Fadoua Balabdaoui

Kaspar Rufibach

Jon A Wellner

LIMIT DISTRIBUTION THEORY FOR MAXIMUM LIKELIHOOD ESTIMATION OF A LOG-CONCAVE DENSITY

Keywords: asymptotic distribution, integral of Brownian motion, invelope process, log-concave density estimation, lower bounds, maximum likelihood, mode estimation, nonparametric estimation, qualitative assumptions, shape constraints, strongly unimodal, unimodal FADOUA BALABDAOUI, KASPAR RUFIBACH

WELLNER

, page 492.

The log-concave shape constraint is appealing for many reasons: (1) Many parametric models, for a certain range of their parameters, are in fact log-concave, e.g. normal, uniform, gamma (r, λ) for r ≥ 1, beta(a, b) for a ≥ 1 and b ≥ 1, generalized Pareto, Gumbel, Fréchet, logistic or Laplace, to mention only some of these models. Therefore, assuming log-concavity offers a flexible non-parametric alternative to purely parametric models. Note that a log-concave density need not be symmetric.

, and Section 7 in Dümbgen et al. (2007). Thus, having an estimator (and its limiting distribution) for f at hand provides almost automatically estimators (and limiting distributions) for those functions. Corollary 2.3 illustrates this for the hazard rate.

.

.

We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, i.e. a density of the form f 0 = exp ϕ 0 where ϕ 0 is a concave function on R. Existence, form, characterizations and uniform rates of convergence of the MLE are given by [START_REF] Rufibach | Log-concave density estimation and bump hunting for I[END_REF] and Dümbgen and Rufibach (2007). The characterization of the log-concave MLE in terms of distribution functions is the same (up to sign) as the characterization of the least squares estimator of a convex density on [0, ∞) as studied by [START_REF] Groeneboom | Estimation of a convex function: characterizations and asymptotic theory[END_REF]. We use this connection to show that the limiting distributions of the MLE and its derivative are, under comparable smoothness assumptions, the same (up to sign) as in the convex density estimation problem. In particular, changing the smoothness assumptions of [START_REF] Groeneboom | Estimation of a convex function: characterizations and asymptotic theory[END_REF] slightly by allowing some higher derivatives to vanish at the point of interest, we find that the pointwise limiting distributions depend on the second and third derivatives at 0 of H k , the "lower invelope" of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of ϕ0 = log f0 at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode M (f 0) and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values.

Introduction.

1.1. Log-concave densities. A probability density f on the real line is called log-concave if it can be written as f (x) = exp ϕ(x) (8) It should be noted that no arbitrary choices such as bandwidth, kernel, or prior are involved in the estimation of a log-concave density; these are all obviated by this shape restriction. (9) We expect good adaptivity properties of the MLE f n in the class LC.

Here is some elaboration on this last point. It is fairly well-known that nonparametric maximum likelihood estimators of monotone functions have desirable adaptation properties with respect to the smoothness of the underlying true monotone function: see [START_REF] Birgé | The Grenander estimator: a nonasymptotic approach[END_REF] for a study of the Grenander estimator of a monotone density; see [START_REF] Low | Estimating monotone functions[END_REF] for the construction of an adaptive estimator (at a point) in the context of the white noise model; and see [START_REF] Low | On adaptive estimation of linear functionals[END_REF] for an illuminating discussion of the issues. [START_REF] Cai | Estimating convex functions[END_REF] have initiated a study of adaptive estimation in the setting of convex function estimation, but much more remains to be done. Dümbgen and Rufibach (2007) show that the MLE ϕ n of ϕ 0 adapts to the local smoothness of ϕ 0 in the following sense: if ϕ 0 is Hölder(β, L) on [A, B] ⊂ [x : f 0 (x) > 0] where 1 ≤ β ≤ 2, then ϕ n -ϕ 0 b a = O p ((n -1 log n) β/(2β+1) ) for any A < a < b < B and g b a := sup a≤x≤b |g(x)|. This carries over similarly to yield local adaptivity properities of f n . Current evidence suggests that the nonparametric MLE's of convex densities as studied in Groeneboom, [START_REF] Groeneboom | Estimation of a convex function: characterizations and asymptotic theory[END_REF] and of log-concave densities as in Dümbgen and Rufibach (2007) are also adaptive to local smoothness in terms of their local limiting distributions. We intend to investigate this in more detail in future work.

For properties of (random variables with) log-concave densities we refer to [START_REF] Dharmadhikari | Unimodality, convexity, and applications[END_REF], [START_REF] Marshall | Inequalities: theory of majorization and its applications[END_REF], and [START_REF] Rufibach | Log-concave density estimation and bump hunting for I[END_REF]. Log-concavity of a density f implies certain shape constraints for functions derived from f , such as the distribution function, the tail or hazard function. See [START_REF] An | Logconcavity versus logconvexity: a complete characterization[END_REF] for comparisons with the related notion of a log-convex density.

1.2. Log-concave density estimation. Now let X (1) < X (2) < • • • < X (n) be the order statistics of n independent random variables X 1 , . . . , X n distributed according to a log-concave probability density f 0 = exp ϕ 0 on R. The distribution function corresponding to f 0 is denoted by F 0 .

The maximum likelihood estimator (MLE) of a log-concave density was introduced in [START_REF] Rufibach | Log-concave density estimation and bump hunting for I[END_REF] and Dümbgen and Rufibach (2007). Algorithmic aspects were treated in [START_REF] Rufibach | Computing maximum likelihood estimators of a log-concave density function[END_REF] and in a more general framework in [START_REF] Dümbgen | Active set and EM algorithms for log-concave densities based on complete and censored data[END_REF], while consistency with respect to the Hellinger metric was established by [START_REF] Pal | Estimating a polya frequency function[END_REF], and rates of convergence of f n and F n were established by Dümbgen and Rufibach (2007). Since the derivation of the MLE of a log-concave density is extensively treated in these references, we only briefly recall its definition and the properties relevant for this paper.

If C denotes the class of all concave functions ϕ : R → [-∞, ∞), the estimator ϕ n of ϕ 0 is the maximizer of the "adjusted" criterion function

L(ϕ) = R ϕ(x)dF n (x) - R exp ϕ(x)dx
over C, where F n is the empirical distribution function of the observations. The log-concave density estimator is then f n := exp ϕ n , which exists and is unique.

1.3. Some properties of ϕ n . For any continuous piecewise linear function h n : [X (1) , X (n) ] → R such that the knots of h n coincide with (some of) the order statistics X (1) , . . . , X (n) , introduce the set of knots S n (h n ) of h n as

S n (h n ) := {t ∈ (X (1) , X (n) ) : h ′ n (t-) > h ′ n (t+)} ∪ {X (1) , X (n) }.
Dümbgen and [START_REF] Rufibach | Computing maximum likelihood estimators of a log-concave density function[END_REF] found that ϕ n is piecewise linear, that

ϕ n = -∞ on R \ [X (1) , X (n)
] and that the knots of ϕ n only occur at (some of the) ordered observations

X (1) < • • • < X (n)
. The latter property is entirely different from the estimation of a k-monotone density for k > 1 (see below), where the knots fall strictly between observations with probability equal to 1. According to Theorem 2.4 in Dümbgen and Rufibach (2007), the estimator ϕ n has the following characterization. For x ≥ X (1) (recall that ϕ n := 0 outside [X (1) , X (n) ]) define the processes

F n (x) := x X (1) exp( ϕ n (t))dt, H n (x) := x X (1)
F n (t)dt,

H n (x) := x X (1) F n (t)dt = x -∞ F n (t)dt.
Then the concave function ϕ n is the MLE of the log-density ϕ 0 if and only if

H n (x) ≤ H n (x) for all x ≥ X (1) = H n (x) if x ∈ S n ( ϕ n ). (1.1)
1.4. Other shape constraints. Maximum likelihood estimation of a monotone density f 0 on [0, ∞) was first studied by [START_REF] Grenander | On the theory of mortality measurement[END_REF] who found that a function fn is the estimator of f 0 if and only if it is equal to the imsart-aos ver. 2007/09/18 file: ModeLogconc-v11.tex date: January 3, 2008 left derivative of the least concave majorant of F n . Prakasa [START_REF] Rao | Estimation of a unimodal density[END_REF] established the asymptotic distribution theory at a point x 0 > 0 such that f ′ 0 (x 0 ) < 0 and f ′ 0 is continuous in a neighborhood of x 0 :

n 1/3 ( fn (x 0 ) -f 0 (x 0 )) → d |f ′ 0 (x 0 )f 0 (x 0 )/2| 1/3 Z
where Z is the slope at zero of the (least) concave majorant of the process W (t) -t 2 , t ∈ R for two-sided Brownian motion W starting at 0. Under the assumption that the true density f 0 is convex and non-increasing on [0, ∞), [START_REF] Groeneboom | Estimation of a convex function: characterizations and asymptotic theory[END_REF] defined and characterized the MLE fn as well as the least squares estimator of f 0 . Here, at any point

x 0 > 0 such that f ′′ 0 (x 0 ) > 0 and f ′′ 0 is continuous in a neighborhood of x 0 , n 2/5 ( fn (x 0 ) -f 0 (x 0 )) → d 24 -1 f 2 0 (x 0 )f ′′ 0 (x 0 ) 1/5 H ′′ (0),
where H is a random cubic spline such that H ′′ is convex and H stays above integrated two-sided Brownian motion +t 4 and touches the Gaussian process exactly at those points where H ′′ changes its slope, see Groeneboom, Jongbloed and Wellner (2001a).

The classes of monotone and convex decreasing densities are particular cases of the class of k-monotone densities. A density function p on [0, ∞) is 1-monotone if it is non-increasing; it is 2-monotone if it is non-increasing and convex; and it is k-monotone for k ≥ 3 if and only if (-1) j p (j) is nonnegative, non-increasing, and convex for j = 0, . . . , k -2. [START_REF] Balabdaoui | Estimation of a k-monotone density: limiting distribution theory and the spline connection[END_REF] were able to adapt the approach of [START_REF] Groeneboom | Estimation of a convex function: characterizations and asymptotic theory[END_REF] to this general class of densities. However, their result depends on the validity of a conjecture about an upper bound for the error in a particular Hermite interpolation via odd-degree splines.

We find that log-concave estimation shares many similarities with the aforementioned shape-constrained estimation problems. In particular the limiting distribution of the MLE, our nonparametric estimator, involves a stochastic process whose second derivative is concave, and which stays below an integrated Brownian motion minus t k+2 . The even integer k determines the number of vanishing derivatives of the true concave function ϕ 0 at the estimation point x 0 . 1.5. Organization of the paper. In Section 2, we establish the limiting distributions of the MLE estimators, ϕ n and f n , at a fixed point x 0 ∈ R under some specified working assumptions. The characterization of either ϕ n or f n given in (1.1) coincides, except for the direction of the inequality, with that of the least squares estimator of a convex decreasing density, studied by [START_REF] Groeneboom | Estimation of a convex function: characterizations and asymptotic theory[END_REF]; see their Lemma 2.2, p. 1657. This enables us to adopt the general scheme of the proof in their paper.

Log-concave densities f and their logarithm ϕ can easily have vanishing second and higher derivatives at fixed points as can be seen from the density function

f 0 (x) = √ 2 Γ(3/4) π exp(-x 4 ), x ∈ R.
In this case ϕ (j) 0 (x 0 ) = 0, j = 1, 2, 3 for x 0 = 0, and ϕ (4) 0 (x 0 ) = 0. The following "tilted" version of f 0 shows that vanishing second derivatives of ϕ 0 can also occur at points other than the mode of f :

f0 (x) = exp(a + bx)f 0 (x) = ã exp(bx -x 4 )
where ã = ã(b) := 1/ R exp(bx -x 4 )dx; in this case φ0 := log f0 satisfies φ′′ 0 (0) = 0, but the mode m0 := M ( f0 ) = (b/4) 1/3 > 0 when b > 0, and φ′′ 0 ( m0 ) = -12(b/4) 2/3 < 0. Thus the formulation of our asymptotic results allows higher derivatives of the concave function ϕ 0 to vanish at the estimation point. This is somewhat more general than the assumptions of [START_REF] Groeneboom | Estimation of a convex function: characterizations and asymptotic theory[END_REF] (where a natural assumption is that the second derivative is positive at the point of interest, but similar vanishing of second derivatives and existence of a non-zero higher order derivative can also easily occur), but it is analogous to the results of [START_REF] Wright | The asymptotic behavior of monotone regression estimates[END_REF] and [START_REF] Leurgans | Asymptotic distributions of slope-of-greatest-convex-minorant estimators[END_REF] for nonparametric estimation of a monotone regression function. Similar results for the Grenander estimator of a monotone density are stated by [START_REF] Anevski | Monotone regression and density function estimation at a point of discontinuity[END_REF]. We find that the respective limiting distributions of the MLE and its first derivative depend on a stochastic process, H k , equal almost surely to the "lower invelope" (or just "invelope") on R of the integrated Brownian motion minus t k+2 where k is the order of the first non-zero derivative of ϕ 0 at the point of interest.

In Section 3, the estimation point x 0 is taken to be equal to the mode, m 0 , defined to be the smallest point in the modal interval of the log-concave density f 0 . A natural estimator of m 0 , which we denote by M n , can be taken to be the smallest number maximizing the MLE ϕ n , or equivalently the smallest number maximizing the MLE f n . In this section, we establish our second main result: the asymptotic distribution of M n . Under the assumption that the second derivative f ′′ 0 (m 0 ) < 0, we show that this distribution depends on the random variable defined to be the argmax or mode of H

(2) 2 on R. When the second, third, and higher derivatives of order k -1 or lower vanish at imsart-aos ver. 2007/09/18 file: ModeLogconc-v11.tex date: January 3, 2008 m 0 but f (k) 0 (m 0 ) < 0, then the limit distribution depends on the mode of H

(2) k . Proofs are deferred to Section 4. To illustrate all the quantities for which we provide limiting distributions, in Figure 1 we give plots of f n , ϕ n , F n , and λ n = f n /(1 -F n ), based on two samples of sizes n = 20 and n = 200 drawn from a Gamma(2, 1) density f 0 (x) = xe -x 1 [0,∞) (x). All these plots were generated using the R-package logcondens, see [START_REF] Rufibach | logcondens: Estimate a Log-Concave Probability Density from iid Observations[END_REF].

2. Limiting distribution theory. To state the main result, we make the following assumptions.

2.1. Assumptions. Fix x 0 ∈ R. We suppose that the true density f 0 = exp ϕ 0 satisfies the following assumptions:

(A1) The density function f 0 ∈ LC. (A2) f 0 (x 0 ) > 0. (A3) The function ϕ 0 is at least twice continuously differentiable in a neigh- borhood of x 0 . (A4) If ϕ ′′ 0 (x 0 ) = 0, then k = 2. Otherwise, suppose that k is the smallest integer such that ϕ (j) 0 (x 0 ) = 0, j = 2, . . . , k -1, ϕ (k) 0 (x 0 ) = 0, and ϕ (k) 0 is continuous in a neighborhood of x 0 .
Note that concavity of ϕ 0 and A3 and A4 imply that k is necessarily even and that ϕ (k) 0 (x 0 ) < 0. Indeed, suppose that k > 2. Using Taylor expansion of ϕ ′′ 0 up to degree k -2, there exists a small h > 0 for which we can write

ϕ ′′ 0 (x) = ϕ (k) 0 (x 0 ) (k -2)! (x -x 0 ) k-2 + o((x -x 0 ) k-2 ), x ∈ [x 0 -h, x 0 + h].
Since ϕ ′′ 0 (x) ≤ 0 for all x ∈ [x 0 -h, x 0 + h], it follows that k -2 is even; i.e. k even and ϕ (k) 0 (x 0 ) < 0. 2.2. Notation. Let W denote two-sided Brownian motion, starting at 0. For t ∈ R, define:

Y k (t) = t 0 W (s)ds -t k+2 if t ≥ 0 0 t W (s)ds -t k+2 if t < 0. (2.2) For the uniform norm of a bounded function f we write f ∞ = sup x∈R |f (x)|. The derivative of ϕ n at x ∈ R is as usual denoted by ϕ ′ n (x). However, if x ∈ S n ( ϕ n ), then we define ϕ ′ n (x)
as the left-derivative. Theorem 2.1. Suppose that A1 -A4 hold. Then 

 n k/(2k+1) f n (x 0 ) -f 0 (x 0 ) n (k-1)/(2k+1) f ′ n (x 0 ) -f ′ 0 (x 0 )   → d c k (x 0 , ϕ 0 ) H (2) k (0) d k (x 0 , ϕ 0 ) H (3) k (0) and   n k/(2k+1) ϕ n (x 0 ) -ϕ 0 (x 0 ) n (k-1)/(2k+1) ϕ ′ n (x 0 ) -ϕ ′ 0 (x 0 )   → d C k (x 0 , ϕ 0 ) H (2) k (0) D k (x 0 , ϕ 0 ) H (3) k (0)
. where H k is the "lower invelope" of the process Y k ; that is,

H k (t) ≤ Y k (t) for all t ∈ R; H (2) k is concave; H k (t) = Y k (t) if the slope of H (2) k decreases strictly at t. The constants c k , d k , C k , and D k are given by c k (x 0 , ϕ 0 ) = f 0 (x 0 ) k+1 |ϕ (k) 0 (x 0 )| (k + 2)! 1/(2k+1) (2.3) d k (x 0 , ϕ 0 ) = f 0 (x 0 ) k+2 |ϕ (k) 0 (x 0 )| 3 [(k + 2)!] 3 1/(2k+1) (2.4) C k (x 0 , ϕ 0 ) = |ϕ (k) 0 (x 0 )| f 0 (x 0 ) k (k + 2)! 1/(2k+1) (2.5) D k (x 0 , ϕ 0 ) = |ϕ (k) 0 (x 0 )| 3 f 0 (x 0 ) k-1 [(k + 2)!] 3 1/(2k+1) . (2.6) Corollary 2.2. Suppose that A1 -A4 hold with k = 2. Then   n 2/5 f n (x 0 ) -f 0 (x 0 ) n 1/5 f ′ n (x 0 ) -f ′ 0 (x 0 )   → d c 2 (x 0 , ϕ 0 ) H (2) 2 (0) d 2 (x 0 , ϕ 0 ) H (3) 2 (0) and   n 2/5 ϕ n (x 0 ) -ϕ 0 (x 0 ) n 1/5 ϕ ′ n (x 0 ) -ϕ ′ 0 (x 0 )   → d C 2 (x 0 , ϕ 0 ) H (2) 2 (0) D 2 (x 0 , ϕ 0 ) H (3) 2 (0)
. where H 2 is the (concave) invelope of the process Y 2 ; that is,

H 2 (t) ≤ Y 2 (t) for all t ∈ R;
imsart-aos ver. 2007/09/18 file: ModeLogconc-v11.tex date: January 3, 2008

H (2) 2 is concave; H 2 (t) = Y 2 (t) if the slope of H (2)
2 decreases strictly at t. The constants c 2 , d 2 , C 2 andD 2 are given by (2.3) -(2.6) with k = 2. Note that the constants C 2 (x 0 , ϕ 0 ) and D 2 (x 0 , ϕ 0 ), up to inversion of f 0 (x 0 ), exhibit a structure very similar to that of the constants given by [START_REF] Groeneboom | Estimation of a convex function: characterizations and asymptotic theory[END_REF] in the problem of estimating a convex density g 0 on [0, ∞). We recall here that in the latter problem, those constants are found to be equal to (we use our notation to make the comparison easy)

c 2 (x 0 , g 0 ) = g 0 (x 0 ) 2 g (2) 0 (x 0 ) 4! 1/5 , d 2 (x 0 , g 0 ) = g 0 (x 0 )(g (2) 0 (x 0 )) 3 (4!) 3 1/5 .
It is clear that ϕ 0 in the log-concave problem plays exactly the same role as f 0 in the problem of estimating a convex density. However, in the first case estimation is based on observations which are distributed according to exp ϕ 0 , whereas in the latter the data come from f 0 itself. A good insight into the difference between the expressions of the asymptotic constants can be gained from the proof of Theorem 4.6 in Section 4. There, we show that the leading coefficient of the drift of the limiting process Y k depends on ϕ

(k) 0 (x 0 )f 0 (x 0 ) = f (k) 0 (x 0 ) -(ϕ ′ 0 (x 0 )) k f 0 (x 0 )
, where the second term is "filtered out" in the Taylor expansion of the estimation error in the neighborhood of x 0 . Hence, |ϕ (k) 0 (x 0 )| • f 0 (x 0 ) can be viewed as the dominating term replacing |g (k) 0 (x 0 )| in the convex estimation problem. For k = 2 the constants c 2 (x 0 , ϕ 0 ) and d 2 (x 0 , ϕ 0 ) given in (2.3) and (2.4) with k = 2 match closely with c 2 (x 0 , g 0 ) and d 2 (x 0 , g 0 ) obtained by [START_REF] Groeneboom | Estimation of a convex function: characterizations and asymptotic theory[END_REF] in the convex estimation problem, with f 0 (x 0 ) in the numerator, whereas f 0 (x 0 ) shows up in the denominator in the asymptotic constants C 2 (x 0 , ϕ 0 ) and D 2 (x 0 , ϕ 0 ). This results from applying the deltamethod to

f n (x 0 ) = exp( ϕ n (x 0 )) and f ′ n (x 0 ) = ϕ ′ n (x 0 ) f n (x 0 ) which yields C 2 (x 0 , ϕ 0 ) and D 2 (x 0 , ϕ 0 )
Finally, and in order to compare also the random parts of the limits in the convex and log-concave estimation problems, we would like to note that for our lower invelope process H k , -H k has the same distribution as the "upper invelope" of -Y k , which was called just the "invelope" in the case k = 2 by [START_REF] Groeneboom | Estimation of a convex function: characterizations and asymptotic theory[END_REF]: The process -Y k has a drift equal to plus t k+2 which specializes to t 4 in the convex density problem with k = 2. This "upper invelope" stays above -Y k and admits a convex second derivative. Since -W has the same distribution as W , it follows that the

k (0) and H (3) k (0) of H k .
As shown by [START_REF] Barlow | Statistical theory of reliability and life testing[END_REF], Lemma 5.8, page 77, (see also Marshall and Olkin (1979), p. 493;Marshall and Olkin (2007), p. 102;[START_REF] An | Logconcavity versus logconvexity: a complete characterization[END_REF]; and [START_REF] Bagnoli | Log-concave probability and its applications[END_REF]), if f 0 is log-concave, then the hazard function

λ 0 (x) = f 0 (x) 1 -F 0 (x) 1 {x<F -1 0 (1)}
is monotone non-decreasing. Defining the estimator of λ 0 based on f n as

λ n (x) = f n (x) 1 -F n (x) 1 {x<X (n) } ,
application of the delta-method yields the following corollary.

Corollary 2.3. Suppose that A1 -A4 hold. Then   n k/(2k+1) λ n (x 0 ) -λ 0 (x 0 ) n (k-1)/(2k+1) λ ′ n (x 0 ) -λ ′ 0 (x 0 )   → d g k (x 0 , ϕ 0 ) H (2) k (0) h k (x 0 , ϕ 0 ) H (3) k (0)
where the constants g k and h k are given by

g k (x 0 , ϕ 0 ) = c k (x 0 , ϕ 0 )/(1 -F 0 (x 0 )) h k (x 0 , ϕ 0 ) = d k (x 0 , ϕ 0 )/(1 -F 0 (x 0 ))
For a more thorough discussion of the implications for the hazard rate if f 0 is log-concave see Dümbgen and Rufibach (2007), at the end of Section 3 and Dümbgen et al. (2007), Section 7.

3. Inference about the mode of f 0 . Estimation of the mode of a unimodal density has been considered by many authors; see e.g. [START_REF] Parzen | On estimation of a probability density function and mode[END_REF], [START_REF] Chernoff | Estimation of the mode[END_REF], [START_REF] Grenander | Some direct estimates of the mode[END_REF], [START_REF] Dalenius | The mode-a neglected statistical parameter[END_REF], [START_REF] Venter | On estimation of the mode[END_REF], Wegman (1970a), [START_REF] Wegman | Maximum likelihood estimation of a unimodal density[END_REF], [START_REF] Wegman | A note on the estimation of the mode[END_REF], [START_REF] Eddy | Optimum kernel estimators of the mode[END_REF], [START_REF] Eddy | The asymptotic distributions of kernel estimators of the mode[END_REF], [START_REF] Hall | Asymptotic theory of Grenander's mode estimator[END_REF], [START_REF] Müller | Adaptive nonparametric peak estimation[END_REF], [START_REF] Romano | On weak convergence and optimality of kernel density estimates of the mode[END_REF], [START_REF] Vieu | A note on density mode estimation[END_REF], and more recently [START_REF] Meyer | An alternative unimodal density estimator with a consistent estimate of the mode[END_REF], and [START_REF] Herrmann | Rates on consistency for nonparametric estimation of the mode in absence of smoothness assumptions[END_REF].

Empirical studies of the performance of various estimators are given by [START_REF] Dalenius | The mode-a neglected statistical parameter[END_REF], [START_REF] Ekblom | A Monte Carlo investigation of mode estimators in small samples[END_REF], [START_REF] Meyer | An alternative unimodal density estimator with a consistent estimate of the mode[END_REF], and [START_REF] Meyer | Consistent maximum likelihood estimation of a unimodal density using shape restrictions[END_REF]. Many of the methods considered for estimating the mode of a unimodal smooth density use kernel estimation, but others are based on the principle of substitution with another choice of estimator of the population density: for example, the estimators of [START_REF] Venter | On estimation of the mode[END_REF], are related to nearest -neighbor estimators of the density f 0 . All the estimators of the mode in the class of unimodal densities known to us involve some more more or less ad-hoc choice essentially because the maximum likelihood estimator of a unimodal density is not well-defined as is nicely explained by [START_REF] Birgé | Estimation of unimodal densities without smoothness assumptions[END_REF]. (Note that [START_REF] Wegman | Maximum likelihood estimation of a unimodal density[END_REF], [START_REF] Wegman | A note on the estimation of the mode[END_REF] discussed the nonparametric MLE of a unimodal density subject to a constraint on the height of the mode; without some constraint of this type, the MLE does not exist.)

For virtually all of the estimators of which we are aware, some choice of a smoothing parameter or bandwidth or constraint is required. Empirical choice of smoothing parameters has been studied by [START_REF] Müller | Adaptive nonparametric peak estimation[END_REF] who studied local methods of choosing the smoothing parameter, [START_REF] Grund | On the minimisation of L p error in mode estimation[END_REF] who studied bootstrap methods, and Ziegler ( 2004) who studied plugin methods. [START_REF] Klemelä | Adaptive estimation of the mode of a multivariate density[END_REF] gave a construction of adaptive estimators based on Lepski's method [START_REF] Lepskiȋ | On problems of adaptive estimation in white Gaussian noise[END_REF]). For nonparametric Bayes estimators of unimodal densities and hence of the mode, see [START_REF] Brunner | Bayes methods for a symmetric unimodal density and its mode[END_REF], Ho (2006a), and[START_REF] Ho | Bayes estimation of a unimodal density via S-paths[END_REF]; for these estimators, choice of a prior is equivalent to a choice of smoothing parameters.

In contrast, estimation in the (large!) subclass of log-concave (or strongly unimodal) densities is much simpler, avoiding bandwidth or smoothing parameter choices completely. Since the maximum likelihood estimator exists, we can simply estimate the mode by the mode (or smallest point in a modal interval) of the MLE f n . Using the notation introduced by [START_REF] Eddy | The asymptotic distributions of kernel estimators of the mode[END_REF] (and also used by [START_REF] Romano | On weak convergence and optimality of kernel density estimates of the mode[END_REF]), we let M n := M ( f n ) where M denotes the mode functional (or "smallest argmax" functional) given by M (g) := min{t : g(t) = max u∈R g(u)}.

Because of the adaptive properties of the MLE's f n of f 0 and ϕ n of ϕ 0 discussed in Section 1, we expect M n to adapt to different local smoothness (or peakedness) hypotheses on f 0 (much as the Grenander estimator is locally adaptive in the case of estimating a monotone density, see e.g. [START_REF] Birgé | The Grenander estimator: a nonasymptotic approach[END_REF], page 1535). Here we study M n as an estimator of the mode M (f 0 ) := m 0 under just the condition that f 0 has a continuous second derivative f ′′ 0 in a neighborhood of m 0 with f ′′ 0 (m 0 ) < 0. We begin in the next subsection with a new asymptotic minimax lower bound for estimation of m 0 under this hypothesis. The following subsection gives our new limiting distribution result for the MLE M n of the mode m 0 .

3.1. New lower bounds for estimating the mode.

Has ′ minskiȋ (1979) established a lower bound for estimation of the mode m 0 of a unimodal density f ∈ U assuming that f satisfies f ′′ (m 0 ) < 0. He showed that the best local asymptotic minimax rate of convergence for any estimator of m 0 is n -1/5 . Has'minskii based his proof on a sequence of parametric submodels of the form

f n (x, θ) = f (x) + θn -2/5 g(n 1/5 (x -m 0 ))
where, for a := -f ′′ (m 0 ),

g(x) := g a (x) = x, if |x| ≤ 1/a, 0, if |x| ≥ K > 1/a,
and g := g a satisfies g(-x) = -g(x) and |g ′′ (x)| < a/2 for all x ∈ R. However, Has ′ minskiȋ (1979) did not study the dependence of the local minimax bound on a = -f ′′ (m 0 ) and f (m 0 ), leaving his bound in terms of c 2 0 := f (m 0 )/ g 2 a (x)dx involving the still unspecified function g = g a . Here we consider different parametric submodels and derive the dependence of the constant in local asymptotic minimax lower bound for estimation of the mode m 0 in the family LC of log-concave (or strongly unimodal) densities.

We want to derive asymptotic lower bounds for the local minimax risks for estimating the mode M (f ). The L 1 -minimax risk for estimating a functional ν of f 0 based on a sample X 1 , . . . , X n of size n from f 0 which is known to be in a subset LC n,τ of LC is defined by

M M R 1 (n, T n , LC n,τ ) := inf Tn sup f ∈LCn,τ E f |T n -ν(f )| (3.7)
where the infimum ranges over all possible measurable functions T n = t n (X 1 , . . . , X n ) mapping R n to R. The shrinking classes LC n,τ used here are Hellinger balls centered at f 0 : Let f 0 ∈ LC and m 0 = M (f 0 ) be fixed such that f 0 is twice continuously differentiable at m 0 and f ′′ 0 (m 0 ) < 0. Consider the family {ϕ ǫ } ǫ>0 and resulting family {f ǫ } ǫ>0 defined as follows:

LC n,τ = f ∈ LC : H 2 (f, f 0 ) = 1 2 ∞ -∞ f (z) -f 0 (z) 2 dz ≤ τ /n . Consider estimation of ν(f ) := M (f ) = inf{t ∈ R : t = sup
ϕ ǫ (x) =                ϕ 0 (x), x < m 0 -ǫc ǫ ϕ 0 (x), x > m 0 + ǫ, ϕ 0 (m 0 + ǫ) + ϕ ′ 0 (m 0 + ǫ)(x -m 0 -ǫ), x ∈ [m 0 -ǫ, m 0 + ǫ] ϕ 0 (m 0 -ǫc ǫ ) + ϕ ′ 0 (m 0 -ǫc ǫ )(x -m 0 + ǫc ǫ ), x ∈ [m 0 -ǫc ǫ , m 0 -ǫ)
where c ǫ is chosen so that ϕ ǫ is continuous at m 0 -ǫ. Note that if ϕ 0 (x) = γ -γ 0 (x -m 0 ) 2 , then c ǫ = 3 for all ǫ, and c ǫ → 3 as ǫ ↓ 0 since f ′′ 0 (m 0 ) < 0. Now define h ǫ (x) := exp(ϕ ǫ (x)), and f ǫ (x) := h ǫ (x) h ǫ (y)dy .

Then f ǫ is log-concave for each ǫ > 0 with mode m 0 -ǫ by construction, so with ν(f ǫ ) := M (f ǫ ) := the mode of f ǫ we have

ν(f ǫ ) -ν(f 0 ) = M (f ǫ ) -M (f 0 ) = m 0 -ǫ -m 0 = -ǫ.
Furthermore, the following lemma holds.

Lemma 3.1. Under the above assumptions

H 2 (f ǫ , f 0 ) = 2f ′′ 0 (m 0 ) 2 5f 0 (m 0 ) ǫ 5 + o(ǫ 5 ) := ρǫ 5 + o(ǫ 5 ).
Proof. Proceeding as in [START_REF] Jongbloed | Three Statistical Inverse Problems[END_REF],

H 2 (f ǫ , f 0 ) = 1 2 ∞ -∞ [ f ǫ (x) -f 0 (x)] 2 dx = 1 2 m 0 +ǫ m 0 -ǫc ǫ [ f ǫ (x) -f 0 (x)] 2 dx = 2 5 f 0 (m 0 )ϕ ′′ 0 (m 0 ) 2 ǫ 5 + o(ǫ 5 ) = 2 5 f ′′ 0 (m 0 ) 2 f 0 (m 0 ) ǫ 5 + o(ǫ 5 )
as ǫ ↓ 0. Calculations similar to those of [START_REF] Jongbloed | Three Statistical Inverse Problems[END_REF] Taking ǫ = cn -1/5 and defining

f n := f cn -1/5 yields ν(f n ) -ν(f 0 ) = M (f n ) -M (f 0 ) = -cn -1/5 and nH 2 (f n , f 0 ) = 2 5 f ′′ 0 (m 0 ) 2 f 0 (m 0 ) c 5 + o(1) := ρc 5 + o(1).
Plugging these into the lower bound Lemma 4.1 of [START_REF] Groeneboom | Lectures on inverse problems[END_REF] with ℓ(x) := |x| yields lim inf

n inf Tn n 1/5 max {E n,Pn |T n -M (f n )|, E n,P |T n -M (f 0 )|} ≥ 1 4 c exp(-2ρc 5 ) = e -1/5 4 • 10 1/5 ρ -1/5 = (.15512) f 0 (m 0 ) f ′′ 0 (m 0 ) 2 1/5
by choosing c = (10ρ) -1/5 . This yields the following proposition.

Proposition 3.2. (Minimax risk lower bound). Suppose that ν(f ) = M (f ) as defined in (3.8), and that LC n,τ is as defined above where

f ′′ 0 is continuous in a neighborhood of m 0 = M (f 0 ) with f ′′ 0 (m 0 ) < 0. Then sup τ >0 lim sup n→∞ n 1/5 inf Tn sup f ∈LCn,τ E f |T n -M (f )| ≥ 5/2 4 5 • e • 10 1/5 f 0 (m 0 ) f ′′ 0 (m 0 ) 2 1/5 = (.15512) f 0 (m 0 ) f ′′ 0 (m 0 ) 2 1/5
. Remark 3.3. Note that the constant b(f 0 , m 0 ) := (f 0 (m 0 )/f ′′ 0 (m 0 ) 2 ) 1/5 appearing on the right side of this lower bound is scale equivariant in exactly the right way: if f c (x) := f 0 (m 0 + (x -m 0 )/c)/c for c > 0, then b(f c , m 0 ) = cb(f 0 , m 0 ) for all c > 0. The constant b(f 0 , m 0 ) will appear in the limit distribution appearing in the next subsection.

Remark 3.4. If LC is replaced by the class U of unimodal densities on R and LC n,τ is replaced by U n,τ defined analogously where f 0 satisfies f ′′ 0 (m 0 ) < 0 and f ′′ 0 continuous in a neighborhood of m 0 , then a minimax lower bound of the same form as Proposition 3.2 holds with exactly the same dependence on b(f 0 , m 0 ) = (f 0 (m 0 )/f ′′ 0 (m 0 ) 2 ) 1/5 , but with the absolute constant .15512... replaced by .19784.... This can be seen by taking the perturbations {f ǫ } ǫ>0 defined by where b ǫ is chosen so that f ǫ (x 0 + ǫ) > f 0 (x 0 + ǫ) and

f ǫ (x) =    f 0 (x), x ≤ x 0 -ǫ, f 0 (x), x > x 0 + ǫ, f 0 (x 0 ) + b ǫ (x -x 0 + ǫ), x 0 -ǫ ≤ x ≤ x 0 + ǫ imsart-
x 0 +ǫ x 0 -ǫ f ǫ (x)dx = x 0 +ǫ x 0 -ǫ f 0 (x)dx.
Remark 3.5. If ϕ 0 is continuously k-times differentiable in a neighborhood of the mode m 0 , ϕ (j) 0 (m 0 ) = 0 for j = 2, . . . , k -1, and ϕ (k) 0 (m 0 ) = 0 (Assumption A4), then it can be shown that the minimax rate of convergence is n 1/(2k+1) and that the minimax lower bound is proportional to

1 f 0 (m 0 )ϕ (k) 0 (m 0 ) 2 1/(2k+1) = f 0 (m 0 ) f (k) 0 (m 0 ) 2 1/(2k+1)
.

where the proportionality constant depends on the largest root of the polynomial

x k -(k/(k -1))x k-1 -(2k -1)/(k -1) (which equals 3 when k = 2).
3.2. Limiting distribution for the MLE M n in LC .

Now let f n be the MLE of f in the class LC of log-concave densities, and let

M n = M ( f n ), m 0 = M (f 0 ).
Here is our result concerning the limiting distribution of M n under the same assumptions on f 0 as in the previous section on lower bounds.

Theorem 3.6. Suppose that f ′′ 0 is continuous in a neighborhood of m 0 = M (f 0 ) and that f ′′ 0 (m 0 ) < 0. Then

n 1/5 ( M n -m 0 ) → d (4!) 2 f 0 (m 0 ) f ′′ 0 (m 0 ) 2 1/5 M (H (2)
2 ).

Note that the limiting distribution depends on a multiple of the same constant b(f 0 , m 0 ) which appears in the asymptotic minimax lower bound of Proposition 3.2, times a universal term M (H

(2)
2 ), the mode of the "estimator" H

(2) 2 (t) of the canonical concave function -12t 2 in the limit Gaussian problem: estimate the mode of f 0 (t) = -12t 2 based on observation of

Y (t) = t 0 X(s)ds when dX(t) = f 0 (t)dt + dW (t).
We expect that this distribution, namely the distribution of

M (H (2) 2 ) = argmax t∈R H (2) 2 (t)
will occur in several other problems involving nonparametric estimation of the mode or antimode of convex or concave functions under similar second imsart-aos ver. 2007/09/18 file: ModeLogconc-v11.tex date: January 3, 2008 derivative hypotheses: for example it seems clear that it will occur as the limiting distribution of the nonparametric estimator of the antimode of a convex bathtub shaped hazard (in the setting of [START_REF] Jankowski | Nonparametric estimation of a convex bathtub-shaped hazard function[END_REF]); as the limiting distribution of the nonparametric estimator of the antimode of of a convex regression function in the setting of [START_REF] Groeneboom | Estimation of a convex function: characterizations and asymptotic theory[END_REF]; and as the limiting distribution of the nonparametric estimator of the mode of a concave regression function.

When ϕ

(j) 0 (m 0 ) = 0 for j = 2, . . . , k -1, ϕ (k)
0 (m 0 ) = 0, and ϕ

(k)
0 is continuous in a neighborhood of m 0 , then an analogous result (with a completely similar proof) holds:

n 1/(2k+1) ( M n -m 0 ) → d (k + 2)! 2 f 0 (m 0 )|ϕ (k) 0 (m 0 )| 2 1/(2k+1) M (H (2) k ).
In particular, when k = 4, the rate of convergence is n 1/9 and the limit distribution becomes that of

6! 2 f 0 (m 0 ) f (4) 0 (m 0 ) 2 1/9 M (H (2) 4 ).
Apparently estimation of m 0 becomes considerably more difficult when the second and possibly higher order derivatives of ϕ 0 vanish. On the other hand, if ϕ 0 (or equivalently f 0 ) is cusp-shaped at m 0 , then the rate of convergence of M n is n 1/3 and the local asymptotic minimax rate of convergence is also n 1/3 ; we will pursue these issues elsewhere.

Proofs for Sections 2 and 3.

Throughout this section we fix k and let

r n := n (k+2)/(2k+1) , s n := n -1/(2k+1) ,
x n (t) := x n,k (t) := x 0 + s n t := x 0 + n -1/(2k+1) t,

I := I(x 0 , n, k, t) := [x 0 , x n (t)], t ≥ 0, [x n (t), x 0 ] , t < 0.
4.1. Preparation: Technical Lemmas and Tightness Results.

First, some notation.

F n (v) -F n (x 0 ) - v x 0 k-1 j=0 f (j) 0 (x 0 ) j! (u -x 0 ) j du dv
and

H loc n (t) := r n xn(t) x 0 v x 0 f n (u) - k-1 j=0 f (j) 0 (x 0 ) j! (u -x 0 ) j dudv + Ân t + Bn where Ân = r n s n F n (x 0 ) -F n (x 0 ) (4.9) Bn = r n H n (x 0 ) -H n (x 0 ) . (4.10)
We also define the "modified" local processes

Y locmod n (t) := r n f 0 (x 0 ) xn(t) x 0 F n (v) -F n (x 0 ) (4.11) - v x 0 k-1 j=0 f (j) 0 (x 0 ) j! (u -x 0 ) j du dv -r n xn(t) x 0 v x 0 Ψ k,n,2 (u)dudv,
and

H locmod n (t) := r n x n (t) x 0 v x 0 ϕ n (u) -ϕ 0 (x 0 ) -(u -x 0 )ϕ ′ 0 (x 0 ) dudv + Ân t + Bn f 0 (x 0 ) (4.12)
where Ψ k,n,2 is defined below in (4.34).

Lemma 4.1. Let F be a collection of functions defined on [x 0 -δ, x 0 + δ], with δ > 0 small and let s > 0. Suppose that for a fixed

x ∈ [x 0 -δ, x 0 + δ] and R > 0 such that [x, x + R] ⊆ [x 0 -δ, x 0 + δ], the collection F x,R = f x,y := f 1 [x,y] , f ∈ F, x ≤ y ≤ x + R
imsart-aos ver. 2007/09/18 file: ModeLogconc-v11.tex date: January 3, 2008 admits an envelope F x,R such that

EF 2 x,R (X 1 ) ≤ KR 2d-1 , R ≤ R 0 ,
for some d ≥ 1/2 and K > 0 depending only on x 0 and δ Moreover, suppose that

sup Q 1 0 log N (η F x,R Q,2 , F x,R , L 2 (Q))dη < ∞. (4.13)
Then, for each ǫ > 0 there exist random variables M n of order O p (1) (not depending on x or y and R 0 > 0 such that

f x,y d(F n -F 0 ) ≤ ǫ|y -x| s+d + n -s+d 2s+1 M n for |y -x| ≤ R 0 .
Proof. See [START_REF] Kim | Cube root asymptotics[END_REF] and [START_REF] Balabdaoui | Estimation of a k-monotone density: limiting distribution theory and the spline connection[END_REF], Lemmas 4.4 and 6.1. The special case s = 1 = d is Lemma 4.1 of [START_REF] Kim | Cube root asymptotics[END_REF]. ✷ Lemma 4.2. If A3 and A4 hold, then

f (j) 0 (x 0 ) = [ϕ ′ 0 (x 0 )] j f 0 (x 0 ) for j = 1, . . . , k -1, (4.14) and for j = k, f (k) 0 (x 0 ) = (ϕ (k) 0 (x 0 ) + [ϕ ′ 0 (x 0 )] k ) f 0 (x 0 ).
Proof. The expressions for f (j) 0 (x 0 ) follow immediately from a recursive argument using the identity f 0 = exp ϕ 0 and the assumption ϕ

(j) 0 (x 0 ) = 0 for j = 2, . . . , k -1 if k > 2. ✷ Now let τ + n := inf{t ∈ S( ϕ n ) : t > x 0 }, and τ - n := sup{t ∈ S( ϕ n ) : t < x 0 }. Theorem 4.3. If A1 -A4 hold, then τ + n -τ - n = O p (n -1/(2k+1) ). (4.15)
Theorem 4.3 should be compared to Theorem 3.3 of Dümbgen and Rufibach (2007). When their Theorem 3.3 is specialized to the case β = 2 so that that ϕ ′′ 0 (x) ≤ C < 0 for all x ∈ T := [A, B], then it yields the following:

If m n denotes the number of elements in S n ( ϕ n ) ∩ T , then for any successive knot points t i-1 and t i in S n ( ϕ n ) ∩ T , sup i=2,...,mn

(t i -t i-1 ) = O p (ρ 1/5 n ) (4.16)
where ρ n = log(n)/n. This is "weaker" in the sense that only the supremum over all knots on a compact interval was considered; but Theorem 3.3 of Dümbgen and Rufibach (2007) is more general in the sense that it provides the correct rate over a whole range of Hölder classes, not only for twice differentiable ϕ 0 . Bounds generalizing (4.16) to 1 ≤ β ≤ 2 are then used to get upper bounds on the uniform rate of convergence for F n -F n on T . However, for the estimation problem treated in this paper the localized version of the gap problem given in Theorem 4.3 provides the results of interest here concerning local limiting distribution theory.

Proof of Theorem 4.3. From the first characterization of the estimator f n in Dümbgen and Rufibach (2007), for every function ∆ such that ϕ n + t∆ is concave for a t > 0 small enough, we know that

R ∆(x)dF n (x) ≤ R ∆(x)d F n (x). (4.17) This is equivalent to R ∆(x)d(F n (x) -F 0 (x)) ≤ R ∆(x)( f n (x) -f 0 (x))dx. (4.18)
Using specific indicator functions for ∆, one can furthermore show that

F n (τ ) ∈ [F n (τ ) -1/n, F n (τ )] (4.19)
for every τ ∈ S n ( ϕ n ), see [START_REF] Rufibach | Log-concave density estimation and bump hunting for I[END_REF] and Corollary 2.5 of Dümbgen and Rufibach (2007). Now, the idea is to choose a particular permissible perturbation function ∆ that satisfies the following two conditions:

1. ∆ is "local", i.e. compactly supported on [τ - n , τ + n ]. 2. ∆ should "filter" out the unknown error f n -f 0 .
The second requirement means that ∆ should be chosen so that (4.20) imsart-aos ver. 2007/09/18 file: ModeLogconc-v11.tex date: January 3, 2008

τ + n τ - n ∆(x)dx = 0 τ + n τ - n ∆(x)(x -τ )dx = 0,
where τ := (τ - n + τ + n )/2 is the mid-point of [τ - n , τ + n ].
If this is guaranteed, then the right side of (4.18) in the end will only depend on the distance τ + n -τ - n and f 0 (x 0 ). Define ∆ 0 by

∆ 0 (x) = (x -τ - n )1 [τ - n ,τ ] (x) + (τ + n -x)1 [τ ,τ + n ] (x)
. Since ϕ n + t∆ 0 is concave for small t > 0, ∆ 0 is permissible. It is also compactly supported. However, since ∆ 0 is nonnegative, there is no hope that it fulfills the second of the requirements above. We therefore introduce a modified perturbation function

∆ 1 (x) = ∆ 0 (x) - 1 4 (τ + n -τ - n )1 [τ - n ,τ + n ] (x), x ∈ R.
Clearly, existence of a t > 0 such that ϕ n + t∆ 1 is concave is no longer guaranteed. However, using (4.19),

∆ 1 (x)d(F n -F 0 )(x) = ∆ 1 (x)d(F n -F n )(x) + ∆ 1 (x)d( F n -F 0 )(x) ≤ τ + n -τ - n 4 τ + n τ - n d(F n -F n )(x) + ∆ 1 (x)d( F n -F 0 )(x) (4.21) ≤ τ + n -τ - n 2n + ∆ 1 (x)( f n -f 0 )(x)dx. (4.22)
To get the inequality in (4.21), we used (4.17) with ∆ = ∆ 0 and (4.19). The next step is to get bounds for the integrals in the crucial inequality (4.22). Define

R 1n := ∆ 1 (x)( f n -f 0 )(x)dx and R 2n := ∆ 1 (x)d(F n -F 0 )(x).
Rearranging the inequality in (4.22) and use these definitions yields

-R 1n ≤ τ + n -τ - n 2n -R 2n .
Consistency of ϕ n together with ϕ This yields the claimed rate, O p (n -1/(2k+1) ), for the distance between τ + n and τ - n . ✷ Lemma 4.4. Suppose A1-A4 hold. Then

(k) 0 (x 0 ) < 0 implies τ + n -τ - n = o p (1). Thus it follows from Lemma 4.4 that M k -ϕ (k) 0 (x 0 ) (τ + n -τ - n ) k+2 (1 + o p (1)) ≤ o p (1)n -1 + O p (r -1 n ) = O p (r -1 n ).
R 2n = O p (r -1 n )
and

R 1n = M k f 0 (x 0 )ϕ (k) 0 (x 0 )(τ + n -τ - n ) k+2 + o p ((τ + n -τ - n ) k+2 )
where M k > 0 depends only on k and ϕ

(k) 0 (x 0 ) < 0. Proof. Define the function p n (t) = ϕ n (t) -ϕ 0 (t) for any t ∈ [τ - n , τ + n ]
. Then, using Taylor expansion of h → exp(h) up to order k, we can find

θ t,n ∈ [τ - n , τ + n ] such that R 1n = τ + n τ - n ∆ 1 (t)f 0 (t) k-1 j=1 p n (t) j j! + 1 k! exp(θ t,n )p n (t) k dt := k j=1 S nj j!
where

S nj := τ + n τ - n ∆ 1 (t)f 0 (t)p n (t) j dt for 1 ≤ j ≤ k -1 and
S nk := τ + n τ - n ∆ 1 (t)f 0 (t) exp(θ t,n )p n (t) k dt.
If we expand f 0 (t) around the mid-point τ of [τ - n , τ + n ] we get for 1 ≤ j ≤ k-1 and a η n,t,j ∈ [τ - n , τ + n ],

S nj = k-1 l=0 f (l) 0 (τ ) l! τ + n τ - n ∆ 1 (t)(t -τ ) l p n (t) j dt + τ + n τ - n f (k) 0 (η n,t,j ) k! ∆ 1 (t)(t -τ ) k p n (t) j dt and for j = k, S nk = k-1 l=0 f (l) 0 (τ ) l! τ + n τ - n ∆ 1 (t) exp(θ t,n )(t -τ ) l p n (t) k dt + τ + n τ - n f (k) 0 (η n,t,k ) k! ∆ 1 (t) exp(θ t,n )(t -τ ) k p n (t) k dt.
Q 1n f 0 (τ ) = τ + n τ - n ∆ 1 (t)p n (t)dt = p n (τ ) τ + n τ - n ∆ 1 (t)dt + p ′ n (τ ) τ + n τ - n ∆ 1 (t)(t -τ )dt - k j=2 ϕ (j) 0 (τ ) j! τ + n τ - n ∆ 1 (t)(t -τ ) j dt - τ + n τ - n ǫ n (t)∆ 1 (t)(t -τ ) k dt
where the first two terms are zero since (4.20) holds when ∆ = ∆ 1 and ǫ n ∞ → p 0 as τ + n -τ - n → p 0. Using the fact that

τ + n τ - n ∆ 1 (t)(t -τ ) j dt =    0,
for j = 0 and j odd,

(τ + n -τ - n ) j+2 -j 2 (j+2) (j+1)(j+2) , for j even, (4.23)
we conclude that

Q 1n = k 2 (k+2) k!(k + 1)(k + 2) f 0 (τ )ϕ (k) 0 (τ ) (τ + n -τ - n ) k+2 + o p (1)
and the claimed form of R 1n in the lemma follows.

For R 2n , we proceed along the lines of the proof of Lemma 4.1 in Groeneboom, [START_REF] Groeneboom | Estimation of a convex function: characterizations and asymptotic theory[END_REF]. This means, we have to line up with the assumption of Theorem 2.14.1 in van der Vaart and Wellner (1996). Therefore, define a generalized version of R 2n :

R x,y 2n = y x ∆ 1 (z)d(F n -F 0 )(z)
z -x - 1 4 (y -x) d(F n -F 0 )(z) = 2 sup y : 0≤y-x≤R h x,y (z)d(F n -F 0 )(z)
where

h x,y (z) = z -x - 1 4 (y -x) 1 [x,(x+y)/2] (z) = h(z)1 [x,(x+y)/2] (z).
Then the collection of functions

F x,R = {h1 [x,(x+y)/2] : x ≤ y ≤ x + R}
is a Vapnik-Chervonenkis subgraph class with envelope function

F x,R (z) = (z -x) + R/4 1 [x,x+R] (z).
Finally, Theorem 2.6.7 in van der Vaart and Wellner (1996) yields the entropy condition (4.13).

A log-concave density is always unimodal and the value at the mode is finite, and hence Lemma 4.5. For any M > 0, we have

K := f 0 ∞ is finite. Therefore EF 2 x,R (X 1 ) = x+R x (z -x) 2 f 0 (z)dz + R 2 x+R x (z -x)f 0 (z)dz + R 2 16 x+R x f 0 (z)dz ≤ K 3 (z -x) 3 + RK 4 (z -x) 2 + R 2 K
sup |t|≤M | ϕ ′ n (x 0 + s n t) -ϕ ′ 0 (x 0 )| = O p (s k-1 n ), (4.24) sup |t|≤M ϕ n (x 0 + s n t) -ϕ 0 (x 0 ) -s n tϕ ′ 0 (x 0 ) = O p (s k n ). (4.25) Furthermore, if we define for any u ∈ R ên (u) = f n (u) - k-1 j=0 f (j) 0 (x 0 ) j! (u -x 0 ) j -f 0 (x 0 ) [ϕ ′ 0 (x 0 )] k k! (u -x 0 ) k , then sup |t|≤M ên (x 0 + s n t) -f 0 (x 0 ) ϕ n (x 0 + s n t) -ϕ 0 (x 0 ) -s n tϕ ′ 0 (x 0 ) = o p (s k n ). (4.26)
Proof. The proof of (4.24) and (4.25) is identical to that of Lemma 4.4 in [START_REF] Groeneboom | Estimation of a convex function: characterizations and asymptotic theory[END_REF] since the characterization of f n given in (1.1) is (up to the direction of the inequality) equivalent to that of the least squares estimator of a convex density. Now, we prove (4.26). Using Taylor expansion of h → exp(h) up to order k around zero, we can write

f n (u) -f 0 (x 0 ) = f 0 (x 0 )[exp( ϕ n (u) -ϕ 0 (x 0 )) -1] = f 0 (x 0 ) k j=1 1 j! ( ϕ n (u) -ϕ 0 (x 0 )) j + f 0 (x 0 ) Ψ k,n,1 (u) (4.27) where Ψ k,n,1 (u) = ∞ j=k+1 1 j! ( ϕ n (u) -ϕ 0 (x 0 )) j .
But for any j ≥ 1, Hence, using (4.25) and (A3), we get on the set {u :

( ϕ n (u) -ϕ 0 (x 0 )) j = [ ϕ n (u) -ϕ 0 (x 0 ) -(u -x 0 )ϕ ′ 0 (x 0 ) + (u -x 0 )ϕ ′ 0 (x 0 )] j = j r=1 j r [ ϕ n (u) -ϕ 0 (x 0 ) -(u -x 0 )ϕ ′ (x 0 )] r [ϕ ′ 0 (x 0 )] j-r (u -x 0 ) j-r + [ϕ ′ 0 (x 0 )] j (u -x 0 ) j . ( 4
|u-x 0 | ≤ M n -1/(2k+1) } ( ϕ n (u) -ϕ 0 (x 0 )) j = o p (n -k/(2k+1) ) for all j ≥ k + 1.
In particular, this implies that 2k+1) ] where |t| ≤ M , and

Ψ k,n,1 (u) = o p (n -k/(2k+1) ) (4.29) uniformly in u ∈ [x 0 -tn -1/(2k+1) , x 0 + tn -1/(
f n (u) -f 0 (x 0 ) -f 0 (x 0 ) ϕ n (u) -ϕ 0 (x 0 ) -(u -x 0 )ϕ ′ 0 (x 0 ) -f 0 (x 0 ) k j=1 ϕ (j) 0 (x 0 ) j! (u -x 0 ) j = o p (n -k/(2k+1) ).
Using Lemma 4.2, the latter can be rewritten as

f n (u) -f 0 (x 0 ) -f 0 (x 0 ) ϕ n (u) -ϕ 0 (x 0 ) -(u -x 0 )ϕ ′ 0 (x 0 ) - k-1 j=1 f (j) 0 (x 0 ) j! (u -x 0 ) j -f 0 (x 0 ) ϕ (k) 0 (x 0 ) k! (u -x 0 ) k = o p (n -k/(2k+1) ), or equivalently ên (x 0 + tn -1/(2k+1) ) -f 0 (x 0 ) ϕ n (x 0 + tn -1/(2k+1) ) -ϕ 0 (x 0 ) -n -1/(2k+1) tϕ ′ 0 (x 0 ) = o p (n -k/(2k+1
(γ 2 •) converges weakly in C[-K, K] to Y k where γ 1 = f 0 (x 0 ) k-1 |ϕ (k) 0 (x 0 )| 3 [(k + 2)!] 3 1/(2k+1) (4.30) γ 2 = f 0 (x 0 )|ϕ (k) 0 (x 0 )| 2 [(k + 2)!] 2 1/(2k+1) . (4.31) Equivalently, Y locmod n converges weakly in C[-K, K] to the "driving process" Y a,k,σ where Y k,a,σ (t) := a t 0 W (s)ds -σt k+2 (4.32)
and where a

= 1/ f 0 (x 0 ), σ = |ϕ (k) 0 (x 0 )|/(k + 2)!. (ii) The localized processes satisfy Y locmod n (t) -H locmod n (t) ≥ 0 for all t ∈ R, with equality for all t such that x n (t) = x 0 + tn -1/(2k+1) ∈ S n ( ϕ n ).
(iii) Both Ân and Bn defined above in (4.9) and (4.10) are tight. (iv) The vector of processes

H locmod n , ( H locmod n ) (1) , ( H locmod n ) (2) , Y locmod n , ( H locmod n ) (3) , (Y locmod n ) (1)
converges weakly in (C[-K, K]) 4 × (D[-K, K]) 2 endowed with the product topology induced by the uniform topology on the spaces C[-K, K] and the Skorohod topology on the spaces D[-K, K] to the process

H k,a,σ , H (1) k,a,σ , H (2) k,a,σ , Y k,a,σ , H (3) k,a,σ , Y (1) k,a,σ where H k,a,σ is the unique process on R satisfying      H k,a,σ (t) ≤ Y k,a,σ (t) for all t ∈ R, (H k,a,σ (t) -Y k,a,σ (t))dH (3) k,a,σ (t) = 0, H (2) k,a,σ is concave. (4.33)
Proof. (i) The first step will be to modify the local processes, i.e. going from the "density" to the "log-density" level, in order to be able to exploit concavity of ϕ 0 and ϕ n and to connect the local process to the limiting distribution obtained by [START_REF] Groeneboom | Estimation of a convex function: characterizations and asymptotic theory[END_REF] for estimating a convex density.

First, by Lemma 4.2, (4.27) and (A3), we can write

f 0 (x 0 ) -1 f n (u) - k-1 j=0 f (j) 0 (x 0 ) j! (u -x 0 ) j = f 0 (x 0 ) -1 f n (u) -f 0 (x 0 ) -f 0 (x 0 ) k-1 j=1 [ϕ ′ 0 (x 0 )] j j! (u -x 0 ) j = Ψ k,n,1 (u) + k j=1 1 j! [ ϕ n (u) -ϕ 0 (x 0 )] j - k-1 j=1 [ϕ ′ 0 (x 0 )] j j! (u -x 0 ) j = Ψ k,n,1 (u) + ϕ n (u) -ϕ 0 (x 0 ) -ϕ ′ 0 (x 0 )(u -x 0 ) + + k j=2 1 j! [ ϕ n (u) -ϕ 0 (x 0 )] j - k-1 j=2 [ϕ ′ 0 (x 0 )] j j! (u -x 0 ) j =: ϕ n (u) -ϕ 0 (x 0 ) -ϕ ′ 0 (x 0 )(u -x 0 ) + Ψ k,n,2 (u)
introducing the new remainder term

Ψ k,n,2 (u) = Ψ k,n,1 (u) + k j=2 1 j! [ ϕ n (u) -ϕ 0 (x 0 )] j - k-1 j=2 [ϕ ′ 0 (x 0 )] j j! (u -x 0 ) j . (4.34)
Using (4.28) and (4.29) yields Note that the process H locmod n is in fact similar to H loc n , except that it is defined in terms of the log-density ϕ 0 instead of the density f 0 . This can be more easily seen from its original expression given in (4.12). The second expression of H locmod n given above is only useful for showing that it stays below Y locmod n while touching it at points t such that x n (t) = x 0 + tn -1/(2k+1) ∈ S n ( ϕ n ). The biggest advantage of considering this modified version is to be able to use concavity of ϕ 0 the same way [START_REF] Groeneboom | Estimation of a convex function: characterizations and asymptotic theory[END_REF]

I v x 0 Ψ k,n,2 (u)dudv = t 2 n -2/(2k+1) sup u∈[x 0 ,v],v∈I | Ψ k,n,1 (u)| + k j=2 1 j! I v x 0 [ ϕ n (u) -ϕ 0 (x 0 )] j dudv - k-1 j=2 1 j! I v x 0 [ϕ ′ 0 (x 0 )] j (u -x 0 ) j dudv imsart-
= o p (r -1 n ) + k j=2 1 j! j l=1 j l I v x 0 [ ϕ n (u) -ϕ 0 (x 0 ) -(u -x 0 )ϕ ′ 0 (x 0 )] l (u -x 0 ) j-l [ϕ ′ 0 (x 0 )] j-l dudv + k j=2 1 j! I v x 0 [ϕ ′ 0 (x 0 )] j (u -x 0 ) j dudv - k-1 j=2 1 j! I v x 0 [ϕ ′ 0 (x 0 )] j (u -x 0 ) j dudv = o p (r -1 n ) + k j=2 1 j! j l=1 j l I v x 0 [ ϕ n (u) -ϕ 0 (x 0 ) -(u -x 0 )ϕ ′ 0 (x 0 )] l (u -x 0 ) j-l [ϕ ′ 0 (x 0 )] j-l dudv + 1 k! I v x 0 (u -x 0 ) k [ϕ ′ 0 (x 0 )] k dudv.
⇒ [f 0 (x 0 )] -1/2 t 0 W (s)ds + f (k) 0 (x 0 ) (k + 2)!f 0 (x 0 ) t k+2 - [ϕ ′ 0 (x 0 )] k (k + 2)! t k+2 = [f 0 (x 0 )] -1/2 t 0 W (s)ds + ϕ (k) 0 (x 0 ) (k + 2)! t k+2 = Y k,a,σ (t) in C[-K, K]
where a := [f 0 (x 0 )] -1/2 , σ := |ϕ (k) 0 (x 0 )|/(k + 2)!, as in (4.32). Now let γ 1 and γ 2 be chosen so that

γ 1 Y k,a,σ (γ 2 t) d = Y k (t)
as processes where Y k is the integrated Gaussian process defined in (2.2). Using the scaling property of Brownian motion; that is, α -1/2 W (αt) = d W (t) for any α > 0, we get γ 1 γ 3/2 2 = a -1 and γ 1 γ k+2 2 = σ -1 . This yields γ 1 and γ 2 as given in (4.30) and (4.31), and hence   n k/(2k+1) ϕ n (x 0 ) -ϕ 0 (x 0 )

n (k-1)/(2k+1) ϕ ′ n (x 0 ) -ϕ ′ 0 (x 0 )   → d f 0 (x 0 ) -1 c k (x 0 , ϕ 0 ) H (2)
k (0) d k (x 0 , ϕ 0 ) H where ǫ n is a function such that ǫ n → p 0 as x 0 -τ → p 0. To bound Ân3 , similar derivations as the ones used for bounding R 2n (see the proof of Lemma 4.4) can be employed where the perturbation function ∆ 1 needs to be replaced by ∆ 2 (x) = 1 [τ, x 0 ] (x).

At "one higher integration level", similar computations can be used to show tightness of Bn .

(iv) The proof of this last part of the theorem is basically identical to that of Theorem 6.2 for the LSE in Groeneboom, Jongbloed and Wellner

Fig 1 .

 1 Fig 1. Examples for log-concave density, log-density, CDF, and hazard rate estimation for n = 20, 200 (--true functions, -estimators). The dotted vertical lines indicate the set > Sn( > ϕn). The • -•-vertical lines are placed at the mode of the estimated density.

  imsart-aos ver. 2007/09/18 file: ModeLogconc-v11.tex date: January 3, 2008 upper and lower invelopes H k and H k (associated with estimation of convex and concave functions respectively) satisfy H k d = -H k . Since the derivatives at zero H (2) k (0) and H (3) k (0) of H k are distributed symmetrically about zero, the same is true of the derivatives at zero H (2)

  imsart-aos ver. 2007/09/18 file: ModeLogconc-v11.tex date: January 3, 2008

  ver. 2007/09/18 file: ModeLogconc-v11.tex date: January 3, 2008

  aos ver. 2007/09/18 file: ModeLogconc-v11.tex date: January 3, 2008

  imsart-aos ver. 2007/09/18 file: ModeLogconc-v11.tex date: January 3, 2008

  imsart-aos ver. 2007/09/18 file: ModeLogconc-v11.tex date: January 3, 2008 for -∞ < x ≤ y. With this function we have for some R > 0 specified later,

  .28) imsart-aos ver. 2007/09/18 file: ModeLogconc-v11.tex date: January 3, 2008

[

  But by Lemma 4.5 one can easily show that for j = 2, . . . , k and l= 1, . . ϕ n (u) -ϕ 0 (x 0 ) -(u -x 0 )ϕ ′ 0 (x 0 )] l (u -x 0 ) j-l [ϕ ′ 0 (x 0 )] j-l dudv = O p (n -[k(l-1)+(j-l)]/(2k+1) ) = o p (1) uniformly in |t| ≤ M . Similarly, n,2 (u)dudv = [ϕ ′ 0 (x 0 )] k (k + 2)! t k+2 + o p (1)as n → ∞ and uniformly in |t| ≤ M .We turn now to the modified local processes,

≤

  ver. 2007/09/18 file: ModeLogconc-v11.tex date: January 3, 2008 large probability. Now using (4.19) we can write| Ân | ≤ r n s n ( F n (x 0 ) -F n (τ )) -(F n (x 0 ) -F n (τ )) + r n /n n -F 0 ) + n -k/(2k+1) := Ân1 + Ân2 + Ân3 + n -k/(2k+1) . du -f 0 (x 0 ) ϕ n (u) -ϕ 0 (x 0 ) -(u -x 0 )ϕ ′ 0 (x 0 ) du + r n s n f 0 (x 0 ) x 0 ) k du + + r n s n f 0 (x 0 ) x 0 τ ϕ n (u) -ϕ 0 (x 0 ) -(u -x 0 )ϕ ′ 0 (x 0 ) du ≤ o p (1) + O p (r n s n (τ -x 0 ) k+1 ) + O p (r n s n (τ -x 0 )n -k/(2k+1) ) = O p(1), where we used (4.26) and (4.25) to bound the first and last terms. To bound Ân2 we use Taylor approximation of f 0 (u) around x 0 x 0 ) k du + r n x 0 τ (u -x 0 ) k ǫ n (u)du = O p (1),

  imsart-aos ver. 2007/09/18 file: ModeLogconc-v11.tex date: January 3, 2008whereγ 2 = f 0 (m 0 )|ϕ using f ′ 0 (m 0 ) = 0 = ϕ ′ 0 (m 0 ) and Lemma 4.2. ✷

  It turns out that the dominating term in R 1n is the first term in the Taylor expansion of S n1 . All the other terms are of smaller order since both p n and (t -τ ) l , l > 0 are o p (1) uniformly in t ∈ [τ - n , τ + n ]. We denote this dominating term by Q n1 . Since ϕ n is linear on [τ -

	n , τ + n ] write ϕ n (t) = ϕ n (τ )+(t-τ ) ϕ ′ n (τ ).
	By Taylor expansion of p n around τ we get
	imsart-aos ver. 2007/09/18 file: ModeLogconc-v11.tex date: January 3, 2008

  If {Y k (t), t ∈ R} is the canonical process defined in (2.2), then the localized process γ 1 Y locmod n

	Theorem 4.6. Let K > 0.
	(i)
	) )
	uniformly in |t| ≤ M .	✷
	imsart-aos ver. 2007/09/18 file: ModeLogconc-v11.tex date: January 3, 2008

  used convexity of the true estimated density g 0 . Their process H loc n resembles H locmod n to a large extent (see p. 1688), and by combining arguments similar to theirs with Lemma 4.2 and the results obtained above, it follows that

	Y locmod n	(t)

imsart-aos ver. 2007/09/18 file: ModeLogconc-v11.tex date: January 3, 2008

Acknowledgments. This research was initiated while Kaspar Rufibach was visiting the Institute for Mathematical Stochastics at the University of Göttingen, Germany and while Fadoua Balabdaoui was visiting the Institute of Mathematical Statistics and Actuarial Science at the University of Bern, Switzerland. We would like to thank both institutions for their hospitality.

Partially supported by the Swiss National Science Foundation † Research supported in part by NSF grant DMS-0503822 and NI-AID grant 2R01 AI291968-04 AMS 2000 subject classifications: Primary 62N01, 62G20; secondary 62G05

We get the explicit expression of the asymptotic constants c k (x 0 , ϕ 0 ) and d k (x 0 , ϕ 0 ) using the following relations:

2 ) -1 and (4.37) (4.38) This is completely analogous to the derivations on p. 1689 in [START_REF] Groeneboom | Estimation of a convex function: characterizations and asymptotic theory[END_REF], precisely

From (4.37) and (4.38) we get c k (x 0 , ϕ 0 ) and d k (x 0 , ϕ 0 ) as given in (2.3) and (2.4), and C k (x 0 , ϕ 0 ) and D k (x 0 , ϕ 0 ) as in (2.5) and (2.6).

(ii) Note that we can write

by making use of (1.1) and the specific choice of Ân and Bn . But since we connect H locmod n and Y locmod n to the "invelope" the latter property needs primarily to hold for the modified processes. This can easily be established by considering (4.35) and (4.36), and hence it follows that

(iii) To show that Ân and Bn are tight. By Theorem 4.3, we know that there exists M > 0 and τ ∈ S( ϕ n ) such that 0 ≤ x 0 -τ ≤ M n -1/(2k+1) with (2001b), and arguments similar to those of Groeneboom, Jongbloed and Wellner (2001a) or, alternatively, tightness plus uniqueness arguments along the lines of Groeneboom, Maathuis, and Wellner (2007). ✷

Proof of Theorem 2.1. The claimed joint convergence involving ϕ n and ϕ ′ n follows from part (iv) of Theorem 4.6 and the relations (4.39) and (4.40). The joint limiting distribution of f n (x 0 ) -f 0 (x 0 ) and f ′ n (x 0 ) -f ′ 0 (x 0 ) follows immediately by applying the delta-method. ✷ 4.3. Proofs for Section 3.

Proof of Theorem 3.6. We first use the simple fact that M n is the only point x ∈ R which satisfies

This follows immediately from concavity of ϕ n and the definition of M n . Note that ϕ n may have a flat region or "modal interval"; in this case, there exists an entire interval of points where the maximum is attained, and M n is the left endpoint of this interval.

A tightness property of the process H

2 , which follows from Lemma 2.7 of [START_REF] Groeneboom | Estimation of a convex function: characterizations and asymptotic theory[END_REF], is also needed to establish the limiting distribution of M n : for any ǫ > 0 and t ∈ R, there exists C = C(ǫ) such that P H

(3)

In other words, one can view H

(3) 2 (t) as an "estimator" of the odd function -24t. Since C is independent of t, it follows that for a fixed ǫ, H

(3) 2 (t) < 0 (resp. H

(3) 2 (t) > 0) for t > 0 (resp. -t < 0) big enough, with probability greater than 1 -ǫ.

The sign of H

(3) 2

and uniqueness of M n turn out to be crucial in determining the limiting distribution of the latter. From Theorem 4.6 and the two derivative relations (4.39) and (4.40) it follows that

for each K > 0 with the product topology induced by the uniform topology on C[-K, K] and the Skorohod topology on D[-K, K]. Here H k,a,σ is is the imsart-aos ver. 2007/09/18 file: ModeLogconc-v11.tex date: January 3, 2008 unique process on R satisfying (4.33). A similar result holds for the MLE of the log-concave density f 0 . When x 0 is replaced by the population mode m 0 = M (f 0 ) and k = 2 the second weak convergence implies that

2,a,σ (-T ), and

2,a,σ (T ).

For T > 0 large enough, this in turn implies that for ǫ > 0, we can find N ∈ N \ {0} such that for all n > N we have that

using the property of M n in (4.41), it follows that

). Then we note that

where

for each K > 0 by (4.42) with k = 2. Thus by the argmax continuous mapping theorem (see e.g. van der Vaart and Wellner (1996), page 286) it follows that

where Z = H

(2) 2,a,σ , a = 1/ f 0 (m 0 ), and σ = |ϕ

(2) 0 (m 0 )|/4!. Note that H 2,a,σ is related to the "driving process" Y 2,a,σ with a =