Abstract
We consider the problem of estimating a mixture of probability measures in an abstract setting. Twelve examples are worked out, in order to show the applicability of the theory.
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References
Abramowitz, M. and Stegun, J. A. (1965), Handbook of Mathematical Functions, Dover, New York.
Andrews, D. F. and Mallows, C. L. (1974), Scale mixture of normal distributions, J. Roy. Statist. Soc., Ser. B 36, 99-102.
Andrews, D. W. K. and Pollard, D. (1994), An introduction to functional central limit theorems for dependent stochastic processes, Internat. Statist. Rev. 62, 119-132.
Bach, A., Plachky, D. and Thomson, W. (1987), A characterization of identifiability of mixtures of distributions, in: M. L. Puri and P. Révész (eds), Mathematical Statistics and Probability Theory, Vol. A, D. Reidel, Dordrecht, pp. 15-21.
Bagirov, E. B. (1988), Some remark on mixtures of normal distributions, Theory Probab. Appl. 33, 709-710.
Barbe, Ph. and Bertail, P. (1995), The Weighted Bootstrap, Lecture Notes in Statistics, Springer, New York.
Barndorff-Nielsen, O. (1965), Identifiability of finite mixtures of exponential families, J. Math. Anal. Appl. 21, 115-121.
Barndorff-Nielsen, O., Kent, J. and Sørensen, M. (1982), Normal variance-mean mixture and z-distributions, Internat. Statist. Rev. 50, 145-159.
Bercovici, H. and Foias, C. (1984), A real variable restatement of Riemann's hypothesis, Israel J. Math. 48, 57-68.
Beurling, A. (1955), A closure problem related to the Riemann Zeta-function, Proc. Acad. Sci. U.S.A. 41, 312-314.
Bickel, P. J., Klaassen, C. A., Ritov, Y. and Wellner, J. A. (1993), Efficiency and Adaptative Estimation for Semiparametric Models, Johns Hopkins University Press, Baltimore.
Birgé, L. (1989), The Grenander estimator: a nonasymptotic approach, Ann. Statist. 17, 1532-1549.
Blum, J. R. and Sursala, V. (1977), Estimating the parameters of a mixing disribution function, Ann. Probab. 5, 200-209.
Bondesson, L. (1992), Generalized Gamma Convolutions and Related Classes of Distributions, Lecture Notes in Statistics 76, Springer, New York.
Borwein, P. and Ederlyi, T. (1995), Polynomials and Polynomial Inequalities, Springer, New York.
Bruni, C. and Koch, G. (1985), Identifiability of continuous mixtures of unknown Gaussian distributions, Ann. Probab. 13, 1341-1357.
Carroll, R. J. and Hall, P. (1988), Optimal rates for deconvolving a density, J. Amer. Statist. Assoc. 83, 1184-1186.
Chandra, S. (1977), On the mixture of probability distributions, Scand. J. Statist. 4, 105-112.
Chen, J. (1995), Optimal rate of convergence for finite mixture models, Ann. Statist. 23, 221-223.
Choi, K. and Bulgren, W. B. (1968), An estimation procedure for mixture of distributions, J. Roy. Statist. Soc., Ser. B, 444-460.
Conway, J. B. (1973), Functions of One Complex Variable, Springer, New York.
Davis, Ph. J. (1963), Interpolation and Approximation, Blaisdell, New York.
Deely, J. J. and Kuse, R. L. (1968), Construction of sequences estimating the mixture distribution, Ann. Math. Statist. 39, 286-288.
Delbaen, F. and Hazendonck, J. (1984), Weighted Markov processes with an application to risk theory, in: F. de Vylder, M. Govaerts and J. Hazendonck (eds), Proc. NATO Advanced Study Institute on Insurance Premium, Louvain, July 18-31, 1983, D. Reidel, Dordrecht, pp. 121-132.
Devroye, L. (1990), Consistent deconvolution in density estimation, Canad. J. Statist. 17, 235-239.
Dharmadhikari, S. and Joag-Dev, K. (1988), Unimodality, Convexity, and Applications, Academic Press, New York.
Dvoretski, A., Kiefer, J. and Wolfowitz, J. (1956), Asymptotic minimax charater of the sample distribution function and of the classical multinomial estimation, Ann. Math. Statist. 27, 642- 669.
Dudley, R. M. (1984), A Course on Empirical Processes, in: École d'Été de Probabilité de Saint Flour, XII-1982, Lecture Notes in Math. 1097, Springer, New York, pp. 1-142.
Dudley, R. M. (1987), Universal Donsker classes and metric entropy, Ann. Probab. 15, 1306-1326.
Eagleson, G. K. (1975), Martingale convergence to mixture of infinitely divisible laws, Ann. Probab. 3, 557-562.
Edelman, D. (1988), Estimation of the mixing distribution for a normal mean with applications to the compound decision problem, Ann. Statist. 16, 1602-1622.
Efron, B. and Olshen, R. (1978), How broad is the class of normal scale mixtures?, Ann. Statist. 6, 1159-1164.
Eggermont, P. and LaRiccia, V. N. (1995), Maximum smoothed likelihood density estimation for inverse problems, Ann. Statist. 23, 199-220.
Everitt, B. S. and Hand, D. J. (1981), Finite Mixture Distributions, Chapman and Hall, London.
Fan, J. (1988), On the optimal rates of convergence for nonparametric deconvolution problem, Ann. Statist. 19, 1257-1272.
Farrell, R. H. (1962), Representation of invariant measures, Illinois J. Math. 6, 447-467.
Feller, W. (1971), Theory of Probability and its Applications, Vol. 2, Wiley, New York.
Fraser, M. D., Hsu, Y. S. and Walker, J. J. (1981), Identifiability of finite mixture of von Mises distributions, Ann. Statist. 9, 1130-1131.
Gallot, S., Hulin, D. and Lafontaine, J. (1993), Riemannian Geometry, 2nd edn, Springer, Berlin.
Giné, E. and Zinn, J. (1990), Bootstrapping general empirical functions, Ann. Probab. 18, 851-869.
Goldie, C. M. (1967), A class of infinitely divisible random variables, Proc. Camb. Philos. Soc. 63, 1141-1143.
Gradshteyn, J. and Ryzhik, J. M. (1980), Table of Integrals, Series and Products, 5th edn, Academic Press, San Diego, Calif.
Grenander, U. (1956), On the theory of mortality measurement, part II, Scand. Actuar. J. 39, 125-153.
Groeneboom, P. (1985), Estimating a monotone density, in: L. M. LeCam and R. A. Olshen (eds), Proc. Berkeley Conf. in Honor of Jerzy Neyman and Jack Kiefer, 2, Wadsworth, Belmont, Calif., pp. 539-555.
Gross, L. (1967), Abstract Wiener spaces, in: L. M. Le Cam and J. Neyman (eds), Proc. Fifth Berkeley Symp. Math. Statist. Probab., Vol II(1), University of California Press, Berkeley, Calif., pp. 31-42.
Hartigan, J. A. and Hartigan, P. M. (1985), The dip test of unimodality, Ann. Statist. 13, 70-80.
Hengartner, N. (1997), Adaptative demixing in Poisson mixture models, Ann. Statist. 25, 917-928.
Hervé, M. (1989), Analycity in Infinite Dimensional Spaces, De Gruyter, Berlin.
Ikeda, N. and Watanabe, S. (1989), Stochastic Differential Equations and Diffusions Processes, 2nd edn, North-Holland/Kodansha.
Jewel, N. (1982), Mixtures of exponential distributions, Ann. Statist. 10, 479-484.
Kamarkar, N. (1984), A new polynomial-time algorithm for linear programming, Combinatorica 4, 373-395.
Keilson, J. and Steutel, F. W. (1972), Families of infinite divisible distributions closed under mixing and convolutions, Ann. Math. Statist. 43, 242-250.
Kelker, D. (1971), Infinite divisibility and variance mixture of the normal distribution, Ann. Math. Statist. 42, 802-808.
Kent, J. T. (1981), Convolution mixture of infinitely divisible distributions, Math. Proc. Camb. Phil. Soc. 90, 141-153.
Kent, J. T. (1983), Identifiability of finite mixture for directional data, Ann. Statist. 11, 984-988.
Khachiyan, L. G. (1979), A polynomial algorithm in linear progamming, Soviet. Math. Dokl. 20, 191-194.
Kiefer, J. and Wolfowitz, J. (1976), Asymptotically minimax estimation of concave and convex distribution functions, Zeit. Wahrsch. Theor. verw. Geb. 34, 73-85.
Kim, J. and Pollard, D. (1990), Cube root asymptotics, Ann. Statist. 18, 191-219.
Kintchine, A. Y. (1938), On unimodal distributions, Izv. Nauchno-Issled. Inst. Mat. Mech. Tomsk. Gos. Univ. 2, 1-7.
Laird, N. (1978), Nonparametric maximum-likelihood estimation of a mixing distribution, J. Amer. Statist. Assoc. 73, 805-811.
Lambert, D. and Tierney, L. (1984), Asymptotic properties of maximum likelihood estimates in the mixed Poisson model, Ann. Statist. 12, 1388-1399.
Ledoux, M. (1994), Isoperimetry and Gaussian Analysis, École d'Été de Probabilités de St. Flour, Lecture Notes in Math., Springer, New York, to appear.
Ledoux, M. (1996), On Talagrand's deviation inequalities for product measures, Preprint.
Lehman, E. L. (1959), Testing Statistical Hypotheses, Wiley, New York.
Lemdani, M. (1995), Tests dans le cas d'un mélange de Lois dans des modèles paramétriques et non paramétriques, PhD thesis, École Polytechnique.
Lindsay, B. G. (1983a), The geometry of mixture likelihood: a general theory, Ann. Statist. 11, 86-94.
Lindsay, B. G. (1983b), Efficiency of the conditional score in a mixture setting, Ann. Statist. 11, 486-497.
Lindsay, B. G. (1995), Mixture Models: Theory, Geometry and Applications, NSF-CBMS Regional Conference Series in Probability and Statistics, Vol. 5, IMS, Hayward, CA.
Lo, A. (1991), Bayesian bootstrap clones and a biometry function, Sankhya Ser. A 53, 320-333.
Lüxmann-Ellingaus, U. (1987), On the identifiability of mixtures of infinitely divisible power series distributions, Statist. Probab. Lett. 5, 375-378.
MacDonald, P. D. M. (1971), Comment on 'An estimation procedure for mixture of distributions' by Choi and Bulgren, J. Roy. Statist. Soc. Ser. B 33, 326-329.
McLachlan, G. J. and Basford, K. E. (1988), Mixture Models: Inference and Applications to Clustering, Marcel Dekker, New York.
Maitra, A. (1977), Integral representations of invariant measures, Trans. Amer. Math. Soc. 229, 209-225.
Mammen, E., Maron, S. and Fisher, N. I. (1992), Some asymptotic for multimodality tests based on kernel density estimator, Probab. Theory Related Fields 91, 115-132.
Martin, R. D. and Schwartz, S. C. (1972), On mixture, quasi-mixture and nearly normal random processes, Ann. Math. Statist. 40, 948-967.
Mason, D. M. and Newton, M. A. (1992), A rank statistic approach to the consistency of a general weighted bootstrap, Ann. Statist. 20, 1611-1624.
Massart, P. (1986), Rates of convergence in the central limit theorem for empirical process, Ann. Inst. Henri Poincaré, Probab. Statist. 22, 381-424.
Massé, J. C. (1993), Nonparametric maximum likelihood estimation in a nonlocally compact parameter setting, Technical report 93-24, Département de Mathématique et de Statistique, Université Laval, Québec.
Milhaud, X. and Mounime, S. (1993), A modified maximum likelihood estimator for finite mixture, preprint.
Miller, K. S. and Ross, B. (1993), An Introduction to the Fractional Calculus and Fractional Differential Equation, Wiley, New York.
Noack, A. (1950), A class of random variables with discrete distribution, Ann. Math. Statist. 21, 127-132.
Olshen, R. A. and Savage, L. J. (1970), Generalized unimodality, J. Appl. Probab. 7, 21-34.
Patil, G. P. and Bildikar, S. (1966), Identifiability of countable mixtures of discrete probability distributions using methods of infinite matrices, Proc. Camb. Philos. Soc. 62, 485-494.
Pearson, K. (1894), Contribution to the mathematical theory of evolution, Philos. Trans. Roy. Soc. London Ser. A 185, 71-110.
Pfanzagl, J. (1988), Consistency of maximum likelihood estimators for certain nonparametric families, in particular mixtures, J. Statist. Plann. Inf. 19, 137-158.
Pollard, D. (1984), Convergence of Stochastic Processes, Springer, New York.
Praestgaard, J. and Wellner, J. A. (1984), Exchangeably weighted bootstraps of the general empirical process, Ann. Probab. 21, 2053-2086.
Prakasa Rao, B. L. S. (1969), Estimating a unimodal density, SankhyáSer. A 31, 23-36.
Prakasa Rao, B. L. (1992), Identifiability in Stochastic Models: Characterization of Probability Distributions, Academic Press, New York.
Redner, R. and Walker, H. F. (1984), Mixture densities, maximum likelihood and the E.M. algorithm, SIAM Rev. 26, 195-239.
Rennie, R. R. (1972), On the independence of the identifiability of finite multivariate mixture and the identifiability of the marginal mixtures, SankhyáSer. A 34, 449-452.
Robbins, H. (1948), Mixture of distributions, Ann. Math. Statist. 19, 360-369.
Rootzén, H. (1977), A note on convergence to mixture of normal distribution, Zeit. Wahrsch. Theor. verw. Geb. 38, 211-216.
Santalò, L. (1984), Integral Geometry and Geometric Probability, Cambridge University Press.
Schwartz, L. (1943), Étude des sommes d'exponentielles réelles, Act. Sci. Ind. 959, Hermann, Paris.
Shanbhag, D. N. and Sreehari, M. (1977), On certain self-decomposable distributions, Zeit. Wahrsch. Theor. verw. Geb. 38, 217-222.
Shorack, G. R. and Wellner, J. A. (1986), Empirical Processes with Applications to Statistics, Wiley, New York.
Sheehy, A. and Wellner, J. A. (1992), Uniform Donsker classes of functions, Ann. Probab. 20, 1983-2030.
Silverman, B. W. (1981), Using kernel density estimates to investigate multimodality, J. Roy. Statist. Soc. Ser. B 43, 97-99.
Simar, L. (1976), Maximum likelihood estimation of a compound Poisson process, Ann. Statist. 4, 1200-1209.
Stefanski, L. A. (1990), Rates of convergence of some estimators in a class of deconvolution problems, Statist. Probab. Lett. 9, 229-235.
Steutel, F. W. (1967), Note on the infinite divisibility of exponential mixtures, Ann. Math. Statist. 38, 1303-1305.
Steutel, F. W. (1968), A class of infinitely divisible mixtures, Ann. Math. Statist. 39, 1153-1157.
Steutel, F. W. (1970), Preservation of infinite divisibility under mixing and related topics, Math. Center Tracts 33, Amsterdam.
Stroock, D. and Varadhan, S. R. S. (1972), On the support of diffusion processes with applications to the strong maximum principle, in: L. M. Le Cam, J. Neyman and E. L. Scott (eds), Proc. Sixth Berkeley Symp. Math. Statist. Probab., University of California Press, Berkeley, CA., pp. 333-359.
Szegö, G. (1939), Orthogonal Polynomials, AMS Coll. Publ., Amer. Math. Ser., Providence, R.I.
Talagrand, M. (1987), Donsker classes and geometry, Ann. Probab. 15, 1327-1338.
Talagrand, M. (1994), Sharper bounds for Gaussian and empirical processes, Ann. Probab. 22, 28-76.
Talagrand, M. (1995), The Glivenko-Cantelli problem, ten years later, Preprint.
Talagrand, M. (1995), Concentration of measure and isoperimetric inequalities in product spaces, Publ. Math. IHES 81, 73-205.
Tallis, G. M. (1969), The identifiability of mixtures of distributions, J. Appl. Probab. 6, 389-398.
Tallis, G. M. and Chesson, P. (1982), Identifiability of mixtures, J. Austral. Math. Soc. 32, 339-348.
Teicher, H. (1960), On the mixture of distributions, Ann. Math. Statist. 31, 55-77.
Teicher, H. (1961), Identifiability of mixtures, Ann. Math. Statist. 32, 244-248.
Teicher, H. (1963), Identifiability of finite mixtures, Ann. Math. Statist. 34, 1265-1269.
Teicher, H. (1967), Identifiability of mixtures of product measures, Ann. Math. Statist. 38, 1300- 1302.
Thierney, L. and Lambert, D. (1984), Asymptotic efficiency of estimators of functional of mixed distributions, Ann. Statist. 12, 1380-1387.
Thorin, O. (1977), On the infinite divisibility of the lognormal distribution, Scand. Actuar. J., 121-148.
Titterington, D. M., Smith, A. F. M. and Makov, U. E. (1985), Statistical Analysis of Finite Mixture Distributions, Wiley, New York.
Van de Geer, S. (1993), Rates of convergence for the maximum likelihood estimator in mixture models, Technical Report, TW 93-09, University of Leiden.
Van de Geer, S. (1994), Asymptotic normality in mixture models, Technical report, TW 94-03, University of Leiden.
Van der Vaart, A. (1991), On differentiable functionals, Ann. Statist. 19, 178-204.
Varadarajan, V. S. (1963), Groups of automorphisms of Borel spaces, Trans. Amer. Math. Soc. 109, 191-220.
Yakowitz, S. I. and Spragins, J. D. (1968), On the identifiability of finite mixture, Ann. Math. Statist. 40, 1728-1735.
Zhang, C.-H. (1990), Fourier methods for estimating mixing densities and distributions, Ann. Statist. 18, 806-831.
Zhang, C.-H. (1995), On estimating mixing densities in discrete exponential family models, Ann. Statist. 23, 929-945.
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Barbe, P. Statistical Analysis of Mixtures and the Empirical Probability Measure. Acta Applicandae Mathematicae 50, 253–340 (1998). https://doi.org/10.1023/A:1005824513112
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DOI: https://doi.org/10.1023/A:1005824513112