The Annals of Statistics

Maximum likelihood estimation of smooth monotone and unimodal densities

P. P. B. Eggermont and V. N. LaRiccia

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Abstract

We study the nonparametric estimation of univariate monotone and unimodal densities usingthe maximum smoothed likelihood approach. The monotone estimator is the derivative of the least concave majorant of the distribution correspondingto a kernel estimator.We prove that the mapping on distributions $\Phi$ with density $\varphi$,

$$\varphi \mapsto \text{the derivative of the least concave majorant of $\Phi},$$

is a contraction in all $L^P$ norms $(1 \leq p \leq \infty)$, and some other “distances” such as the Hellinger and Kullback–Leibler distances. The contractivity implies error bounds for monotone density estimation. Almost the same error bounds hold for unimodal estimation.

Article information

Source
Ann. Statist., Volume 28, Number 3 (2000), 922-947.

Dates
First available in Project Euclid: 12 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1015952005

Digital Object Identifier
doi:10.1214/aos/1015952005

Mathematical Reviews number (MathSciNet)
MR1792794

Zentralblatt MATH identifier
1105.62332

Subjects
Primary: 62G07: Density estimation

Keywords
Maximum likelihood estimation monotone and unimodal densities least concave majorants contractions $L^1$ error bounds

Citation

Eggermont, P. P. B.; LaRiccia, V. N. Maximum likelihood estimation of smooth monotone and unimodal densities. Ann. Statist. 28 (2000), no. 3, 922--947. doi:10.1214/aos/1015952005. https://projecteuclid.org/euclid.aos/1015952005


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References

  • Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. (1972). Statistical Inference under Order Restrictions. Wiley, New York.
  • Bickel, P. J. and Fan, I. (1996). Some problems on the estimation of unimodal densities. Statist. Sinica 6 23-45. Birg´e, L. (1987a). Estimatinga density under restrictions: nonasymptotic minimax risk. Ann. Statist. 15 995-1012. Birg´e, L. (1987b). On the risk of histograms for estimatingdecreasingdensities. Ann. Statist. 15 1113-1022.
  • Birg´e, L. (1989). The Grenander estimator: a nonasymptotic approach. Ann. Statist. 17 1532-1549.
  • Birg´e, L. (1997). Estimation of unimodal densities without smoothness assumptions. Ann. Statist. 25 970-981.
  • Brunk, H. D. (1965). Conditional expectation given a lattice and applications. Ann. Math. Statist. 36 1339-1350.
  • Chernoff, H. (1964). Estimation of the mode. Ann. Inst. Statist. Math. 16 31-41.
  • Devroye, L. (1987). A Course in Density Estimation. Birkh¨auser, Boston.
  • Devroye, L. (1991). Exponential inequalities in nonparametric estimation. In Nonparametric Functional Estimation and Related Topics (G. Roussas, ed.) 31-44. Kluwer, Dordrecht.
  • Devroye, L. and Gy ¨orfi, L. (1985). Density Estimation: the L1-View. Wiley, New York.
  • Devroye, L. Gy ¨orfi, L. and Lugosi, G. (1996). A Probabilistic Theory of Pattern Recognition. Springer, New York.
  • Eddy, W. F. (1980). Optimal kernel estimators of the mode. Ann. Statist. 8 870-882.
  • Eggermont, P. P. B. and LaRiccia, V. N. (1995). Maximum smoothed likelihood density estimation. J. Nonparametr. Statist. 4 211-222.
  • Eggermont, P. P. B. and LaRiccia, V. N. (1999). Best asymptotically normal kernel density entropy estimators for smooth densities. IEEE Trans Inform. Theory 45 1321-1326.
  • Foug eres, A.-L. (1997). Estimation de densit´es unimodales. Canad. J. Statist. 25 375-387.
  • Grenander, U. (1956). On the theory of mortality measurements II. Skand. Akt. 39 125-153.
  • Groeneboom, P. (1985). Estimatinga monotone density. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer (L. M. Le Cam and R. A. Olshen, eds.) 539-555. Wadsworth, Belmont, CA.
  • Groeneboom, P., Hooghiemstra, G. and Lopuha¨a, H. P. (1999). Asymptotic normality of the L1 error of the Grenander estimator. Ann. Statist. 27 1316-1347.
  • Grund, B. and Hall, P. (1995). On the minimisation of Lp error in mode estimation. Ann. Statist. 23 2264-2284.
  • Hall, P. and Wehrly, T. E. (1991). A geometrical method for removing edge effects from kerneltype nonparametric regression estimators. J. Amer. Statist. Assoc. 86 665-672.
  • Hardy, G. H., Littlewood, G. E. and P ´olya, G. (1951). Inequalities. Cambridge Univ. Press.
  • Ibragimov, I. A. (1956). On the composition of unimodal distributions. Theory Probab. Appl. 1 255-260.
  • Jones, M. C. (1993). Simple boundary corrections for kernel density estimation. Statist. Comput. 3 135-146.
  • Kemperman, J. H. B. (1967). On the optimum rate of transmittinginformation. Lecture Notes in Math. 23 126-169. Springer, Berlin.
  • Lieb, E. H. and Loss, M. (1996). Analysis. Amer. Math. Soc., Providence, RI.
  • Mammen, E. (1991). Estimatinga smooth monotone regression function. Ann. Statist. 19 724-740.
  • Marshall, A. W. (1970). Discussion on Barlow and van Zwet's paper. In Nonparametric Techniques in Statistical Inference (M. L. Puri, ed.) 175-176. Cambridge Univ. Press.
  • M ¨uller, H.-G. (1993). On the boundary kernel method for nonparametric curve estimation near endpoints. Scand. J. Statist. 20 313-328.
  • Parzen, E. (1962). On estimation of a probability density function and mode. Ann. Math. Statist. 33 1065-1076.
  • Silverman, B. W. (1978). Weak and stronguniform consistency of the kernel estimate of a density and its derivatives. Ann. Statist. 6 177-184.
  • Silverman, B. W. (1982). On the estimation of a probability density function by the maximum penalized likelihood method. Ann. Statist. 10 795-810.
  • Wegman, E. I. (1970). Maximum likelihood estimation of a unimodal density II. Ann. Math. Statist. 6 2169-2174.
  • Wheeden, R. and Zygmund, A. (1977). Measure and Integral. North-Holland, Amsterdam.
  • Woodroofe, M. and Sun, J. (1993). A penalized maximum likelihood estimate of f 0+ when f is non-increasing Statist. Sinica 3 501-515.