Canonical correlation analysis based on information theory

In this article, we propose a new canonical correlation method based on information theory. This method examines potential nonlinear relationships between p×1 vector Y-set and q×1 vector X-set. It finds canonical coefficient vectors a and b by maximizing a more general measure, the mutual information, between a^TX and b^TY. We use a permutation test to determine the pairs of the new canonical correlation variates, which requires no specific distributions for X and Y as long as one can estimate the densities of a^TX and b^TY nonparametrically. Examples illustrating the new method are presented.     

Stability of best rational Chebyshev approximation

In this paper we consider best Chebyshev approximation to continuous functions by generalized rational functions using an optimization theoretical approach introduced in [[5.]]. This general approach includes, in a unified way, usual, weighted, one-sided, unsymmetric, and also more general rational Chebychev approximation problems with side-conditions. We derive various continuity conditions for the optimal value, for the feasible set, and the optimal set of the corresponding optimization problem. From these results we derive conditions for the upper semicontinuity of the metric projection, which include some of the results of Werner [On the rational Tschebyscheff operator, Math. Z. 86 (1964), 317–326] and Cheney and Loeb [On the continuity of rational approximation operators, Arch. Rational Mech. Anal. 21 (1966), 391–401].     

On equivalence of moduli of smoothness of polynomials in ,

It is well known that for functions , 1p∞. For general functions fL_p, it does not hold for 0<p<1, and its inverse is not true for any p in general. It has been shown in the literature, however, that for certain classes of functions the inverse is true, and the terms in the inequalities are all equivalent. Recently, Zhou and Zhou proved the equivalence for polynomials with p=∞. Using a technique by Ditzian, Hristov and Ivanov, we give a simpler proof to their result and extend it to the L_p space for 0<p∞. We then show its analogues for the Ditzian–Totik modulus of smoothness and the weighted Ditzian–Totik modulus of smoothness for polynomials with .     

A Wide Class of Ultrabornological Spaces of Measurable Functions

Our main result states that a bornological locally convex space having a suitable Boolean algebra of projections is ultrabornological. This general theorem, whose proof is a variation of the sliding-hump techniques used in [Díaz et al., Arch. Math. (Basel)60 (1993), 73-78; Díaz et al., Resultate Math.23 (1993), 242-250; Drewnowski el al., Proc. Amer. Math. Sec.114 (1992), 687-694; Drewnowski et al., Atti. Sem. Mat. Fis. Univ. Modena41 (1993), 317-329], is applied to prove that some non-complete normed spaces such as the spaces of Dunford, Pettis, or McShane integrable functions, as well as other interesting spaces of weakly or strongly measurable functions, are ultrabornological. We also give applications to vector-valued sequence spaces; in particular, we prove that ℓ^p{X} (1 ≤ p < ∞) is an ultrabornological DF-space when X is.     

The central limit theorem on spaces of positive definite matrices

A central limit theorem is obtained for orthogonally invariant random variables on _n, the space of n × n real, positive definite symmetric matrices. The derivation requires the Taylor expansion of the spherical functions for the general linear group GL(n, R). This extends from the case n = 3 a result of Terras (J. Multivariate Anal. 23 (1987), 13–36).     

Noninformative Priors for Multivariate Linear Calibration

This paper derives a class of first order probability matching priors and a complete catalog of the reference priors for the general multivariate linear calibration problem. In an important special case, a complete characterization of first order probability matching priors is given, and a fairly general class of second order probability matching priors is also provided. Orthogonal transformations (1987, D. R. Cox and N. Reid, J. Roy. Statist. Soc. Ser. B49, 1–18) are found to facilitate the derivations. It turns out that under orthogonal parameterization, reference priors (including Jeffreys' prior) are first order probability matching priors for unidimensional multivariate linear calibration. Also, in univariate linear calibration, the prior of W. G. Hunter and W. F. Lamboy (1981, Technometrics23, 323–350) is a second order probability matching prior.     

Estimation of multivariate binary density using orthogonal functions

In this paper, the authors studied certain properties of the estimate of Liang and Krishnaiah (1985, J. Multivariate Anal. 16, 162–172) for multivariate binary density. An alternative shrinkage estimate is also obtained. The above results are generalized to general orthonormal systems.     

Copulae of probability measures on product spaces

It has been proved (Sklar, 1959, Publ. Inst. Statist. Univ. Paris 8 229–231) that any multivariate distribution function depends on its arguments only through its marginal distributions. An analogous result will be proved in the general framework of probability measures on (Polish) product spaces. Many properties, holding for distribution functions, still hold in the more general situation. Some results related to convergence in probability will be examined.     

Approximation ofk-Monotone Functions

It is shown that an algebraic polynomial of degree k−1 which interpolates ak-monotone functionfatkpoints, sufficiently approximates it, even if the points of interpolation are close to each other. It is well known that this result is not true in general for non-k-monotone functions. As an application, we prove a (positive) result on simultaneous approximation of ak-monotone function and its derivatives inL_p, 0<p<1, metric, and also show that the rate of the best algebraic approximation ofk-monotone functions (with bounded (k−2)nd derivatives inL_p, 1<p<∞, iso(n^−k/p).     

Some new approaches to multivariate probability distributions

We extend and generalize to the multivariate set-up our earlier investigations related to expected remaining life functions and general hazard measures including representations and stability theorems for arbitrary probability distributions in terms of these concepts. (The univariate case is discussed in detail in Kotz and Shanbhag, Advan. Appl. Probab. 12 (1980), 903–921.)