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On the Convergence of Monte Carlo Maximum Likelihood Calculations
 Journal of the Royal Statistical Society B
, 1992
"... Monte Carlo maximum likelihood for normalized families of distributions (Geyer and Thompson, 1992) can be used for an extremely broad class of models. Given any family f h ` : ` 2 \Theta g of nonnegative integrable functions, maximum likelihood estimates in the family obtained by normalizing the the ..."
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Cited by 67 (5 self)
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Monte Carlo maximum likelihood for normalized families of distributions (Geyer and Thompson, 1992) can be used for an extremely broad class of models. Given any family f h ` : ` 2 \Theta g of nonnegative integrable functions, maximum likelihood estimates in the family obtained by normalizing the the functions to integrate to one can be approximated by Monte Carlo, the only regularity conditions being a compactification of the parameter space such that the the evaluation maps ` 7! h ` (x) remain continuous. Then with probability one the Monte Carlo approximant to the log likelihood hypoconverges to the exact log likelihood, its maximizer converges to the exact maximum likelihood estimate, approximations to profile likelihoods hypoconverge to the exact profile, and level sets of the approximate likelihood (support regions) converge to the exact sets (in Painlev'eKuratowski set convergence). The same results hold when there are missing data (Thompson and Guo, 1991, Gelfand and Carlin, 19...
APPROXIMATION ISSUES ABSTRACT
"... Averaging has a smoothing and convexifying effect. So expectation functionals are ‘usually ’ convex. However, for an important class of expectation functionals that arise in stochastic programs with chance constraints one can obtain no more than quasiconvexity. Approximation questions for this clas ..."
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Averaging has a smoothing and convexifying effect. So expectation functionals are ‘usually ’ convex. However, for an important class of expectation functionals that arise in stochastic programs with chance constraints one can obtain no more than quasiconvexity. Approximation questions for this class of expectation functionals are also being considered. 1 1
DIRECTIONS DE MAJORATION D’UNE FONCTION QUASICONVEXE ET APPLICATIONS
"... Abstract. We introduce the convex cone constituted by the directions of majoration of a quasiconvex function. This cone is used to formulate a qualification condition ensuring the epiconvergence of a sequence of general quasiconvex marginal functions in finite dimensional spaces. ..."
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Abstract. We introduce the convex cone constituted by the directions of majoration of a quasiconvex function. This cone is used to formulate a qualification condition ensuring the epiconvergence of a sequence of general quasiconvex marginal functions in finite dimensional spaces.
DOI: (will be inserted later) DUALITY METHODS FOR THE STUDY OF HAMILTON–JACOBI EQUATIONS
"... Abstract. We present a survey of recent results about explicit solutions of the firstorder Hamilton– Jacobi equation. We take advantage of the methods of asymptotic analysis, convex analysis and of nonsmooth analysis to shed a new light on classical results. We use formulas of the Hopf and Lax–Olei ..."
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Abstract. We present a survey of recent results about explicit solutions of the firstorder Hamilton– Jacobi equation. We take advantage of the methods of asymptotic analysis, convex analysis and of nonsmooth analysis to shed a new light on classical results. We use formulas of the Hopf and Lax–Oleinik types. In the quasiconvex case the usual Fenchel conjugacy is replaced by quasiconvex conjugacies known for some years and the usual infconvolution is replaced by a sublevel convolution. Inasmuch we use weak generalized convexity and continuity assumptions, some of our results are new; in particular, we do not assume that the data are finitevalued, so that equations derived from attainability or obstacle problems could be considered. Résumé. Nous présentons un survol de quelques résultats récents relatifs aux solutions explicites des équations d’HamiltonJacobi du premier ordre. Nous nous servons des méthodes de l’analyse asymptotique, de l’analyse convexe et de l’analyse non lisse pour apporter une lumière nouvelle sur des résultats classiques. Nous utilisons des formules de type Hopf et LaxOleinik. Dans le cas quasiconvexe, la conjugaison classique de Fenchel est remplacée par des conjugaisons connues depuis quelques années et l’infconvolution est remplacée par la convolution en sousniveaux. Comme nous faisons des hypothèses de convexite ́ généralisée et de continuite ́ assez générales, certains des résultats présentés