On s-Convex Approximations

Let Bs([a,b];μ1,μ2,...,μs-1) be the class of all distribution functions of random variables with support in [a,b] having μ1,μ2,...,μs-1as their first s-1 moments. In this paper we examine some aspects of the structure of Bs([a,b];μ1,μ2,...,μs-1) and of the s-convex stochastic extrema in it. Using re...

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Bibliographische Detailangaben
Veröffentlicht in:	Advances in Applied Probability. - Applied Probability Trust. - 32(2000), 4, Seite 994-1010
1. Verfasser:	Denuit, Michel (VerfasserIn)
Weitere Verfasser:	Lefèvre, Claude, Shaked, Moshe
Format:	Online-Aufsatz
Sprache:	English
Veröffentlicht:	2000
Zugriff auf das übergeordnete Werk:	Advances in Applied Probability
Schlagworte:	Moment spaces s-convex orders Stochastic extrema Stochastic approximations Tchebycheff-type inequalities Risk theory Lundberg's coefficient Life insurance Mathematics


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520			\|a Let Bs([a,b];μ1,μ2,...,μs-1) be the class of all distribution functions of random variables with support in [a,b] having μ1,μ2,...,μs-1as their first s-1 moments. In this paper we examine some aspects of the structure of Bs([a,b];μ1,μ2,...,μs-1) and of the s-convex stochastic extrema in it. Using representation results of moment matrices à la Lindsay (1989a), we provide conditions for the admissibility of moment sequences in Bs([a,b];μ1,μ2,...,μs-1) in terms of lower bounds on the number of support points of the corresponding distribution functions. We point out two special distributions with a minimal number of support points that are the s-convex extremal distributions. It is shown that the support points of these extrema are the roots of some polynomials, and an efficient method for the complete determination of the distribution functions of these extrema is described. A study of the goodness of fit, of the approximation of an arbitrary element in Bs([a,b];μ1,μ2,...,μs-1) by one of the stochastic s-convex extrema, is then given. Using standard ideas from linear regression, we derive Tchebycheff-type inequalities which extend previous results of Lindsay (1989b), and we establish some limit theorems involving the moment matrices. Finally, we describe some applications in insurance theory, namely, we provide bounds on Lundberg's coefficient in risk theory, and on the actual interest rate relating to a life insurance policy. These bounds are obtained with the aid of the s-convex extrema, and are determined only by the support and the first few moments of the underlying distribution.
540			\|a Copyright 2000 Applied Probability Trust
650		4	\|a Moment spaces
650		4	\|a s-convex orders
650		4	\|a Stochastic extrema
650		4	\|a Stochastic approximations
650		4	\|a Tchebycheff-type inequalities
650		4	\|a Risk theory
650		4	\|a Lundberg's coefficient
650		4	\|a Life insurance
650		4	\|a Mathematics \|x Pure mathematics \|x Probability theory \|x Random variables
650		4	\|a Mathematics \|x Applied mathematics \|x Statistics \|x Applied statistics \|x Descriptive statistics \|x Statistical distributions \|x Distribution functions \|x Probability distributions \|x Mathematical moments
650		4	\|a Mathematics \|x Applied mathematics \|x Statistics \|x Applied statistics \|x Descriptive statistics \|x Statistical distributions \|x Distribution functions \|x Probability distributions \|x Cumulative distribution functions
650		4	\|a Mathematics \|x Mathematical values \|x Critical values \|x Extrema
650		4	\|a Mathematics \|x Mathematical procedures \|x Approximation
650		4	\|a Mathematics \|x Pure mathematics \|x Algebra \|x Polynomials
650		4	\|a Mathematics \|x Mathematical values \|x Equation roots
650		4	\|a Mathematics \|x Pure mathematics \|x Algebra \|x Coefficients
650		4	\|a Mathematics \|x Pure mathematics \|x Discrete mathematics \|x Number theory \|x Numbers \|x Real numbers \|x Rational numbers \|x Integers \|x General Applied Probability
655		4	\|a research-article
700	1		\|a Lefèvre, Claude \|e verfasserin \|4 aut
700	1		\|a Shaked, Moshe \|e verfasserin \|4 aut
773	0	8	\|i Enthalten in \|t Advances in Applied Probability \|d Applied Probability Trust \|g 32(2000), 4, Seite 994-1010 \|w (DE-627)269247009 \|w (DE-600)1474602-5 \|x 00018678 \|7 nnns
773	1	8	\|g volume:32 \|g year:2000 \|g number:4 \|g pages:994-1010
856	4	0	\|u https://www.jstor.org/stable/1428513 \|3 Volltext
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952			\|d 32 \|j 2000 \|e 4 \|h 994-1010