalized Pareto distributions

It is well-known that the univariate generalized Pareto distributions (GPD) are characterized by their peaks-over-threshold (POT) stability. We extend this result to multivariate GPDs.It is also shown that this POT stability is asymptotically shared by distributions which are in a certain neighborhood of a multivariate GPD. A multivariate extreme value distribution is a typical example.The usefulness of the results is demonstrated by various applications. We immediately obtain, for example, that the excess distribution of a linear portfolio with positive weights a_i, i≤d, is independent of the weights, if (U₁,…,U_d) follows a multivariate GPD with identical univariate polynomial or Pareto margins, which was established by Macke [On the distribution of linear combinations of multivariate EVD and GPD distributed random vectors with an application to the expected shortfall of portfolios, Diploma Thesis, University of Würzburg, 2004, (in German)] and Falk and Michel [Testing for tail independence in extreme value models. Ann. Inst. Statist. Math. 58 (2006) 261-290]. This implies, for instance, that the expected shortfall as a measure of risk fails in this case.     

Three general multivariate semi-Pareto distributions and their characterizations

Three general multivariate semi-Pareto distributions are developed in this paper. First one—GMP^(k)(III) has univariate Pareto (III) marginals, it is characterized by the minimum of two independent and identically distributed random vectors. Second one—GMSP has univariate semi-Pareto marginals and it is characterized by finite sample minima. Third one—MSP is characterized through a geometric minimization procedure. All these three characterizations are based on the general and the particular solutions of the Euler's functional equations of k-variates.     

Third-order power comparisons for a class of tests for multivariate linear hypothesis under general distributions

The purpose of this paper is, in multivariate linear regression model (Part I) and GMANOVA model (Part II), to investigate the effect of nonnormality upon the nonnull distributions of some multivariate test statistics under normality. It is shown that whatever the underlying distributions, the difference of local powers up to order N⁻¹ after either Bartlett’s type adjustment or Cornish-Fisher’s type size adjustment under nonnormality coincides with that in Anderson [An Introduction to Multivariate Statistical Analysis, 2nd ed. and 3rd ed., Wiley, New York, 1984, 2003] under normality. The derivation of asymptotic expansions is based on the differential operator associated with the multivariate linear regression model under general distributions. The performance of higher-order results in finite samples, including monotone Bartlett’s type adjustment and monotone Cornish-Fisher’s type size adjustment, is examined using simulation studies.     

A class of weighted multivariate normal distributions and its properties

This article proposes a class of weighted multivariate normal distributions whose probability density function has the form of a product of a multivariate normal density and a weighting function. The class is obtained from marginal distributions of various doubly truncated multivariate normal distributions. The class strictly includes the multivariate normal and multivariate skew-normal. It is useful for selection modeling and inequality constrained normal mean vector analysis. We report on a study of some distributional properties and the Bayesian perspective of the class. A probabilistic representation of the distributions is also given. The representation is shown to be straightforward to specify the distribution and to implement computation, with output readily adapted for the required analysis. Necessary theories and illustrative examples are provided.     

Some recent developments on complex multivariate distributions

In this paper, the author gives a review of the literature on complex multivariate distributions. Some new results on these distributions are also given. Finally, the author discusses the applications of the complex multivariate distributions in the area of the inference on multiple time series.     

Admissible estimator of the eigenvalues of the variance-covariance matrix for multivariate normal distributions

An admissible estimator of the eigenvalues of the variance-covariance matrix is given for multivariate normal distributions with respect to the scale-invariant squared error loss.     

Asymptotic expansions for distributions of latent roots in multivariate analysis

Asymptotic expansions are given for the distributions of latent roots of matrices in three multivariate situations. The distribution of the roots of the matrix S₁(S₁ + S₂)^?1, where S₁ is $\text{W}{}_{\mathrm{<mtext>m</mtext>}}\text{(n}{}_1\text{, Σ, Ω)}$ and S₂ is W_m(n₂, Σ), is studied in detail and asymptotic series for the distribution are obtained which are valid for some or all of the roots of the noncentrality matrix Ω large. These expansions are obtained using partial-differential equations satisfied by the distribution. Asymptotic series are also obtained for the distributions of the roots of n^?1S, where S in W_m(n, Σ), for large n, and S₁S₂^?1, where S₁ is W_m(n₁, Σ) and S₂ is W_m(n₂, Σ), for large n₁ + n₂.