The Annals of Statistics

Convergence of linear functionals of the Grenander estimator under misspecification

Hanna Jankowski

Abstract

Under the assumption that the true density is decreasing, it is well known that the Grenander estimator converges at rate $n^{1/3}$ if the true density is curved [Sankhyā Ser. A 31 (1969) 23–36] and at rate $n^{1/2}$ if the density is flat [Ann. Probab. 11 (1983) 328–345; Canad. J. Statist. 27 (1999) 557–566]. In the case that the true density is misspecified, the results of Patilea [Ann. Statist. 29 (2001) 94–123] tell us that the global convergence rate is of order $n^{1/3}$ in Hellinger distance. Here, we show that the local convergence rate is $n^{1/2}$ at a point where the density is misspecified. This is not in contradiction with the results of Patilea [Ann. Statist. 29 (2001) 94–123]: the global convergence rate simply comes from locally curved well-specified regions. Furthermore, we study global convergence under misspecification by considering linear functionals. The rate of convergence is $n^{1/2}$ and we show that the limit is made up of two independent terms: a mean-zero Gaussian term and a second term (with nonzero mean) which is present only if the density has well-specified locally flat regions.

Article information

Source
Ann. Statist., Volume 42, Number 2 (2014), 625-653.

Dates
First available in Project Euclid: 20 May 2014

https://projecteuclid.org/euclid.aos/1400592172

Digital Object Identifier
doi:10.1214/13-AOS1196

Mathematical Reviews number (MathSciNet)
MR3210981

Zentralblatt MATH identifier
1302.62045

Citation

Jankowski, Hanna. Convergence of linear functionals of the Grenander estimator under misspecification. Ann. Statist. 42 (2014), no. 2, 625--653. doi:10.1214/13-AOS1196. https://projecteuclid.org/euclid.aos/1400592172

References

• Anevski, D. and Hössjer, O. (2002). Monotone regression and density function estimation at a point of discontinuity. J. Nonparametr. Stat. 14 279–294.
• 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.
• Balabdaoui, F., Jankowski, H., Pavlides, M., Seregin, A. and Wellner, J. (2011). On the Grenander estimator at zero. Statist. Sinica 21 873–899.
• Balabdaoui, F., Jankowski, H., Rufibach, K. and Pavlides, M. (2013). Asymptotics of the discrete log-concave maximum likelihood estimator and related applications. J. R. Stat. Soc. Ser. B Stat. Methodol. 75 769–790.
• Beirlant, J., Dudewicz, E. J., Györfi, L. and van der Meulen, E. C. (1997). Nonparametric entropy estimation: An overview. Int. J. Math. Stat. Sci. 6 17–39.
• Birgé, L. (1987). On the risk of histograms for estimating decreasing densities. Ann. Statist. 15 1013–1022.
• Carolan, C. and Dykstra, R. (1999). Asymptotic behavior of the Grenander estimator at density flat regions. Canad. J. Statist. 27 557–566.
• Carolan, C. and Dykstra, R. (2001). Marginal densities of the least concave majorant of Brownian motion. Ann. Statist. 29 1732–1750.
• Cator, E. (2011). Adaptivity and optimality of the monotone least-squares estimator. Bernoulli 17 714–735.
• Chen, Y. and Samworth, R. J. (2013). Smoothed log-concave maximum likelihood estimation with applications. Statist. Sinica 23 1373–1398.
• Cule, M. and Samworth, R. (2010). Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density. Electron. J. Stat. 4 254–270.
• Cule, M., Samworth, R. and Stewart, M. (2010). Maximum likelihood estimation of a multi-dimensional log-concave density. J. R. Stat. Soc. Ser. B Stat. Methodol. 72 545–607.
• Dümbgen, L., Samworth, R. and Schuhmacher, D. (2011). Approximation by log-concave distributions, with applications to regression. Ann. Statist. 39 702–730.
• Dunford, N. and Schwartz, J. T. (1958). Linear Operators. I. General Theory. With the Assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics 7. Interscience, New York.
• Durot, C., Kulikov, V. N. and Lopuhaä, H. P. (2012). The limit distribution of the $L_\infty$-error of Grenander-type estimators. Ann. Statist. 40 1578–1608.
• Durot, C. and Lopuhaä, H. (2013). A Kiefer–Wolfowitz type of result in a general setting, with an application to smooth monotone estimation. Available at arXiv:1308.0417.
• Grenander, U. (1956). On the theory of mortality measurement. II. Skand. Aktuarietidskr. 39 125–153 (1957).
• Groeneboom, P. (1983). The concave majorant of Brownian motion. Ann. Probab. 11 1016–1027.
• Groeneboom, P. (1985). Estimating a monotone density. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, Vol. II (Berkeley, Calif., 1983). 539–555. Wadsworth, Belmont, CA.
• Groeneboom, P. (1986). Some current developments in density estimation. In Mathematics and Computer Science (Amsterdam, 1983). CWI Monogr. 1 163–192. North-Holland, Amsterdam.
• Groeneboom, P., Hooghiemstra, G. and Lopuhaä, H. P. (1999). Asymptotic normality of the $L_1$ error of the Grenander estimator. Ann. Statist. 27 1316–1347.
• Groeneboom, P. and Pyke, R. (1983). Asymptotic normality of statistics based on the convex minorants of empirical distribution functions. Ann. Probab. 11 328–345.
• Huang, Y. and Zhang, C.-H. (1994). Estimating a monotone density from censored observations. Ann. Statist. 22 1256–1274.
• Jankowski, H. (2014). Supplement to “Convergence of linear functionals of the Grenander estimator under misspecification.” DOI:10.1214/13-AOS1196SUPP.
• Jankowski, H. K. and Wellner, J. A. (2009). Estimation of a discrete monotone distribution. Electron. J. Stat. 3 1567–1605.
• Kiefer, J. and Wolfowitz, J. (1976). Asymptotically minimax estimation of concave and convex distribution functions. Z. Wahrsch. Verw. Gebiete 34 73–85.
• Kulikov, V. N. and Lopuhaä, H. P. (2008). Distribution of global measures of deviation between the empirical distribution function and its concave majorant. J. Theoret. Probab. 21 356–377.
• Le Cam, L. (1960). Locally asymptotically normal families of distributions. Certain approximations to families of distributions and their use in the theory of estimation and testing hypotheses. Univ. California Publ. Statist. 3 37–98.
• Marshall, A. W. (1970). Discussion on Barlow and van Zwet’s paper. In Nonparametric Techniques in Statistical Inference (Proc. Sympos., Indiana Univ., Bloomington, Ind., 1969) 174–176. Cambridge Univ. Press, London.
• Patilea, V. (1997). Convex models, NPMLE and misspecification. Ph.D. thesis, Univ. Catholique de Louvain.
• Patilea, V. (2001). Convex models, MLE and misspecification. Ann. Statist. 29 94–123.
• Prakasa Rao, B. L. S. (1969). Estimation of a unimodal density. Sankhyā Ser. A 31 23–36.
• Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York.
• Simon, J. (1987). Compact sets in the space $L^p(0,T;B)$. Ann. Mat. Pura Appl. (4) 146 65–96.
• van de Geer, S. A. (2000). Applications of Empirical Process Theory. Cambridge Series in Statistical and Probabilistic Mathematics 6. Cambridge Univ. Press, Cambridge.
• van de Geer, S. (2003). Asymptotic theory for maximum likelihood in nonparametric mixture models. Comput. Statist. Data Anal. 41 453–464.
• van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer, New York.
• Wasserman, L. (2006). All of Nonparametric Statistics. Springer Texts in Statistics. Springer, New York.

Supplemental materials

• Supplementary material: Supplement to “Convergence of linear functionals of the Grenander estimator under misspecification”. We provide some proofs and technical details, as well as additional discussions of the assumptions in Theorems 3.1 and 4.1.