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Michel Denuit - One of the best experts on this subject based on the ideXlab platform.
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Comonotonicity, orthant Convex order and sums of random variables
Statistics & Probability Letters, 2015Co-Authors: Mhamed Mesfioui, Michel DenuitAbstract:This paper extends a useful property of the Increasing Convex order to the multivariate orthant Convex order. Specifically, it is shown that vectors of sums of comonotonic random variables dominate in the orthant Convex order vectors of sums of random variables that are smaller in the Increasing Convex sense, whatever their dependence structure. This result is then used to derive orthant Convex order bounds on random vectors of sums of random variables. Extensions to vectors of compound sums are also discussed.
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A separation theorem for the weak s-Convex orders
Insurance: Mathematics and Economics, 2014Co-Authors: Michel Denuit, Liqun Liu, Jack MeyerAbstract:The present paper extends to higher degrees the well-known separation theorem decomposing a shift in the Increasing Convex order into a combination of a shift in the usual stochastic order followed by another shift in the Convex order. An application in decision making under risk is provided to illustrate the interest of the result.
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Ordering Functions of Random Vectors, with Application to Partial Sums
Journal of Theoretical Probability, 2012Co-Authors: Michel Denuit, Mhamed MesfiouiAbstract:It is known that the sums of the components of two random vectors (X1,X2,...,Xn) and (Y1,Y2,...,Yn) ordered in the multivariate (s1,s2,...,sn)-Increasing Convex order are ordered in the univariate (s1+s2+...+sn)-Increasing Convex order. More generally, real-valued functions of (X1,X2,...,Xn) and (Y1,Y2,...,Yn) are ordered in the same sense as long as these functions possess some specified non-negative cross derivatives. This note extends these results to multivariate functions. In particular, we consider vectors of partial sums (S1,S2,...,Sn) and (T1,T2,...,Tn) where Sj = X1+...+Xj and Tj = Y1 +...+Yj and we show that these random vectors are ordered in the multivariate (s1,s1+s2,...,s1+...+sn)-Increasing Convex order. The consequences of these general results for the upper orthant order and the orthant Convex order are discussed.
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Generalized Increasing Convex and Directionally Convex Orders
Journal of Applied Probability, 2010Co-Authors: Michel Denuit, Mhamed MesfiouiAbstract:In this paper, the componentwise Increasing Convex order, the upper orthant order, the upper orthant Convex order, and the Increasing directionally Convex order for random vectors are generalized to hierarchical classes of integral stochastic order relations. The elements of the generating classes of functions possess nonnegative partial derivatives up to some given degrees. Some properties of these new stochastic order relations are studied. Particular attention is paid to the comparison of weighted sums of the respective components of ordered random vectors. By providing a unified derivation of standard multivariate stochastic orderings, the present paper shows how some well-known results derive from a common principle.
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Positive Dependence of Signals
Journal of Applied Probability, 2010Co-Authors: Michel DenuitAbstract:In this paper we further investigate the problem considered by Mizuno (2006) in the special case of identically distributed signals. Specifically, we first propose an alternative sufficient condition of crossing type for the Convex order to hold between the conditional expectations given signal. Then, we prove that the bivariate (2,1)-Increasing Convex order ensures that the conditional expectations are ordered in the Convex sense. Finally, the L-2 distance between the quantity of interest and its conditional expectation given signal (or expected conditional variance) is shown to decrease when the strength of the dependence increases (as measured by the (2,1)-Increasing Convex order).
Tatjana Ostrogorski - One of the best experts on this subject based on the ideXlab platform.
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Decomposition of Convex Additively Slowly Varying Functions * Supported by Grant No. 1835 of the Mntr
Integral Transforms and Special Functions, 2003Co-Authors: Slobodanka Janković, Tatjana OstrogorskiAbstract:We find conditions which imply that the difference of two slowly varying functions is slowly varying. Given an additively slowly varying Increasing Convex function l , we consider the class K_l of Increasing functions F such that F/l is Increasing Convex. If an additively slowly varying function L belongs to K_l , we find conditions under which, if we decompose L into a sum L = F + G , where F , G\in K_l , then it follows that F and G are necessarily slowly varying. As an auxiliary result, we find some properties of additively slowly varying functions with remainder term which are also of independent interest.
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Convex additively slowly varying functions
Journal of Mathematical Analysis and Applications, 2002Co-Authors: Slobodanka Janković, Tatjana OstrogorskiAbstract:Abstract We study the problem of subtraction of slowly varying functions. It is well-known that the difference of two slowly varying functions need not be slowly varying and we look for some additional conditions which guarantee the slow variation of the difference. To this end we consider all possible decompositions L = F + G of a given Increasing Convex additively slowly varying function L into a sum of two Increasing Convex functions F and G . We characterize the class of functions L for which in every such decomposition the summands are necessarily additively slowly varying. The class O Π 2 + we obtain is related to the well-known class O Π g where, instead of first order differences as in O Π g , we have second order differences.
Félix Belzunce - One of the best experts on this subject based on the ideXlab platform.
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On sufficient conditions for the comparison in the excess wealth order and spacings
Journal of Applied Probability, 2016Co-Authors: Félix Belzunce, Carolina Martínez-riquelme, José-maría Ruiz, Miguel A. SordoAbstract:The purpose of this paper is twofold. On the one hand, we provide sufficient conditions for the excess wealth order. These conditions are based on properties of the quantile functions which are useful when the dispersive order does not hold. On the other hand, we study sufficient conditions for the comparison in the Increasing Convex order of spacings of generalized order statistics. These results will be combined to show how we can provide comparisons of quantities of interest in reliability and insurance.
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Increasing directionally Convex orderings of random vectors having the same copula, and their use in comparing ordered data
Journal of Multivariate Analysis, 2012Co-Authors: Narayanaswamy Balakrishnan, Miguel A. Sordo, Félix Belzunce, Alfonso Suárez-llorensAbstract:In this paper, we establish some results for the Increasing Convex comparisons of generalized order statistics. First, we prove that if the minimum of two sets of generalized order statistics are ordered in the Increasing Convex order, then the remaining generalized order statistics are also ordered in the Increasing Convex order. This result is extended to the Increasing directionally Convex comparisons of random vectors of generalized order statistics. For establishing this general result, we first prove a new result in that two random vectors with a common conditionally Increasing copula are ordered in the Increasing directionally Convex order if the marginals are ordered in the Increasing Convex order. This latter result is, of course, of interest in its own right.
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Wiley Encyclopedia of Operations Research and Management Science - Block Replacement Policies
Wiley Encyclopedia of Operations Research and Management Science, 2011Co-Authors: Félix Belzunce, Moshe ShakedAbstract:In this article, we describe block replacement policies and we list several results that compare block replacement policies with some other policies. Mainly, we present results that compare the number of unplanned failures in a block replacement policy in the sense of several stochastic orders, with the number of unplanned failures under the usual replacement policy and under an age replacement policy. Some other results for the Increasing Convex order, Laplace order, and comparisons of expectations for the arrival times are considered. We first list univariate results, and later we list some extensions of the block replacement policy and comparisons of entire processes. We conclude with a basic description of the cost analysis that is associated with block replacement policies. Keywords: preventive maintenance; stochastic orders; aging properties; Laplace transform order; Increasing failure rate (IFR); new better than used (NBU) Poisson process; minimal repair; imperfect repair; Increasing Convex order; cost analysis
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Stochastic comparisons of mixtures of Convexly ordered distributions with applications in reliability theory
Statistics & Probability Letters, 2001Co-Authors: Félix Belzunce, Moshe ShakedAbstract:In this paper we first obtain several results which compare mixtures of distributions in the (Increasing) Convex and concave stochastic orders. We employ these results to derive relatively weak conditions on the intensity functions of a pair of nonhomogeneous Poisson processes (in fact, on the distribution functions that are associated with these intensity functions) under which the corresponding epoch times of the two nonhomogeneous Poisson processes are ordered in the Increasing Convex stochastic order. Applications include bounds on the epoch times of a nonhomogeneous Poisson process whose intensity function is the hazard rate function of a new better than used in expectation (new worse than used in expectation) random variable, and the Increasing Convex ordering of times to the first perfect repair in a Bayesian imperfect repair model.
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On a characterization of right spread order by the Increasing Convex order
Statistics & Probability Letters, 1999Co-Authors: Félix BelzunceAbstract:The purpose of this paper is to give a characterization of a new variability order called the right spread order. This characterization is given in terms of the Increasing Convex order. Also we provide a characterization of DMRL [IMRL] class based on right spread order of residual lives. Some interpretations and applications are given in the last section.
Mhamed Mesfioui - One of the best experts on this subject based on the ideXlab platform.
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Comonotonicity, orthant Convex order and sums of random variables
Statistics & Probability Letters, 2015Co-Authors: Mhamed Mesfioui, Michel DenuitAbstract:This paper extends a useful property of the Increasing Convex order to the multivariate orthant Convex order. Specifically, it is shown that vectors of sums of comonotonic random variables dominate in the orthant Convex order vectors of sums of random variables that are smaller in the Increasing Convex sense, whatever their dependence structure. This result is then used to derive orthant Convex order bounds on random vectors of sums of random variables. Extensions to vectors of compound sums are also discussed.
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Ordering Functions of Random Vectors, with Application to Partial Sums
Journal of Theoretical Probability, 2012Co-Authors: Michel Denuit, Mhamed MesfiouiAbstract:It is known that the sums of the components of two random vectors (X1,X2,...,Xn) and (Y1,Y2,...,Yn) ordered in the multivariate (s1,s2,...,sn)-Increasing Convex order are ordered in the univariate (s1+s2+...+sn)-Increasing Convex order. More generally, real-valued functions of (X1,X2,...,Xn) and (Y1,Y2,...,Yn) are ordered in the same sense as long as these functions possess some specified non-negative cross derivatives. This note extends these results to multivariate functions. In particular, we consider vectors of partial sums (S1,S2,...,Sn) and (T1,T2,...,Tn) where Sj = X1+...+Xj and Tj = Y1 +...+Yj and we show that these random vectors are ordered in the multivariate (s1,s1+s2,...,s1+...+sn)-Increasing Convex order. The consequences of these general results for the upper orthant order and the orthant Convex order are discussed.
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Generalized Increasing Convex and Directionally Convex Orders
Journal of Applied Probability, 2010Co-Authors: Michel Denuit, Mhamed MesfiouiAbstract:In this paper, the componentwise Increasing Convex order, the upper orthant order, the upper orthant Convex order, and the Increasing directionally Convex order for random vectors are generalized to hierarchical classes of integral stochastic order relations. The elements of the generating classes of functions possess nonnegative partial derivatives up to some given degrees. Some properties of these new stochastic order relations are studied. Particular attention is paid to the comparison of weighted sums of the respective components of ordered random vectors. By providing a unified derivation of standard multivariate stochastic orderings, the present paper shows how some well-known results derive from a common principle.
Slobodanka Janković - One of the best experts on this subject based on the ideXlab platform.
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Decomposition of Convex Additively Slowly Varying Functions * Supported by Grant No. 1835 of the Mntr
Integral Transforms and Special Functions, 2003Co-Authors: Slobodanka Janković, Tatjana OstrogorskiAbstract:We find conditions which imply that the difference of two slowly varying functions is slowly varying. Given an additively slowly varying Increasing Convex function l , we consider the class K_l of Increasing functions F such that F/l is Increasing Convex. If an additively slowly varying function L belongs to K_l , we find conditions under which, if we decompose L into a sum L = F + G , where F , G\in K_l , then it follows that F and G are necessarily slowly varying. As an auxiliary result, we find some properties of additively slowly varying functions with remainder term which are also of independent interest.
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Convex additively slowly varying functions
Journal of Mathematical Analysis and Applications, 2002Co-Authors: Slobodanka Janković, Tatjana OstrogorskiAbstract:Abstract We study the problem of subtraction of slowly varying functions. It is well-known that the difference of two slowly varying functions need not be slowly varying and we look for some additional conditions which guarantee the slow variation of the difference. To this end we consider all possible decompositions L = F + G of a given Increasing Convex additively slowly varying function L into a sum of two Increasing Convex functions F and G . We characterize the class of functions L for which in every such decomposition the summands are necessarily additively slowly varying. The class O Π 2 + we obtain is related to the well-known class O Π g where, instead of first order differences as in O Π g , we have second order differences.
