Characterizations of bivariate models using dynamic Kullback-Leibler discrimination measures

In this paper, the residual Kullback-Leibler discrimination information measure is extended to conditionally specified models. The extension is used to characterize some bivariate distributions. These distributions are also characterized in terms of proportional hazard rate models and weighted distributions. Moreover, we also obtain some bounds for this dynamic discrimination function by using the likelihood ratio order and some preceding results.     

Multivariate maximum entropy identification, transformation, and dependence

This paper shows that multivariate distributions can be characterized as maximum entropy (ME) models based on the well-known general representation of density function of the ME distribution subject to moment constraints. In this approach, the problem of ME characterization simplifies to the problem of representing the multivariate density in the ME form, hence there is no need for case-by-case proofs by calculus of variations or other methods. The main vehicle for this ME characterization approach is the information distinguishability relationship, which extends to the multivariate case. Results are also formulated that encapsulate implications of the multiplication rule of probability and the entropy transformation formula for ME characterization. The dependence structure of multivariate ME distribution in terms of the moments and the support of distribution is studied. The relationships of ME distributions with the exponential family and with bivariate distributions having exponential family conditionals are explored. Applications include new ME characterizations of many bivariate distributions, including some singular distributions.     

Nonlinear weighted least squares estimation of a three-parameter Weibull density with a nonparametric start

This paper is concerned with the parameter estimation problem for the three-parameter Weibull density which is widely employed as a model in reliability and lifetime studies. Our approach is a combination of nonparametric and parametric methods. The basic idea is to start with an initial nonparametric density estimate which needs to be as good as possible, and then apply the nonlinear least squares method to estimate the unknown parameters. As a main result, a theorem on the existence of the least squares estimate is obtained. Some simulations are given to show that our approach is satisfactory if the initial density is of good enough quality.     

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.     

On the weighted multivariate Wilcoxon rank regression estimate

Zhou (2010) introduced a multivariate Wilcoxon regression estimate which possesses some nice properties: computational ease, asymptotic normality and high efficiency. However, it is sensitive to the leverage points. To circumvent this problem, we propose a weighted multivariate Wilcoxon regression estimate. Under some regularity conditions, the asymptotic normality is established. We further study the robustness of the proposed estimate through the influence function. By properly choosing the weight functions, our results show that the corresponding estimate can have bounded influence function on both response and covariates.     

On fundamental skew distributions

A new class of multivariate skew-normal distributions, fundamental skew-normal distributions and their canonical version, is developed. It contains the product of independent univariate skew-normal distributions as a special case. Stochastic representations and other main properties of the associated distribution theory of linear and quadratic forms are considered. A unified procedure for extending this class to other families of skew distributions such as the fundamental skew-symmetric, fundamental skew-elliptical, and fundamental skew-spherical class of distributions is also discussed.     

On the spectrum of resolvable orthogonal arrays invariant under the Klein group K4

It is shown that there exists a resolvablen ² by 4 orthogonal array which is invariant under the Klein 4-groupK ₄ for all positive integersn congruent to 0 modulo 4 except possibly forn {12, 24, 156, 348}.     

On efficient estimators of two seemingly unrelated regressions

In this paper, following the results presented in Liu’s work [Liu, A.Y., 2002. Efficient estimation of two seemingly unrelated regression equations. Journal of Multivariate Analysis 82, 445-456], we first represent the Gauss-Markov estimator of the regression parameter as a matrix series, and hence we conclude that the observation vectors should appear in any efficient estimator in pairs. Second, we prove that the simpler form of the two-stage Aitken estimator is unique. Finally we generalize our results to the system of two seemingly unrelated regressions with unequal numbers of observations and briefly summarize our conclusions.     

Nonexistence of complete sufficient statistics for stationaryk-state Markov chains,k≧3

It is shown that there does not exist a complete sufficient statistic for the class of allk-state Markov chains,k≧3. Whenk=2 there is a complete sufficient statistic.     

On the minimization of multinomial tails and the Gupta-Nagel conjecture

This paper is primarily concerned with the open problem of minimizing the lower tail of the multinomial distribution. During the study of that specific problem, we have developed an approach which reveals itself useful for solving a general class of problems involving multinomial probabilities. Concerning the main problem, we provide a self-contained proof that the minimum of the multinomial lower tail is reached, as conjectured by Gupta and Nagel (Sankhya Ser. B 29 (1967) 1) (within the framework of subset-selection problems) at the equal probability configuration, i.e., when the cell probabilities are equal to one another. We also point out some novel inequalities and general properties involving multinomial probabilities and multinomial coefficients.     

A generalized Hollander-Proschan type test for NBUE alternatives

In this note we develop a family of test statistics for testing exponentiality against NBUE alternatives. The asymptotic distribution of the test statistics is derived. The test statistics are shown to be asymptotically normal and consistent. This family of test statistics includes the test proposed by Hollander and Proschan (1975) as a special case. Efficiency studies have also been done.     

On the singularity of multivariate skew-symmetric models

In recent years, the skew-normal models introduced by Azzalini (1985) [1]-and their multivariate generalizations from Azzalini and Dalla Valle (1996) [4]-have enjoyed an amazing success, although an important literature has reported that they exhibit, in the vicinity of symmetry, singular Fisher information matrices and stationary points in the profile log-likelihood function for skewness, with the usual unpleasant consequences for inference. It has been shown (DiCiccio and Monti (2004) [23], DiCiccio and Monti (2009) [24] and Gómez et al. (2007) [25]) that these singularities, in some specific parametric extensions of skew-normal models (such as the classes of skew-t or skew-exponential power distributions), appear at skew-normal distributions only. Yet, an important question remains open: in broader semiparametric models of skewed distributions (such as the general skew-symmetric and skew-elliptical ones), which symmetric kernels lead to such singularities? The present paper provides an answer to this question. In very general (possibly multivariate) skew-symmetric models, we characterize, for each possible value of the rank of Fisher information matrices, the class of symmetric kernels achieving the corresponding rank. Our results show that, for strictly multivariate skew-symmetric models, not only Gaussian kernels yield singular Fisher information matrices. In contrast, we prove that systematic stationary points in the profile log-likelihood functions are obtained for (multi)normal kernels only. Finally, we also discuss the implications of such singularities on inference.     

On the mode-change problem for random measures

The classical change-point problem in modern terms, i.e., the mode-change problem, is stated for sufficiently general set-indexed random processes, namely for random measures. A method is shown for solving this problem both in the general form and for the intensity of compound Poisson random measures. The results obtained are novel for the change-point problem, too.     

On the unlinking conjecture of independent polynomial functions

The celebrated U-conjecture states that under the N_n(0,I_n) distribution of the random vector X=(X₁,…,X_n) in Rⁿ, two polynomials P(X) and Q(X) are unlinkable if they are independent [see Kagan et al., Characterization Problems in Mathematical Statistics, Wiley, New York, 1973]. Some results have been established in this direction, although the original conjecture is yet to be proved in generality. Here, we demonstrate that the conjecture is true in an important special case of the above, where P and Q are convex nonnegative polynomials with P(0)=0.     

On the spanning trees of weighted graphs

Given a weighted graph, letW ₁,W ₂,W ₃,... denote the increasing sequence of all possible distinct spanning tree weights. Settling a conjecture due to Kano, we prove that every spanning tree of weightW ₁ is at mostk–1 edge swaps away from some spanning tree of weightW _k . Three other conjectures posed by Kano are proven for two special classes of graphs. Finally, we consider the algorithmic complexity of generating a spanning tree of weightW _k .This work was supported in part by a grant from the AT&T foundation and NSF grant DCR-8351757.Primarily supported by a 1967 Science and Engineering Scholarship from the Natural Sciences and Engineering Research Council of Canada.     

Constructions for resolvable and related designs

Methods are given for constructing block designs, using resolvable designs. These constructions yield methods for generating resolvable and affine designs and also affine designs with affine duals. The latter are transversal designs or semi-regular group divisible designs with ₁=0 whose duals are also designs of the same type and parameters. The paper is a survey of some old and some recent constructions.     

The polytopal association scheme

Summary The polytopal association scheme for PBIB designs is introduced and studied utilizing the concept of clustering of treatments.     

A note on partially balanced ternary designs

Two new methods of constructing a series of partially balanced ternary designs are presented. One from a BIB design and a PBIB design, and the second from a PBIB design alone, obtained by method of differences in both the cases.