## The Annals of Statistics

### Inference for the mode of a log-concave density

#### Abstract

We study a likelihood ratio test for the location of the mode of a log-concave density. Our test is based on comparison of the log-likelihoods corresponding to the unconstrained maximum likelihood estimator of a log-concave density and the constrained maximum likelihood estimator where the constraint is that the mode of the density is fixed, say at $m$. The constrained estimation problem is studied in detail in Doss and Wellner (2018). Here, the results of that paper are used to show that, under the null hypothesis (and strict curvature of $-\log f$ at the mode), the likelihood ratio statistic is asymptotically pivotal: that is, it converges in distribution to a limiting distribution which is free of nuisance parameters, thus playing the role of the $\chi_{1}^{2}$ distribution in classical parametric statistical problems. By inverting this family of tests, we obtain new (likelihood ratio based) confidence intervals for the mode of a log-concave density $f$. These new intervals do not depend on any smoothing parameters. We study the new confidence intervals via Monte Carlo methods and illustrate them with two real data sets. The new intervals seem to have several advantages over existing procedures. Software implementing the test and confidence intervals is available in the R package \verb+logcondens.mode+.

#### Article information

Source
Ann. Statist., Volume 47, Number 5 (2019), 2950-2976.

Dates
Revised: October 2018
First available in Project Euclid: 3 August 2019

https://projecteuclid.org/euclid.aos/1564797869

Digital Object Identifier
doi:10.1214/18-AOS1770

Mathematical Reviews number (MathSciNet)
MR3988778

#### Citation

Doss, Charles R.; Wellner, Jon A. Inference for the mode of a log-concave density. Ann. Statist. 47 (2019), no. 5, 2950--2976. doi:10.1214/18-AOS1770. https://projecteuclid.org/euclid.aos/1564797869

#### References

• Abraham, C., Biau, G. and Cadre, B. (2003). Simple estimation of the mode of a multivariate density. Canad. J. Statist. 31 23–34.
• Bahadur, R. R. and Savage, L. J. (1956). The nonexistence of certain statistical procedures in nonparametric problems. Ann. Math. Stat. 27 1115–1122.
• 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.
• Banerjee, M. (2007). Likelihood based inference for monotone response models. Ann. Statist. 35 931–956.
• Banerjee, M. and Wellner, J. A. (2001). Likelihood ratio tests for monotone functions. Ann. Statist. 29 1699–1731.
• Birgé, L. (1997). Estimation of unimodal densities without smoothness assumptions. Ann. Statist. 25 970–981.
• Blum, J. R. and Rosenblatt, J. (1966). On some statistical problems requiring purely sequential sampling schemes. Ann. Inst. Statist. Math. 18 351–355.
• Chen, Y. and Samworth, R. J. (2016). Generalized additive and index models with shape constraints. J. R. Stat. Soc. Ser. B. Stat. Methodol. 78 729–754.
• 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.
• Dharmadhikari, S. and Joag-Dev, K. (1988). Unimodality, Convexity, and Applications. Probability and Mathematical Statistics. Academic Press, Boston, MA.
• Donoho, D. L. (1988). One-sided inference about functionals of a density. Ann. Statist. 16 1390–1420.
• Donoho, D. L. and Liu, R. C. (1991). Geometrizing rates of convergence. II, III. Ann. Statist. 19 633–667, 668–701.
• Doss, C. R. (2013a). logcondens.mode: Compute MLE of log-concave density on R with fixed mode, and perform inference for the mode. R package version 1.0.1. Available at http://CRAN.R-project.org/package=logcondens.mode.
• Doss, C. R. (2013b). Shape-constrained inference for concave-transformed densities and their modes. Ph.D. thesis, Univ. Washington, Seattle, WA.
• Doss, C. R. (2018). Concave regression: Value-constrained estimation and likelihood ratio-based inference. Math. Program. To appear. DOI:10.1007/s10107-018-1338-5.
• Doss, C. R. and Wellner, J. A. (2016a). Global rates of convergence of the MLEs of log-concave and $s$-concave densities. Ann. Statist. 44 954–981.
• Doss, C. R. and Wellner, J. A. (2016b). Inference for the mode of a log-concave density. Available at arXiv:1611.10348v2.
• Doss, C. R. and Wellner, J. A. (2018). Log-concave density estimation with symmetry or modal constraints. Available at arXiv:1611.10335v3.
• Doss, C. R. and Wellner, J. A. (2019). Supplement to “Inference for the mode of a log-concave density.” DOI:10.1214/18-AOS1770SUPP.
• 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.
• Dümbgen, L., Samworth, R. and Schuhmacher, D. (2011). Approximation by log-concave distributions, with applications to regression. Ann. Statist. 39 702–730.
• Eddy, W. F. (1980). Optimum kernel estimators of the mode. Ann. Statist. 8 870–882.
• Ehm, W. (1996). Adaptive kernel estimation of a cusp-shaped mode. In Applied Mathematics and Parallel Computing 109–120. Physica, Heidelberg.
• Groeneboom, P. and Jongbloed, G. (2015). Nonparametric confidence intervals for monotone functions. Ann. Statist. 43 2019–2054.
• Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001a). A canonical process for estimation of convex functions: The “invelope” of integrated Brownian motion $+t^{4}$. Ann. Statist. 29 1620–1652.
• Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001b). Estimation of a convex function: Characterizations and asymptotic theory. Ann. Statist. 29 1653–1698.
• Hall, P. (1984). On unimodality and rates of convergence for stable laws. J. Lond. Math. Soc. (2) 30 371–384.
• Hall, P. and Johnstone, I. (1992). Empirical functional and efficient smoothing parameter selection. J. Roy. Statist. Soc. Ser. B 54 475–530. With discussion and a reply by the authors.
• Han, Q. and Wellner, J. A. (2016). Approximation and estimation of $s$-concave densities via Rényi divergences. Ann. Statist. 44 1332–1359.
• Hanson, D. L. and Pledger, G. (1976). Consistency in concave regression. Ann. Statist. 4 1038–1050.
• Härdle, W., Marron, J. S. and Wand, M. P. (1990). Bandwidth choice for density derivatives. J. Roy. Statist. Soc. Ser. B 52 223–232.
• Has’minskii, R. Z. (1979). Lower bound for the risks of nonparametric estimates of the mode. In Contribution to Statistics: Jaroslav Hájek Memorial Volume 91–97. Reidel, Dordrecht.
• Hazelton, M. (1996a). Bandwidth selection for local density estimators. Scand. J. Stat. 23 221–232.
• Hazelton, M. L. (1996b). Optimal rates for local bandwidth selection. J. Nonparametr. Stat. 7 57–66.
• Herrmann, E. and Ziegler, K. (2004). Rates on consistency for nonparametric estimation of the mode in absence of smoothness assumptions. Statist. Probab. Lett. 68 359–368.
• Hildreth, C. (1954). Point estimates of ordinates of concave functions. J. Amer. Statist. Assoc. 49 598–619.
• Hoffleit, D. and Warren, W. H. Jr. (1991). Bright Star Catalogue. Yale Univ. Observatory, New Haven, CT. Available at http://adsabs.harvard.edu/abs/1995yCat.5050....0H.
• Hogg, R. V. and Craig, A. T. (1970). Introduction to Mathematical Statistics. 3rd ed. Macmillan, New York.
• Ibragimov, I. A. (1956). On the composition of unimodal distributions. Theory Probab. Appl. 1 225–260.
• Jankowski, H. K. and Wellner, J. A. (2009). Nonparametric estimation of a convex bathtub-shaped hazard function. Bernoulli 15 1010–1035.
• Kim, J. (1994). Asymptotic theory for multi-dimensional mode estimator. J. Korean Statist. Soc. 23 251–269.
• Kim, A. K. H. and Samworth, R. J. (2016). Global rates of convergence in log-concave density estimation. Ann. Statist. 44 2756–2779.
• Klemelä, J. (2005). Adaptive estimation of the mode of a multivariate density. J. Nonparametr. Stat. 17 83–105.
• Koenker, R. and Mizera, I. (2010). Quasi-concave density estimation. Ann. Statist. 38 2998–3027.
• Konakov, V. D. (1973). The asymptotic normality of the mode of multivariate distributions. Teor. Veroyatn. Primen. 18 836–842.
• Kuchibhotla, A. K., Patra, R. K. and Sen, B. (2017). Efficient estimation in convex single index models. Available at arXiv:1708.00145v2.
• Léger, C. and Romano, J. P. (1990). Bootstrap choice of tuning parameters. Ann. Inst. Statist. Math. 42 709–735.
• Mallows, C. L. (1972). A note on asymptotic joint normality. Ann. Math. Stat. 43 508–515.
• Mammen, E. (1991). Nonparametric regression under qualitative smoothness assumptions. Ann. Statist. 19 741–759.
• Mammen, E. and Park, B. U. (1997). Optimal smoothing in adaptive location estimation. J. Statist. Plann. Inference 58 333–348.
• Müller, H.-G. (1989). Adaptive nonparametric peak estimation. Ann. Statist. 17 1053–1069.
• Pal, J. K., Woodroofe, M. and Meyer, M. (2007). Estimating a Pólya frequency function${}_{2}$. In Complex Datasets and Inverse Problems. Institute of Mathematical Statistics Lecture Notes—Monograph Series 54 239–249. IMS, Beachwood, OH.
• Parzen, E. (1962). On estimation of a probability density function and mode. Ann. Math. Stat. 33 1065–1076.
• Pfanzagl, J. (1998). The nonexistence of confidence sets for discontinuous functionals. J. Statist. Plann. Inference 75 9–20.
• Pfanzagl, J. (2000). On local uniformity for estimators and confidence limits. J. Statist. Plann. Inference 84 27–53.
• R Core Team (2016). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.
• Rockafellar, R. T. (1970). Convex Analysis. Princeton Mathematical Series 28. Princeton Univ. Press, Princeton, NJ.
• Romano, J. P. (1988a). Bootstrapping the mode. Ann. Inst. Statist. Math. 40 565–586.
• Romano, J. P. (1988b). On weak convergence and optimality of kernel density estimates of the mode. Ann. Statist. 16 629–647.
• Rufibach, K. (2006). Log-concave density estimation and bump hunting for IID observations. Ph.D. thesis, Univ. Bern and Gottingen.
• Sager, T. W. (1978). Estimation of a multivariate mode. Ann. Statist. 6 802–812.
• Samanta, M. (1973). Nonparametric estimation of the mode of a multivariate density. South African Statist. J. 7 109–117.
• Samworth, R. J. and Wand, M. P. (2010). Asymptotics and optimal bandwidth selection for highest density region estimation. Ann. Statist. 38 1767–1792.
• Saumard, A. and Wellner, J. A. (2014). Log-concavity and strong log-concavity: A review. Stat. Surv. 8 45–114.
• 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.
• Silverman, B. W. (1982). On the estimation of a probability density function by the maximum penalized likelihood method. Ann. Statist. 10 795–810.
• Singh, R. (1963). Existence of bounded length confidence intervals. Ann. Math. Stat. 34 1474–1485.
• Singh, R. (1968). Non-existence of estimates of prescribed accuracy in fixed sample size. Canad. Math. Bull. 11 135–139.
• Tsybakov, A. B. (1990). Recurrent estimation of the mode of a multidimensional distribution. Problemy Peredachi Informatsii 26 38–45.
• Walther, G. (2002). Detecting the presence of mixing with multiscale maximum likelihood. J. Amer. Statist. Assoc. 97 508–513.
• Wilks, S. S. (1938). The large-sample distribution of the likelihood ratio for testing composite hypotheses. Ann. Math. Stat. 9 60–62.
• Ziegler, K. (2001). On bootstrapping the mode in the nonparametric regression model with random design. Metrika 53 141–170.

#### Supplemental materials

• Supplement to “Inference for the mode of a log-concave density”. In the supplement, we provide additional proofs and technical details that were omitted from the main paper.