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Volumn 38, Issue 6, 2010, Pages 3751-3781

Nonparametric estimation of multivariate convex-transformed densities

Author keywords

Consistency; Log concave density estimation; Lower bounds; Maximum likelihood; Mode estimation; Nonparametric estimation; Qualitative assumptions; Shape constraints; Strongly unimodal; Unimodal

Indexed keywords


EID: 78650124411     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/10-AOS840     Document Type: Article
Times cited : (61)

References (33)
  • 1
    • 0001582531 scopus 로고    scopus 로고
    • Logconcavity versus logconvexity: A complete characterization
    • MR1637480
    • AN, M. Y. (1998). Logconcavity versus logconvexity: A complete characterization. J. Econom. Theory 80 350-369. MR1637480
    • (1998) J. Econom. Theory , vol.80 , pp. 350-369
    • An, M.Y.1
  • 2
    • 0000553335 scopus 로고
    • R-convex functions
    • MR0301151
    • AVRIEL, M. (1972). r-convex functions. Math. Program. 2 309-323. MR0301151
    • (1972) Math. Program. , vol.2 , pp. 309-323
    • Avriel, M.1
  • 3
    • 68649100192 scopus 로고    scopus 로고
    • Limit distribution theory for maximum likelihood estimation of a log-concave density
    • MR2509075
    • BALABDAOUI, F., RUFIBACH, K. andWELLNER, J. A. (2009). Limit distribution theory for maximum likelihood estimation of a log-concave density. Ann. Statist. 37 1299-1331. MR2509075
    • (2009) Ann. Statist. , vol.37 , pp. 1299-1331
    • Balabdaoui, F.1    Rufibach, K.2    Wellner, J.A.3
  • 4
    • 21344492381 scopus 로고
    • Rates of convergence for minimum contrast estimators
    • Available at, MR1240719
    • BIRGÉ, L. and MASSART, P. (1993). Rates of convergence for minimum contrast estimators. Probab. Theory Related Fields 97 113-150. Available at http://dx.doi.org/10.1007/BF01199316. MR1240719
    • (1993) Probab. Theory Related Fields , vol.97 , pp. 113-150
    • Birgé, L.1    Massart, P.2
  • 5
    • 0000130180 scopus 로고
    • Convex set functions in d-space
    • MR0404559
    • BORELL, C. (1975). Convex set functions in d-space. Period. Math. Hungar. 6 111-136. MR0404559
    • (1975) Period. Math. Hungar. , vol.6 , pp. 111-136
    • Borell, C.1
  • 6
    • 49549132663 scopus 로고
    • On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation
    • MR0450480
    • BRASCAMP, H. J. and LIEB, E. H. (1976). On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22 366-389. MR0450480
    • (1976) J. Funct. Anal. , vol.22 , pp. 366-389
    • Brascamp, H.J.1    Lieb, E.H.2
  • 7
    • 0003795688 scopus 로고
    • E-entropy of convex sets and functions
    • MR0415155
    • BRONŠTE?IN, E. M. (1976). e-entropy of convex sets and functions. Sibirsk. Mat. Ž. 17 508-514, 715. MR0415155
    • (1976) Sibirsk. Mat. Ž. , vol.17 , Issue.508-514 , pp. 715
    • Bronšteǐn, E.M.1
  • 8
    • 0035602628 scopus 로고    scopus 로고
    • A Riemannian interpolation inequality à la Borell, Brascamp and Lieb
    • DOI 10.1007/s002220100160
    • CORDERO-ERAUSQUIN, D., MCCANN, R. J. and SCHMUCKENSCHLÄGER, M. (2001). A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math. 146 219-257. MR1865396 (Pubitemid 33725759)
    • (2001) Inventiones Mathematicae , vol.146 , Issue.2 , pp. 219-257
    • Cordero-Erausquin, D.1    McCann, R.J.2    Schmuckenschlager, M.3
  • 9
    • 77957584324 scopus 로고    scopus 로고
    • Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density
    • CULE, M. and SAMWORTH, R. (2010). Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density. Electron. J. Statist. 4 254-270.
    • (2010) Electron. J. Statist. , vol.4 , pp. 254-270
    • Cule, M.1    Samworth, R.2
  • 10
    • 84871658599 scopus 로고    scopus 로고
    • Maximum likelihood estimation of a multidimensional log-concave density (with discussion)
    • CULE, M., SAMWORTH, R. and STEWART, M. (2010). Maximum likelihood estimation of a multidimensional log-concave density (with discussion). J. Roy. Statist. Soc. Ser. B 72 1-32.
    • (2010) J. Roy. Statist. Soc. Ser. B , vol.72 , pp. 1-32
    • Cule, M.1    Samworth, R.2    Stewart, M.3
  • 12
    • 0000251164 scopus 로고
    • Geometrizing rates of convergence. II, III
    • 668-701. MR1105839
    • DONOHO, D. L. and LIU, R. C. (1991). Geometrizing rates of convergence. II, III. Ann. Statist. 19 633-667, 668-701. MR1105839
    • (1991) Ann. Statist. , vol.19 , pp. 633-667
    • Donoho, D.L.1    Liu, R.C.2
  • 14
    • 50249107779 scopus 로고    scopus 로고
    • Active set and EM algorithms for logconcave densities based on complete and censored data
    • Univ. Bern. Available at arXiv: 0707.4643
    • DÜMBGEN, L., HÜSLER, A. and RUFIBACH, K. (2007). Active set and EM algorithms for logconcave densities based on complete and censored data. Technical report, Univ. Bern. Available at arXiv:0707.4643.
    • (2007) Technical Report
    • Dümbgen, L.1    Hüsler, A.2    Rufibach, K.3
  • 15
    • 62749189131 scopus 로고    scopus 로고
    • Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency
    • MR2546798
    • DÜMBGEN, L. and RUFIBACH, K. (2009). Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency. Bernoulli 15 40-68. MR2546798
    • (2009) Bernoulli , vol.15 , pp. 40-68
    • Dümbgen, L.1    Rufibach, K.2
  • 16
    • 0035528158 scopus 로고    scopus 로고
    • Estimation of a convex function: Characterizations and asymptotic theory
    • MR1891742
    • GROENEBOOM, P., JONGBLOED, G. andWELLNER, J. A. (2001). Estimation of a convex function: Characterizations and asymptotic theory. Ann. Statist. 29 1653-1698. MR1891742
    • (2001) Ann. Statist. , vol.29 , pp. 1653-1698
    • Groeneboom, P.1    Jongbloed, G.2    Wellner, J.A.3
  • 17
    • 0000907205 scopus 로고
    • On the composition of unimodal distributions
    • MR0087249
    • IBRAGIMOV, I. A. (1956). On the composition of unimodal distributions. Teor. Veroyatnost. i Primenen. 1 283-288. MR0087249
    • (1956) Teor. Veroyatnost. i Primenen. , vol.1 , pp. 283-288
    • Ibragimov, I.A.1
  • 19
    • 0007259913 scopus 로고    scopus 로고
    • Minimax lower bounds and moduli of continuity
    • MR1792307
    • JONGBLOED, G. (2000). Minimax lower bounds and moduli of continuity. Statist. Probab. Lett. 50 279-284. MR1792307
    • (2000) Statist. Probab. Lett. , vol.50 , pp. 279-284
    • Jongbloed, G.1
  • 20
    • 77957585646 scopus 로고    scopus 로고
    • Quasi-concave density estimation
    • KOENKER, R. and MIZERA, I. (2010). Quasi-concave density estimation. Ann. Statist. 38 2998-3027.
    • (2010) Ann. Statist. , vol.38 , pp. 2998-3027
    • Koenker, R.1    Mizera, I.2
  • 21
    • 0000168012 scopus 로고
    • Distinctness of the eigenvalues of a quadratic form in a multivariate sample
    • MR0331643
    • OKAMOTO, M. (1973). Distinctness of the eigenvalues of a quadratic form in a multivariate sample. Ann. Statist. 1 763-765. MR0331643
    • (1973) Ann. Statist. , vol.1 , pp. 763-765
    • Okamoto, M.1
  • 23
    • 0000485147 scopus 로고
    • On logarithmic concave measures and functions
    • MR0404557
    • PRÉKOPA, A. (1973). On logarithmic concave measures and functions. Acta Sci. Math. (Szeged) 34 335-343. MR0404557
    • (1973) Acta Sci. Math. (Szeged) , vol.34 , pp. 335-343
    • Prékopa, A.1
  • 24
    • 0001377577 scopus 로고
    • On convexity of measures
    • MR0428540
    • RINOTT, Y. (1976). On convexity of measures. Ann. Probab. 4 1020-1026. MR0428540
    • (1976) Ann. Probab. , vol.4 , pp. 1020-1026
    • Rinott, Y.1
  • 27
    • 34547741997 scopus 로고    scopus 로고
    • Computing maximum likelihood estimators of a log-concave density function
    • MR2407642
    • RUFIBACH, K. (2007). Computing maximum likelihood estimators of a log-concave density function. J. Stat. Comput. Simul. 77 561-574. MR2407642
    • (2007) J. Stat. Comput. Simul. , vol.77 , pp. 561-574
    • Rufibach, K.1
  • 28
    • 74849099592 scopus 로고    scopus 로고
    • Consistency of multivariate log-concave density estimators
    • MR2593576
    • SCHUHMACHER, D. and DUEMBGEN, L. (2010). Consistency of multivariate log-concave density estimators. Statist. Probab. Lett. 80 376-380. MR2593576
    • (2010) Statist. Probab. Lett. , vol.80 , pp. 376-380
    • Schuhmacher, D.1    Duembgen, L.2
  • 29
    • 78650086541 scopus 로고    scopus 로고
    • Multivariate log-concave distributions as a nearly parametric model
    • Univ. Bern., Available at arXiv:0907.0250v1
    • SCHUHMACHER, D., HÜSLER, A. and DUEMBGEN, L. (2009). Multivariate log-concave distributions as a nearly parametric model. Technical report, Univ. Bern. Available at arXiv:0907.0250v1.
    • (2009) Technical Report
    • Schuhmacher, D.1    Hüsler, A.2    Duembgen, L.3
  • 31
    • 0001095650 scopus 로고
    • Some remarks about the convolution of unimodal functions
    • MR0735860
    • UHRIN, B. (1984). Some remarks about the convolution of unimodal functions. Ann. Probab. 12 640-645. MR0735860
    • (1984) Ann. Probab. , vol.12 , pp. 640-645
    • Uhrin, B.1
  • 33
    • 77955142547 scopus 로고    scopus 로고
    • Inference and modeling with log-concave distributions
    • WALTHER, G. (2010). Inference and modeling with log-concave distributions. Statist. Sci. 24 319-327.
    • (2010) Statist. Sci. , vol.24 , pp. 319-327
    • Walther, G.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.