Showing papers by "Barry C. Arnold published in 2018"

Analytic Expressions for Multivariate Lorenz Surfaces

Abstract: The Lorenz curve is a much used instrument in economic analysis. It is typically used for measuring inequality and concentration. In insurance, it is used to compare the riskiness of portfolios, to order reinsurance contracts and to summarize relativity scores (see Frees et al. J. Am. Statist. Assoc.106, 1085–1098, 2011; J. Risk Insur.81, 335–366, 2014 and Samanthi et al. Insur. Math. Econ.68, 84–91, 2016). It is sometimes called a concentration curve and, with this designation, it attracted the attention of Mahalanobis (Econometrica28, 335–351, 1960) in his well known paper on fractile graphical analysis. The extension of the Lorenz curve to higher dimensions is not a simple task. There are three proposed definitions for a suitable Lorenz surface, proposed by Taguchi (Ann. Inst. Statist. Math.24, 355–382, 1972a, 599–619, 1972b; Comput. Stat. Data Anal.6, 307–334, 1988) and Lunetta (1972), Arnold (1987, 2015) and Koshevoy and Mosler (J. Am. Statist. Assoc.91, 873–882, 1996). In this paper, using the definition proposed by Arnold (1987, 2015), we obtain analytic expressions for many multivariate Lorenz surfaces. We consider two general classes of models. The first is based on mixtures of Lorenz surfaces and the second one is based on some simple classes of bivariate mixture distributions.

The power piecewise exponential model

Abstract: In this paper an extension of the piecewise exponential distribution based on the distribution of the maximum of a random sample is considered. Properties of its density and hazard function are investigated. Maximum likelihood inference is discussed and the Fisher information matrix is identified. Results of two real data applications are reported, where model fitting is implemented by using maximum likelihood. The applications illustrate the better performance of the new distribution when compared with other recently proposed alternative models.

On Parameter Dependence and Related Topics: The Impact of Jerzy Filus from Genesis to Recent Developments

Abstract: This chapter discusses via an interview with Jerzy Filus about his career from his early years in Poland all the way up to today, we will unfold and explore the genesis and development of multivariate pseudonormal distributions, parameter dependence, and finally stochastic dependence in general. As a kind of fusion of the two approaches, the common paper Filus, Filus, and Arnold demonstrated a method of extension of the bivariate normal density where only six parameters were needed; in the Arnold et al. approach, eight parameters were needed as a minimum. In Filus and Filus, the authors gave a reliability motivation for pseudonormal distributions using the parameter dependence paradigm. In this area, Jerzy Filus and coauthors have made substantial recent developments on this topic and propose an interesting new model. In Filus, Filus and Stehlik, a statistical study was conducted related to bivariate pseudoexponential distribution via its survival function, which allows to model other multiple failures.

Majorization in \(\mathbb {R}_n^+\)

Abstract: As mentioned in Chap. 1, the name majorization seems to have appeared first in HLP (1959) , The, idea had appeared earlier (HLP, 1929) although unchristened. Muirhead who dealt with \(\mathbb {Z}_n^+\) (i.e., vectors of nonnegative integers) already had identified the partial order defined in ( 1.1) (i.e., majorization). But he, when he needed to refer to it, merely called it “ordering.” Perhaps it took the insight of HLP to recognize that little of Muirhead ’s work need necessarily be restricted to integers, but the key ideas including Dalton’s transfer principle were already present in Muirhead’s paper . If there was anything lacking in Muirhead’s development, it was motivation for the novel results he obtained. He did exhibit the arithmetic-geometric mean inequality as an example of his general results, but proofs of that inequality are legion. If that was the only use of his “inequalities of symmetric algebraic functions of n letters,” then they might well remain buried in the Edinburgh proceedings. HLP effectively rescued Muirhead’s work from such potential obscurity. In the present book theorems will be stated in generality comparable to that achieved by HLP and will be ascribed to those authors. Muirhead’s priority will not be repeatedly asserted. HLP restricted attention to \(\mathbb {R}_n^+\), but the restriction to the positive orthant can and will be often dispensed with. First let us establish the relationship between majorization as defined by HLP (i.e., ( 1.1)) and averaging as defined by Schur. Recall that \( \underline {x}\) is an average of \( \underline {y}\) in the Schur sense if \( \underline {x}=P \underline {y}\) for some doubly stochastic matrix P.