×

Nonparametric estimation of multivariate scale mixtures of uniform densities. (English) Zbl 1352.62049

Summary: Suppose that \(\boldsymbol U=(U_1,\dots, U_d)\) has a Uniform\(([0,1]^{d})\) distribution, that \(\boldsymbol Y=(Y_1,\dots, Y_d)\) has the distribution \(G\) on \(\mathbb R^d_+\), and let \(\boldsymbol X=(X_1,\dots, X_d)=(U_1Y_1,\dots, U_dY_d)\). The resulting class of distributions of \(\boldsymbol X\) (as \(G\) varies over all distributions on \(\mathbb R^d_+\)) is called the Scale Mixture of Uniforms class of distributions, and the corresponding class of densities on \(\mathbb R^d_+\) is denoted by \(\mathcal F_{\mathrm{SMU}}(d)\). We study maximum likelihood estimation in the family \(\mathcal F_{\mathrm{SMU}}(d)\). We prove existence of the MLE, establish Fenchel characterizations, and prove strong consistency of the almost surely unique maximum likelihood estimator (MLE) in \(\mathcal F_{\mathrm{SMU}}(d)\). We also provide an asymptotic minimax lower bound for estimating the functional \(f \to f (\boldsymbol x)\) under reasonable differentiability assumptions on \(f\in \mathcal F_{\mathrm{SMU}}(d)\) in a neighborhood of \(\boldsymbol x\). We conclude the paper with discussion, conjectures and open problems pertaining to global and local rates of convergence of the MLE.

MSC:

62G05 Nonparametric estimation
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62H12 Estimation in multivariate analysis
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] D. Anevski, Estimating the derivative of a convex density. Technical Report, dept. of Math. Statistics, Univ. of Lund, 1994. · Zbl 1090.62524
[2] Anevski, D., Estimating the derivative of a convex density, Stat. neerl., 57, 2, 245-257, (2003) · Zbl 1090.62524
[3] Ayer, M.; Brunk, H.D.; Ewing, G.M.; Reid, W.T.; Silverman, E., An empirical distribution function for sampling with incomplete information, Ann. math. statist., 26, 641-647, (1955) · Zbl 0066.38502
[4] Balabdaoui, F.; Jankowski, H.; Pavlides, M.; Seregin, A.; Wellner, J.A., On the grenander estimator at zero, Statist. sinica, 21, 873-879, (2011) · Zbl 1214.62037
[5] Barlow, R.E.; Bartholomew, D.J.; Bremner, J.M.; Brunk, H.D., Statistical inference under order restrictions, () · Zbl 0246.62038
[6] Biau, G.; Devroye, L., On the risk of estimates for block decreasing densities, J. multivariate anal., 86, 1, 143-165, (2003) · Zbl 1025.62015
[7] Billingsley, P., ()
[8] Birgé, L., Estimating a density under order restrictions: nonasymptotic minimax risk, Ann. statist., 15, 3, 995-1012, (1987) · Zbl 0631.62037
[9] Birgé, L., The grenander estimator: a nonasymptotic approach, Ann. statist., 17, 4, 1532-1549, (1989) · Zbl 0703.62042
[10] Blei, R.; Gao, F.; Li, W.V., Metric entropy of high dimensional distributions, Proc. amer. math. soc., 135, 12, 4009-4018, (2007), (electronic) · Zbl 1147.46018
[11] Brunk, H.D., On the estimation of parameters restricted by inequalities, Ann. math. statist., 29, 437-454, (1958) · Zbl 0087.34302
[12] Brunk, H.D., Estimation of isotonic regression, (), 177-197
[13] Devroye, L., Nonuniform random variate generation, (1986), Springer-Verlag New York · Zbl 0584.65002
[14] Devroye, L., ()
[15] Doob, J.L., ()
[16] Feller, W., An introduction to probability theory and its applications, vol. II, (1971), John Wiley & Sons Inc. New York · Zbl 0219.60003
[17] Glick, N., Consistency conditions for probability estimators and integrals of density estimators, Util. math., 6, 61-74, (1974) · Zbl 0295.62045
[18] Grenander, U., On the theory of mortality measurement, I. skand. aktuarietidskr., 39, 70-96, (1956) · Zbl 0073.15404
[19] Grenander, U., On the theory of mortality measurement. II, Skand. aktuarietidskr., 39, 125-153, (1957) · Zbl 0077.33715
[20] Groeneboom, P., Estimating a monotone density, (), 539-555, Belmont, CA · Zbl 1373.62144
[21] Groeneboom, P., Brownian motion with a parabolic drift and Airy functions, Probab. theory related fields, 81, 1, 79-109, (1989)
[22] Groeneboom, P., Lectures on inverse problems, (), 67-164, Saint-Flour, 1994 · Zbl 0907.62042
[23] Groeneboom, P.; Jongbloed, G., Isotonic estimation and rates of convergence in wicksell’s problem, Ann. statist., 23, 5, 1518-1542, (1995) · Zbl 0843.62034
[24] Groeneboom, P.; Jongbloed, G.; Wellner, J.A., A canonical process for estimation of convex functions: the invelope of integrated Brownian motion \(+ t^4\), Ann. statist., 29, 6, 1620-1652, (2001) · Zbl 1043.62026
[25] Groeneboom, P.; Jongbloed, G.; Wellner, J.A., Estimation of a convex function: characterizations and asymptotic theory, Ann. statist., 29, 6, 1653-1698, (2001) · Zbl 1043.62027
[26] Hampel, F.R., Design, modelling, and analysis of some biological data sets, (), 93-128
[27] Jewell; Nicholas, P.; van der Laan; Mark, Current status data: review, recent developments and open problems, (), 625-642
[28] G. Jongbloed, Three statistical inverse problems. Ph.D. Thesis, Delft University, 1995.
[29] Jongbloed, G., Minimax lower bounds and moduli of continuity, Statist. probab. lett., 50, 3, 279-284, (2000) · Zbl 0965.60083
[30] Kim, J.; Pollard, D., Cube root asymptotics, Ann. statist., 18, 1, 191-219, (1990) · Zbl 0703.62063
[31] Lang, R., A note on the measurability of convex sets, Arch. math. (basel), 47, 1, 90-92, (1986) · Zbl 0607.28003
[32] Lavee, D.; Safrie, U.N.; Meilijson, I., For how long do trans-saharan migrants stop over at an oasis?, Ornis scandinavica, 22, 33-44, (1991)
[33] Le Cam, L., ()
[34] Lindsay, B.G., The geometry of mixture likelihoods: a general theory, Ann. statist., 11, 1, 86-94, (1983) · Zbl 0512.62005
[35] Lindsay, B.G., ()
[36] Müller, D.W.; Sawitzki, G., Excess mass estimates and tests for multimodality, J. amer. statist. assoc., 86, 415, 738-746, (1991) · Zbl 0733.62040
[37] M. Pavlides, Nonparametric estimation of multivariate monotone densities. Ph.D. Thesis, University of Washington, 2008.
[38] M. Pavlides, Local asymptotic minimax theory for block-decreasing densities. Tech. rep., Frederick University, Nicosia, Cyprus, 2009. · Zbl 1244.62053
[39] Pfanzagl, J., Consistency of maximum likelihood estimators for certain nonparametric families, in particular: mixtures, J. statist. plann. inference, 19, 2, 137-158, (1988) · Zbl 0656.62044
[40] Polonik, W., Density estimation under qualitative assumptions in higher dimensions, J. multivariate anal., 55, 1, 61-81, (1995) · Zbl 0847.62027
[41] Polonik, W., Measuring mass concentrations and estimating density contour clusters—an excess mass approach, Ann. statist., 23, 3, 855-881, (1995) · Zbl 0841.62045
[42] Polonik, W., Minimum volume sets and generalized quantile processes, Stochastic process. appl., 69, 1, 1-24, (1997) · Zbl 0905.62053
[43] Polonik, W., The silhouette, concentration functions and ML-density estimation under order restrictions, Ann. statist., 26, 5, 1857-1877, (1998) · Zbl 1073.62523
[44] Prakasa Rao, B.L.S., Estimation of a unimodal density, Sankhyā ser. A, 31, 23-36, (1969) · Zbl 0181.45901
[45] Robertson, T., On estimating a density which is measurable with respect to a \(\sigma\)-lattice, Ann. math. statist., 38, 482-493, (1967) · Zbl 0157.48001
[46] Robertson, T.; Wright, F.T.; Dykstra, R.L., Order restricted statistical inference, () · Zbl 0645.62028
[47] Rockafellar, R.T., Convex analysis. Princeton mathematical series, (1970), Princeton University Press Princeton, N.J, No. 28 · Zbl 0229.90020
[48] Sager, T.W., An iterative method for estimating a multivariate mode and isopleth, J. amer. statist. assoc., 74, 329-339, (1979), 366, part 1 · Zbl 0428.62040
[49] Sager, T.W., Nonparametric maximum likelihood estimation of spatial patterns, Ann. statist., 10, 4, 1125-1136, (1982) · Zbl 0506.62025
[50] Shorack, G.R., Probability for statisticians, () · Zbl 0726.60079
[51] C. van Eeden, Maximum likelihood estimation of ordered probabilities. Statist. Afdeling S 188 (VP 5). Math. Centrum Amsterdam, 1956. · Zbl 0086.12802
[52] van Eeden, C., Maximum likelihood estimation of ordered probabilities, Nederl. akad. wetensch. proc. ser. A. 59 = indag. math., 18, 444-455, (1956) · Zbl 0086.12802
[53] C. van Eeden, 1956. Maximum likelihood estimation of ordered probabilities. II. Statist. Afdeling Rep. S 196 (VP7). Math. Centrum Amsterdam. · Zbl 0086.12802
[54] van Eeden, C., Maximum likelihood estimation of partially or completely ordered parameters. I, Nederl. akad. wetensch. proc. ser. A. 60 = indag. math., 19, 128-136, (1957) · Zbl 0086.12803
[55] van Eeden, C., Maximum likelihood estimation of partially or completely ordered parameters. II, Nederl. akad. wetensch. proc. ser. A. 60 = indag. math., 19, 201-211, (1957) · Zbl 0086.12803
[56] Wegman, E.J., A note on estimating a unimodal density, Ann. math. statist., 40, 1661-1667, (1969) · Zbl 0182.51902
[57] Wegman, E.J., Maximum likelihood estimation of a unimodal density function, Ann. math. statist., 41, 457-471, (1970) · Zbl 0195.48601
[58] Wegman, E.J., Maximum likelihood estimation of a unimodal density. II, Ann. math. statist., 41, 2169-2174, (1970) · Zbl 0223.62036
[59] Williamson, R.E., Multiply monotone functions and their Laplace transforms, Duke math. J., 23, 189-207, (1956) · Zbl 0070.28501
[60] Woodroofe, M.; Sun, J., A penalized maximum likelihood estimate of \(f(0 +)\) when \(f\) is nonincreasing, Statist. sinica, 3, 2, 501-515, (1993) · Zbl 0822.62020
[61] Wong, G.Y.C.; Yu, Q., Generalized MLE of a joint distribution function with multivariate interval-censored data, J. multivariate anal., 69, 155-166, (1999) · Zbl 0931.62084
[62] Shaohua, Yu; Qiqing, Yu; Wong, George, Y.C., Consistency of the generalized MLE of a joint distribution function with multivariate interval-censored data, J. multivariate anal., 97, 720-732, (2005) · Zbl 1333.62129
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.