Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency

arXiv:0709.0334v4 [math.ST] 9 Feb 2009 Bernoulli 15(1), 2009, 40–68 DOI: 10.3150/08-BEJ141 Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency LUTZ D¨UMBGEN1 and KASPAR RUFIBACH2 1Institute of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland. E-mail: duembgen@stat.unibe.ch 2Abteilung Biostatistik, Institut f¨ur Sozial- und Pr¨aventivmedizin, Universit¨at Z¨urich, Hirschen- graben 84, CH-8001 Z¨urich, Switzerland. E-mail: kaspar.rufibach@ifspm.uzh.ch We study nonparametric maximum likelihood estimation of a log-concave probability density and its distribution and hazard function. Some general properties of these estimators are derived from two characterizations. It is shown that the rate of convergence with respect to supremum norm on a compact interval for the density and hazard rate estimator is at least (log(n)/n)1/3 and typically (log(n)/n)2/5, whereas the difference between the empirical and estimated distribution function vanishes with rate op(n−1/2) under certain regularity assumptions. Keywords: adaptivity; bracketing; exponential inequality; gap problem; hazard function; method of caricatures; shape constraints 1. Introduction Two common approaches to nonparametric density estimation are smoothing methods and qualitative constraints. The former approach includes, among others, kernel density estimators, estimators based on discrete wavelets or other series expansions and estima- tors based on roughness penalization. Good starting points for the vast literature in this field are Silverman (1982, 1986) and Donoho et al. (1996). A common feature of all of these methods is that they involve certain tuning parameters, for example, the order of a kernel and the bandwidth. A proper choice of these parameters is far from trivial since optimal values depend on unknown properties of the underlying density f. The second approach avoids such problems by imposing qualitative properties on f, for example, monotonicity or convexity on certain intervals in the univariate case. Such assumptions are often plausible or even justified rigorously in specific applications. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2009, Vol. 15, No. 1, 40–68. This reprint differs from the original in pagination and typographic detail. 1350-7265 c⃝ 2009 ISI/BS Estimating log-concave densities 41 Density estimation under shape constraints was first considered by Grenander (1956), who found that the nonparametric maximum likelihood estimator (NPMLE) ˆf mon n of a non-increasing density function f on [0,∞) is given by the left derivative of the least concave majorant of the empirical cumulative distribution function on [0,∞). This work was continued by Rao (1969) and Groeneboom (1985, 1988), who established asymptotic distribution theory for n1/3(f −ˆf mon n )(t) at a fixed point t > 0 under certain regular- ity conditions and analyzed the non-Gaussian limit distribution. For various estimation problems involving monotone functions, the typical rate of convergence is Op(n−1/3) pointwise. The rate of convergence with respect to supremum norm is further deceler- ated by a factor of log(n)1/3 (Jonker and van der Vaart (2001)). For applications of monotone density estimation, consult, for example, Barlow et al. (1972) or Robertson et al. (1988). Monotone estimation can be extended to cover unimodal densities. Remember that a density f on the real line is unimodal if there exists a number M = M(f) such that f is non-decreasing on (−∞,M] and non-increasing on [M,∞). If the true mode is known a priori, unimodal density estimation boils down to monotone estimation in a straightfor- ward manner, but the situation is different if M is unknown. In that case, the likelihood is unbounded, problems being caused by observations too close to a hypothetical mode. Even if the mode was known, the density estimator is inconsistent at the mode, a phe- nomenon called “spiking”. Several methods were proposed to remedy this problem (see Wegman (1970), Woodroofe and Sun (1993), Meyer and Woodroofe (2004) or Kulikov and Lopuha¨a (2006)), but all of them require additional constraints on f. The combination of shape constraints and smoothing was assessed by Eggermont and La-Riccia (2000). To improve the slow rate of convergence of n−1/3 in the space L1(R) for arbitrary unimodal densities, they derived a Grenander-type estimator by taking the derivative of the least concave majorant of an integrated kernel density estimator rather than the empirical distribution function directly, yielding a rate of convergence of Op(n−2/5). Estimation of a convex decreasing density on [0,∞) was pioneered by Anevski (1994, 2003). The problem arose in a study of migrating birds discussed by Hampel (1987). Groeneboom et al. (2001) provide a characterization of the estimator, as well as con- sistency and limiting

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