## The Annals of Statistics

### Nonparametric estimation of multivariate convex-transformed densities

#### Abstract

We study estimation of multivariate densities p of the form p(x)=h(g(x)) for x∈ℝd and for a fixed monotone function h and an unknown convex function g. The canonical example is h(y)=ey for y∈ℝ;; in this case, the resulting class of densities $$\mathcal {P}(e^{-y})=\{p=\exp(-g): g\mbox{ is convex}\}$$ is well known as the class of log-concave densities. Other functions h allow for classes of densities with heavier tails than the log-concave class.

We first investigate when the maximum likelihood estimator exists for the class $\mathcal {P}(h)$ for various choices of monotone transformations h, including decreasing and increasing functions h. The resulting models for increasing transformations h extend the classes of log-convex densities studied previously in the econometrics literature, corresponding to h(y)=exp(y).

We then establish consistency of the maximum likelihood estimator for fairly general functions h, including the log-concave class $\mathcal {P}(e^{-y})$ and many others. In a final section, we provide asymptotic minimax lower bounds for the estimation of p and its vector of derivatives at a fixed point x0 under natural smoothness hypotheses on h and g. The proofs rely heavily on results from convex analysis.

#### Article information

Source
Ann. Statist., Volume 38, Number 6 (2010), 3751-3781.

Dates
First available in Project Euclid: 30 November 2010

https://projecteuclid.org/euclid.aos/1291126972

Digital Object Identifier
doi:10.1214/10-AOS840

Mathematical Reviews number (MathSciNet)
MR2766867

Zentralblatt MATH identifier
1204.62058

Subjects
Primary: 62G07: Density estimation 62H12: Estimation
Secondary: 62G05: Estimation 62G20: Asymptotic properties

#### Citation

Seregin, Arseni; Wellner, Jon A. Nonparametric estimation of multivariate convex-transformed densities. Ann. Statist. 38 (2010), no. 6, 3751--3781. doi:10.1214/10-AOS840. https://projecteuclid.org/euclid.aos/1291126972

#### References

• An, M. Y. (1998). Logconcavity versus logconvexity: A complete characterization. J. Econom. Theory 80 350–369.
• Avriel, M. (1972). r-convex functions. Math. Program. 2 309–323.
• Balabdaoui, F., Rufibach, K. and Wellner, J. A. (2009). Limit distribution theory for maximum likelihood estimation of a log-concave density. Ann. Statist. 37 1299–1331.
• 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.
• Borell, C. (1975). Convex set functions in d-space. Period. Math. Hungar. 6 111–136.
• 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.
• Bronšteĭn, E. M. (1976). ɛ-entropy of convex sets and functions. Sibirsk. Mat. Ž. 17 508–514, 715.
• 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.
• 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.
• 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.
• Dharmadhikari, S. and Joag-Dev, K. (1988). Unimodality, Convexity and Applications. Academic Press, Boston, MA.
• Donoho, D. L. and Liu, R. C. (1991). Geometrizing rates of convergence. II, III. Ann. Statist. 19 633–667, 668–701.
• Dudley, R. M. (1999). Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics 63. Cambridge Univ. Press, Cambridge.
• Dümbgen, L., Hüsler, A. and Rufibach, K. (2007). Active set and EM algorithms for log-concave densities based on complete and censored data. Technical report, Univ. Bern. Available at arXiv:0707.4643.
• 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.
• Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001). Estimation of a convex function: Characterizations and asymptotic theory. Ann. Statist. 29 1653–1698.
• Ibragimov, I. A. (1956). On the composition of unimodal distributions. Teor. Veroyatnost. i Primenen. 1 283–288.
• Johnson, N. L. and Kotz, S. (1972). Distributions in Statistics: Continuous Multivariate Distributions. Wiley, New York.
• Jongbloed, G. (2000). Minimax lower bounds and moduli of continuity. Statist. Probab. Lett. 50 279–284.
• Koenker, R. and Mizera, I. (2010). Quasi-concave density estimation. Ann. Statist. 38 2998–3027.
• Okamoto, M. (1973). Distinctness of the eigenvalues of a quadratic form in a multivariate sample. Ann. Statist. 1 763–765.
• Pal, J. K., Woodroofe, M. B. and Meyer, M. C. (2007). Estimating a Polya frequency function. In Complex Datasets and Inverse Problems: Tomography, Networks and Beyond. Institute of Mathematical Statistics Lecture Notes—Monograph Series 54 239–249. IMS, Beachwood, OH.
• Prékopa, A. (1973). On logarithmic concave measures and functions. Acta Sci. Math. (Szeged) 34 335–343.
• Rinott, Y. (1976). On convexity of measures. Ann. Probab. 4 1020–1026.
• Rockafellar, R. T. (1970). Convex Analysis. Princeton Mathematical Series 28. Princeton Univ. Press, Princeton.
• Rufibach, K. (2006). Log-concave density estimation and bump hunting for I.I.D. observations. Ph.D. thesis, Univ. Bern and Göttingen.
• Rufibach, K. (2007). Computing maximum likelihood estimators of a log-concave density function. J. Stat. Comput. Simul. 77 561–574.
• Schuhmacher, D. and Duembgen, L. (2010). Consistency of multivariate log-concave density estimators. Statist. Probab. Lett. 80 376–380.
• 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.
• Seregin, A. and Wellner, J. A. (2010). Supplement to “Nonparametric estimation of multivariate convex-transformed densities.” DOI: 10.1214/10-AOS840SUPP.
• Uhrin, B. (1984). Some remarks about the convolution of unimodal functions. Ann. Probab. 12 640–645.
• van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes, with Applications to Statistics. Springer, New York.
• Walther, G. (2010). Inference and modeling with log-concave distributions. Statist. Sci. 24 319–327.

#### Supplemental materials

• Supplementary material: Omitted Proofs and Some Facts from Convex Analysis. In the supplement, we provide omitted proofs and some basic facts from convex analysis used in this paper.