Comparisons on aggregate risks from two sets of heterogeneous portfolios

In this paper, we stochastically compare the aggregate risks from two heterogeneous portfolios. It is shown that under suitable conditions the more heterogeneities among aggregate risks would result in larger aggregate risks in the sense of the stochastic order. The stochastic properties of aggregate risks when the claims follow proportional hazard rates models or scale models are studied. We also provide sufficient conditions for comparing the aggregate risks arising from two sets of heterogeneous portfolios with claims having gamma distributions. In particular, the aggregate risks of portfolios from dependent samples with comonotonic dependence structures or arrangement increasing density functions are discussed. The new results established strengthen and generalize several results known in the literature including Ma (2000), Khaledi and Ahmadi (2008), Xu and Hu (2011), Xu and Balakrishnan (2011), Pan et al. (2013) and Barmalzan et al. (2015).     

Quadratic Majorization for Nonconvex Loss with Applications to the Boosting Algorithm

Classical robust statistical methods dealing with noisy data are often based on modifications of convex loss functions. In recent years, nonconvex loss-based robust methods have been increasingly popular. A nonconvex loss can provide robust estimation for data contaminated with outliers. The significant challenge is that a nonconvex loss can be numerically difficult to optimize. This article proposes quadratic majorization algorithm for nonconvex (QManc) loss. The QManc can decompose a nonconvex loss into a sequence of simpler optimization problems. Subsequently, the QManc is applied to a powerful machine learning algorithm: quadratic majorization boosting algorithm (QMBA). We develop QMBA for robust classification (binary and multi-category) and regression. In high-dimensional cancer genetics data and simulations, the QMBA is comparable with convex loss-based boosting algorithms for clean data, and outperforms the latter for data contaminated with outliers. The QMBA is also superior to boosting when directly implemented to optimize nonconvex loss functions. Supplementary material for this article is available online.     

Feedback invariants of pairs of matrices and restrictions

If (A,B) εF^n×n×F^×mis a given pair and S is an (A,B)-invariant subspace we investigate the relationship between the feedback invariants of (A, B) and those of its restrictions

to S.     

Mean residual life order of convolutions of heterogeneous exponential random variables

In this paper, we study convolutions of heterogeneous exponential random variables with respect to the mean residual life order. By introducing a new partial order (reciprocal majorization order), we prove that this order between two parameter vectors implies the mean residual life order between convolutions of two heterogeneous exponential samples. For the 2-dimensional case, it is shown that there exists a stronger equivalence. We discuss, in particular, the case when one convolution involves identically distributed variables, and show in this case that the mean residual life order is actually associated with the harmonic mean of parameters. Finally, we derive the “best gamma bounds” for the mean residual life function of any convolution of exponential distributions under this framework.     

On the structure invariants of proper rational matrices with prescribed finite poles

Abstract

The algebraic structure of matrices defined over arbitrary fields whose elements are rational functions with no poles at infinity and prescribed finite poles is studied. Under certain very general conditions, they are shown to be matrices over an Euclidean domain that can be classified according to the corresponding invariant factors. The relationship between these invariants and the local Wiener–Hopf factorization indices will be clarified. This result can be seen as an extension of the classical theorem on pole placement by Rosenbrock in control theory.     

Constrained graph layout by stress majorization and gradient projection

The adoption of the stress-majorization method from multi-dimensional scaling into graph layout has provided an improved mathematical basis and better convergence properties for so-called “force-directed placement” techniques. In this paper we explore algorithms for augmenting such stress-majorization techniques with simple linear constraints using gradient-projection optimization techniques. Our main focus is a particularly simple class of constraints called “orthogonal-ordering constraints” but we also discuss how gradient-projection methods may be extended to solve more general linear “separation constraints”. In addition, we demonstrate several graph-drawing applications where these types of constraints can be very useful.     

Some results on the majorization theorem of connected graphs

Let π = (d ₁, d ₂, ..., d _n ) and π′ = (d′ ₁, d′ ₂, ..., d′ _n ) be two non-increasing degree sequences. We say π is majorizated by π′, denoted by π ⊲ π′, if and only if π ≠ π′, Σ _i=1 ⁿ d _i = Σ _i=1 ⁿ d′ _i , and Σ _i=1 ^j d _i ≤ Σ _i=1 ^j d′ _i for all j = 1, 2, ..., n. Weuse C _π to denote the class of connected graphs with degree sequence π. Let ρ(G) be the spectral radius, i.e., the largest eigenvalue of the adjacent matrix of G. In this paper, we extend the main results of [Liu, M. H., Liu, B. L., You, Z. F.: The majorization theorem of connected graphs. Linear Algebra Appl., 431(1), 553–557 (2009)] and [Bıyıkoğlu, T., Leydold, J.: Graphs with given degree sequence and maximal spectral radius. Electron. J. Combin., 15(1), R119 (2008)]. Moreover, we prove that if π and π′ are two different non-increasing degree sequences of unicyclic graphs with π ⊲ π′, G and G′ are the unicyclic graphs with the greatest spectral radii in C _π and C′ _π , respectively, then ρ(G) < ρ(G′).     

Bounds and a majorization for the real parts of the zeros of polynomials

We apply some eigenvalue inequalities to the real parts of the Frobenius companion matrices of monic polynomials to establish new bounds and a majorization for the real parts of the zeros of these polynomials.

     

On orthogonal projectors induced by compact groups and Haar measures

We study the difference of two orthogonal projectors induced by compact groups of linear operators acting on a vector space. An upper bound for the difference is derived using the Haar measures of the groups. A particular attention is paid to finite groups. Some applications are given for complex matrices and unitarily invariant norms. Majorization inequalities of Fan and Hoffmann and of Causey are rediscovered.