The Annals of Statistics

Multiscale maximum likelihood analysis of a semiparametric model, with applications

Guenther Walther

Full-text: Open access

Abstract

A special semiparametric model for a univariate density is introduced that allows analyzing a number of problems via appropriate transformations. Two problems treated in some detail are testing for the presence of a mixture and detecting a wear-out trend in a failure rate. The analysis of the semiparametric model leads to an approach that advances the maximum likelihood theory of the Grenander estimator to a multiscale analysis. The construction of the corresponding test statistic rests on an extension of a result on a two-sided Brownian motion with quadratic drift to the simultaneous control of “excursions under parabolas” at various scales of a Brownian bridge. The resulting test is shown to be asymptotically optimal in the minimax sense regarding both rate and constant, and adaptive with respect to the unknown parameter in the semiparametric model. The performance of the method is illustrated with a simulation study for the failure rate problem and with data from a flow cytometry experiment for the mixture analysis.

Article information

Source
Ann. Statist. Volume 29, Number 5 (2001), 1297-1319.

Dates
First available in Project Euclid: 8 February 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1013203455

Digital Object Identifier
doi:10.1214/aos/1013203455

Mathematical Reviews number (MathSciNet)
MR1873332

Zentralblatt MATH identifier
1043.62043

Subjects
Primary: 62G10: Hypothesis testing
Secondary: 60F15: Strong theorems

Keywords
Multiscale analysis Grenander estimator minimax adaptive mixture log-concave failure rate penalized maximum likelihood

Citation

Walther, Guenther. Multiscale maximum likelihood analysis of a semiparametric model, with applications. Ann. Statist. 29 (2001), no. 5, 1297--1319. doi:10.1214/aos/1013203455. https://projecteuclid.org/euclid.aos/1013203455.


Export citation

References

  • Brooks, S. P. (1998). MCMC convergence diagnosis via multivariate bounds on log-concave densities. Ann. Statist. 26 398-433.
  • Chaudhuri, P. and Marron, J. S. (1999). SiZer for exploration of structures in curves. J. Amer. Statist. Assoc. 94 807-823.
  • Chernoff, H. (1964). Estimation of the mode. Ann. Inst. Statist. Math. 16 31-41.
  • Chow, Y. S. and Teicher, H. (1988). Probability Theory, 2nd ed. Springer, NewYork.
  • Dellaportas, P. and Smith, A. F. M. (1993). Bayesian inference for generalized linear and proportional hazards models via Gibbs sampling. J. Roy. Statist. Soc. Ser. C 42 443-460.
  • D ¨umbgen, L. and Spokoiny, V. G. (2001). Multiscale testing of qualitative hypotheses. Ann. Statist. 29 124-152.
  • Gilks, W. R. and Wild, P. (1992). Adaptive rejection sampling for Gibbs sampling. J. Roy. Statist. Soc. Ser. C 41 337-348.
  • Glaser, R. E. (1980). Bathtub and related failure rate characteristics. J. Amer. Statist. Assoc. 75 667-672.
  • Green, P. J. and Silverman, B. W. (1994). Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. Chapman and Hall, London.
  • Groeneboom, P. (1985). Estimating a monotone density. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer (L. Le Cam and R. A. Olshen, eds.) 2 539- 555. Wadsworth, Belmont, CA.
  • Groeneboom, P. (1989). Brownian motion with a parabolic drift and Airy functions. Probab. Theory related Fields 81 79-109.
  • Groeneboom, P., Hooghiemstra, G. and Lopua¨a, H. P. (1999). Asymptotic normality of the L1 error of the Grenander estimator. Ann. Statist. 27 1316-1347.
  • Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001). Estimation of a convex function: characterizations and asymptotic theory. Ann. Statist. To appear.
  • Hildenbrand, K. and Hildenbrand W. (1985). On the mean income effect: a data analysis of the U.K. family expenditure survey. In Contributions to Mathematical Economics (W. Hildenbrand and A. Mas-Colell, eds.) 247-263. North-Holland, Amsterdam.
  • Ingster, Y. I. (1993). Asymptotically minimax hypothesis testing for nonparametric alternatives I-III. Math. Methods Statist. 2 85-114, 171-189, 249-337.
  • Jongbloed, G. (1998). The iterative convex minorant algorithm for nonparametric estimation. J. Comput. Graph. Statist. 7 310-321.
  • Kim, J. and Pollard, D. (1990). Cube root asymptotics. Ann. Statist. 18 191-219.
  • Koml ´os, J., Major, P. and Tusn´ady, G. (1975). An approximation of partial sums of independent r. v.'s and the sample d.f. Z. Wahrsch. Verw. Gebiete 32 111-131.
  • Korostelev, A. and Nussbaum, M. (1995). Density estimation in the uniform norm and white noise approximation. Preprint 154, Weierstrass Institute, Berlin.
  • Liero, H., L¨auter, H. and Konakov, V. (1998). Nonparametric versus parametric goodness of fit. Statistics 31 115-149.
  • Lindsay, B. G. and Roeder, K. (1992). Residual diagnostics for mixture models. J. Amer. Statist. Assoc. 87 785-794.
  • Lindsay, B. G. and Roeder, K. (1997). Moment-based oscillation properties of mixture models. Ann. Statist. 25 378-386.
  • Miller, R. G. (1981). Survival Analysis. Wiley, NewYork.
  • Neumann, M. H. (1998). Strong approximation of density estimators from weakly dependent observations by density estimators from independent observations. Ann. Statist. 26 2014-2048.
  • Prakasa rao, B. L. S. (1969). Estimation of a unimodal density. Sankhy ¯a Ser. A 31 23-36.
  • Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference. Wiley, NewYork.
  • Roeder, K. (1994). A graphical technique for determining the number of components in a mixture of normals. J. Amer. Statist. Assoc. 89 487-495.
  • Sheather, S. J. and Jones, M. C. (1991). A reliable data-based bandwidth selection method for kernel density-estimation. J. Roy. Statist. Soc. Ser. B 53 683-690.
  • Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with applications to statistics. Wiley, NewYork.
  • Silverman, B. W. (1982). On the estimation of a probability density function by the maximum penalized likelihood method. Ann. Statist 10 795-810.
  • Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman and Hall, London.
  • Titterington, D. M., Smith, A. F. M. and Makov, U. E. (1985). Statistical Analysis of Finite Mixture Distributions. Wiley, NewYork. Walther, G. (2000a). Multiscale maximum likelihood analysis of a semiparametric model, with applications. Technical report, Dept. Statistics, Stanford Univ. Walther, G. (2000b). Detecting the presence of mixing with multiscale maximum likelihood. J. Amer. Statist. Assoc. To appear.
  • Woodroofe, M. and Sun, J. (1999). Testing uniformity versus a monotone density. Ann. Statist. 27 338-360.
  • Zhang, Y. L. and Newton, M. A. (1997). On calculating the nonparametric maximum likelihood estimator of a distribution given interval censored data. J. Comput. Graph. Statist. To appear.