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Bounds for the uniform deviation of empirical measures
 Journal of Multivariate Analysis
, 1982
"... If x,)...) X, are independent identically distributed Rdvalued random vectors with probability measure p and empirical probability measure p,, and if QZ is a subset of the Bore1 sets on Rd, then we show that P{sup,, ~ ..."
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Cited by 25 (4 self)
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If x,)...) X, are independent identically distributed Rdvalued random vectors with probability measure p and empirical probability measure p,, and if QZ is a subset of the Bore1 sets on Rd, then we show that P{sup,, ~
Stability analysis for stochastic programs
 ANNALS OF OPERATIONS RESEARCH
, 1991
"... For stochastic programs with recourse and with (several joint) probabilistic constraints, respectively, we derive quantitative continuity properties of the relevant expectation functionals and constraint set mappings. This leads to qualitative and quantitative stability results for optimal values an ..."
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Cited by 25 (15 self)
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For stochastic programs with recourse and with (several joint) probabilistic constraints, respectively, we derive quantitative continuity properties of the relevant expectation functionals and constraint set mappings. This leads to qualitative and quantitative stability results for optimal values and optimal solutions with respect to perturbations of the underlying probability distributions. Earlier stability results for stochastic programs with recourse and for those with probabilistic constraints are refined and extended, respectively. Emphasis is placed on equipping sets of probability measures with metrics that one can handle in specific situations. To illustrate the general stability results we present possible consequences when estimating the original probability measure via empirical ones.
Distribution sensitivity in stochastic programming
, 1991
"... In this paper, stochastic programming problems are viewed as parametric programs with respect to the probability distributions of the random coefficients. General results on quantitative stability in parametric optimization are used to study distribution sensitivity of stochastic programs. For recou ..."
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Cited by 12 (6 self)
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In this paper, stochastic programming problems are viewed as parametric programs with respect to the probability distributions of the random coefficients. General results on quantitative stability in parametric optimization are used to study distribution sensitivity of stochastic programs. For recourse and chance constrained models quantitative continuity results for optimal values and optimal solution sets are proved (with respect to suitable metrics on the space of probability distributions). The results are useful to study the effect of approximations and of incomplete information in stochastic programming.
Distribution sensitivity for certain classes of chance constrained models with application to power dispatch
 J. OPTIM. THEORY APPL
, 1991
"... Using results from parametric optimization, we derive for chanceconstrained stochastic programs quantitative stability properties for locally optimal values and sets of local minimizers when the underlying probability distribution is subjected to perturbations in a metric space of probability meas ..."
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Cited by 7 (2 self)
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Using results from parametric optimization, we derive for chanceconstrained stochastic programs quantitative stability properties for locally optimal values and sets of local minimizers when the underlying probability distribution is subjected to perturbations in a metric space of probability measures. Emphasis is placed on verifiable sufficient conditions for the constraintset mapping to fulfill a Lipschitz property which is essential for the stability results. Both convex and nonconvex problems are investigated. For a chanceconstrained model of power dispatch, where the power demand enters as a random vector with incompletely known probability distribution, we discuss consequences of our general results for the stability of optimal generation costs and optimal generation policies.
From Uniform Laws of Large Numbers to Uniform Ergodic Theorems
"... The purpose of these lectures is to present three different approaches with their own methods for establishing uniform laws of large numbers and uniform ergodic theorems for dynamical systems. The presentation follows the principle according to which the i.i.d. case is considered first in great deta ..."
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Cited by 4 (1 self)
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The purpose of these lectures is to present three different approaches with their own methods for establishing uniform laws of large numbers and uniform ergodic theorems for dynamical systems. The presentation follows the principle according to which the i.i.d. case is considered first in great detail, and then attempts are made to extend these results to the case of more general dependence structures. The lectures begin (Chapter 1) with a review and description of classic laws of large numbers and ergodic theorems, their connection and interplay, and their infinite dimensional extensions towards uniform theorems with applications to dynamical systems. The first approach (Chapter 2) is of metric entropy with bracketing which relies upon the BlumDeHardt law of large numbers and HoffmannJørgensen’s extension of it. The result extends to general dynamical systems using the uniform ergodic lemma (or Kingman’s subadditive ergodic theorem). In this context metric entropy and majorizing measure type conditions are also considered. The second approach (Chapter 3) is of Vapnik and Chervonenkis. It relies
Estimation of Multivariate Conditional Tail Expectation using Kendall’s Process
, 2012
"... This paper deals with the problem of estimating the Multivariate version of the ConditionalTailExpectation, proposed by Cousin and Di Bernardino (2012). We propose a new nonparametric estimator for this multivariate riskmeasure, which is essentially based on the Kendall’s process (see Genest and ..."
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Cited by 1 (0 self)
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This paper deals with the problem of estimating the Multivariate version of the ConditionalTailExpectation, proposed by Cousin and Di Bernardino (2012). We propose a new nonparametric estimator for this multivariate riskmeasure, which is essentially based on the Kendall’s process (see Genest and Rivest, 1993). Using the Central Limit Theorem for the Kendall’s process, proved by Barbe et al. (1996), we provide a functional Central Limit Theorem for our estimator. We illustrate the practical properties of our estimator on simulations. A real case in environmental framework is also analyzed. The performances of our new estimator are compared to the ones of the level setsbased estimator, previously proposed in Di Bernardino et al. (2011). Keywords:
Automatic Pattern Recognition: A Study of the Probability of Error
"... AbstractA test sequence is used to select the best rule from a rich class of discrimination rules defined in terms of the training sequence. The VapnikChervonenkis and related inequalities are used to obtain distributionfree bounds on the difference between the probability of error o € the select ..."
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AbstractA test sequence is used to select the best rule from a rich class of discrimination rules defined in terms of the training sequence. The VapnikChervonenkis and related inequalities are used to obtain distributionfree bounds on the difference between the probability of error o € the selected rule and the probability of error of the best rule in the given class. The bounds are used to prove the consistency and asymptotic optimality for several popular classes, including linear discriminators, nearest neighbor rules, kernelbased rules, histogram rules, binary tree classifiers, and Fourier series classifiers. In particular, the method can be used to choose the smoothing parameter in kernelbased rules, to choose k in the knearest neighbor rule, and to choose between parametric and nonparametric rules. Index TermsAutomatic parameter selection, empirical risk, error estimation, nonparametric rule, probability of error, statistical pattern recognition, VapnikChervonenkis inequality.
Self Organizing Map algorithm and distortion measure
, 2008
"... 1 Self Organizing Map algorithm and distortion measure We study the statistical meaning of the minimization of distortion measure and the relation between the equilibrium points of the SOM algorithm and the minima of distortion measure. If we assume that the observations and the map lie in an compac ..."
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1 Self Organizing Map algorithm and distortion measure We study the statistical meaning of the minimization of distortion measure and the relation between the equilibrium points of the SOM algorithm and the minima of distortion measure. If we assume that the observations and the map lie in an compact Euclidean space, we prove the strong consistency of the map which almost minimizes the empirical distortion. Moreover, after calculating the derivatives of the theoretical distortion measure, we show that the points minimizing this measure and the equilibria of the Kohonen map do not match in general. We illustrate, with a simple example, how this occurs.