Bayesian Bandwidths in Semiparametric Modelling for Nonnegative Orthant Data with Diagnostics

Multivariate nonnegative orthant data are real vectors bounded to the left by the null vector, and they can be continuous, discrete or mixed. We first review the recent relative variability indexes for multivariate nonnegative continuous and count distributions. As a prelude, the classification of two comparable distributions having the same mean vector is done through under-, equi- and over-variability with respect to the reference distribution. Multivariate associated kernel estimators are then reviewed with new proposals that can accommodate any nonnegative orthant dataset. We focus on bandwidth matrix selections by adaptive and local Bayesian methods for semicontinuous and counting supports, respectively. We finally introduce a flexible semiparametric approach for estimating all these distributions on nonnegative supports. The corresponding estimator is directed by a given parametric part, and a nonparametric part which is a weight function to be estimated through multivariate associated kernels. A diagnostic model is also discussed to make an appropriate choice between the parametric, semiparametric and nonparametric approaches. The retention of pure nonparametric means the inconvenience of parametric part used in the modelization. Multivariate real data examples in semicontinuous setup as reliability are gradually considered to illustrate the proposed approach. Concluding remarks are made for extension to other multiple functions.

Karamardian Matrices: An Analogue of Q-Matrices

A real square matrix $A$ is called a $Q$-matrix if the linear complementarity problem LCP$(A,q)$ has a solution for all $q \in \mathbb{R}^n$. This means that for every vector $q$ there exists a vector $x$ such that $x \geq 0, y=Ax+q\geq 0$, and $x^Ty=0$. A well-known result of Karamardian states that if the problems LCP$(A,0)$ and LCP$(A,d)$ for some $d\in \mathbb{R}^n, d >0$ have only the zero solution, then $A$ is a $Q$-matrix. Upon relaxing the requirement on the vectors $d$ and $y$ so that the vector $y$ belongs to the translation of the nonnegative orthant by the null space of $A^T$, $d$ belongs to its interior, and imposing the additional condition on the solution vector $x$ to be in the intersection of the range space of $A$ with the nonnegative orthant, in the two problems as above, the authors introduce a new class of matrices called Karamardian matrices, wherein these two modified problems have only zero as a solution. In this article, a systematic treatment of these matrices is undertaken. Among other things, it is shown how Karamardian matrices have properties that are analogous to those of $Q$-matrices. A subclass of a recently introduced notion of $P_{\#}$-matrices is shown to possess the Karamardian property, and for this reason we undertake a thorough study of $P_{\#}$-matrices and make some fundamental contributions.

A new family of positive recurrent semimartingale reflecting Brownian motions in an orthant

Semimartingale reflecting Brownian motions (SRBMs) are diffusion processes, which arise as approximations for open d-station queueing networks of various kinds. The data for such a process are a drift vector θ, a nonsingular {d\times d} covariance matrix Δ, and a {d\times d} reflection matrix R. The state space is the d-dimensional nonnegative orthant, in the interior of which the processes evolve according to a Brownian motions, and that reflect against the boundary in a specified manner. A standard problem is to determine under what conditions the process is positive recurrent. Necessary and sufficient conditions are formulated for some classes of reflection matrices and in two- and three-dimensional cases, but not more. In this work, we identify a new family of reflection matrices R for which the process is positive recurrent if and only if the drift {\theta\in\mathring{\Gamma}}, where {\mathring{\Gamma}} is the interior of the convex wedge generated by the opposite column vectors of R.

Matrices of minimum norm satisfying certain prescribed band and spectral restrictions -- an extremal characterization of the discrete periodic Laplacian

This paper has been motivated by the curiosity that the circulant matrix ${\rm Circ }(1/2, -1/4, 0, \dots, 0,-1/4)$ is the $n\times n$ positive semidefinite, tridiagonal matrix $A$ of smallest Euclidean norm having the property that $Ae = 0$ and $Af = f$, where $e$ and $f$ are, respectively, the vector of all $1$s and the vector of alternating $1$ and $-1$s. It then raises the following question (minimization problem): What should be the matrix $A$ if the tridiagonal restriction is replaced by a general bandwidth $2r + 1$ ($1\leq r \leq \tfrac{n}{2 } -1$)? It is first easily shown that the solution of this problem must still be a circulant matrix. Then the determination of the first row of this circulant matrix consists in solving a least-squares problem having $\tfrac{n}{2} \, - 1$ nonnegative variables (Nonnegative Orthant) subject to $\tfrac{n}{2} - r$ linear equations. Alternatively, this problem can be viewed as the minimization of the norm of an even function vanishing at the points $|i|>r$ of the set $\left\{-\tfrac{n}{2} + 1, \dots, -1, 0, 1, \dots ,\tfrac{n}{2} \right\}$, and whose Fourier-transform is nonnegative, vanishes at zero, and assumes the value one at $\tfrac{n}{2}$. Explicit solutions are given for the special cases of $r=\tfrac{n}{2}$, $r=\tfrac{n}{2} -1$, and $r=2$. The solution for the particular case of $r=2$ can be physically interpreted as the vibrational mode of a ring-like chain of masses and springs in which the springs link both the nearest neighbors (with positive stiffness) and the next-nearest neighbors (with negative stiffness). The paper ends wiih a numerical illustration of the six cases ($1\leq r \leq 6$)corresponding to $n=12$.