Ahmad Younso
email: ahmad.younso@inrae.fr

Consistency of the simple mode of a density for spatial processes

Keywords: Spatial process, Mode, Simple mode, Mixing condition, Bandwidth, consistency

We are concerned with estimating the mode of a density of a spatial process by kernel method under some dependency conditions. A simple estimate of the mode based only on data is considered. The approach consists of maximizing the density kernel estimate on data. An optimal rate of uniform consistency of the density estimate will be established. Strong consistency of the simple estimate of the mode with the rate of consistency will be investigated when data are spatially dependent by some general mixing conditions.

Introduction

There are many elds where the knowledge of modes is of great interest. For example, in unsupervised problems where modes are used as measure of typicality of a set of data. In particular, in modern applications, mode estimation is often used in clustering, with the modes representing cluster centers. There is an extensive literature on mode estimation in the independent case, see the key references: [START_REF] Parzen | On estimation of a probability density function and mode[END_REF], [START_REF] Devroye | Recursive estimation of the mode of a multivariate density[END_REF], [START_REF] Romano | On weak convergence and optimality of kernel density estimates of the mode[END_REF]), and Abraham et al. (2004, 2003). Most of the existing works are concerned with consistency of the estimators and rates achievable by various approaches, with much less emphasis on the ease of implementation of these approaches. The common approaches consist of estimating the density mode by maximizing an estimate of the unknown density (usually a kernel estimate) on R d . Despite the easy computation, there is only a very few literature dedicated to so-called direct mode estimator (or simple mode) which estimate the mode from practical data (see [START_REF] Devroye | Recursive estimation of the mode of a multivariate density[END_REF] and Abraham et al. (2003)). Abraham et al. (2003) show that the simple estimator is consistent. In the spatially dependent case, there is an extensive literature on nonparametric estimation of density and regression function, and classication using kernel method, see for example: [START_REF] Carbon | Kernel density estimation for random elds[END_REF][START_REF] Carbon | Kernel regression estimation for random elds[END_REF], [START_REF] Tran | Kernel density estimation on random elds[END_REF] and [START_REF] Younso | On the consistency of a new kernel rule for spatially dependent data[END_REF]. Except the work of [START_REF] Amiri | Nonparametric recursive density estimation for spatial data[END_REF], consistency of the mode estimators remains unexplored in the spatial case. [START_REF] Amiri | Nonparametric recursive density estimation for spatial data[END_REF] establish a mean and almost complete consistency results of the conditional kernel mode estimator under some general conditions. For more practical results in the spatial case, our aim in this paper is to extend the results of Abraham et al. (2003) on the strong consistency of the simple mode estimate to the spatial case. Before that we will investigate rate of the uniform consistency of the kernel density estimate.

2. Mode estimation for spatial process Let (X i , i ∈ Z N , N ≥ 1) be a R d -valued measurable strictly stationary spatial process (d ≥ 1) with the same distribution as X and dened on a probability space (Ω, A, P). Assume that X has an unknown unimodal density f with support in R d . We will also assume that f is bounded and that X is squareintegrable. A point i = (i 1 , ..., i n ) ∈ Z N will be referred to as a site. For n = (n 1 , ..., n n ) ∈ (N * ) N , we denote I n the rectangular region

I n = {i ∈ Z N : 1 ≤ i k ≤ n k , ∀k = 1, ..., N }
and we set n = n 1 ×...×n n = card(I n ). We write n → ∞ if min k=1,...,N n k → ∞ and max 1≤i,j≤N |n i /n j | ≤ C for a constant C such that 0 < C < ∞. In this paper, we consider the problem of estimating the mode θ of f . We will dene an estimate of the mode using a kernel density estimate extended by [START_REF] Tran | Kernel density estimation on random elds[END_REF] (see also [START_REF] Carbon | Kernel density estimation for random elds[END_REF]) to the spatial case. Given a bounded and symmetric kernel K and a bandwidth b n > 0 such that b n → 0 as n → ∞, the kernel estimate is given by

f n (x) = 1 nb d n i∈In K x -X i b n , x ∈ R d .
Most of the existing works on the mode estimate consist of estimating the mode θ by any maximizer θn of f n on R d , i.e., θn ∈ arg max

R d f n .
This estimate is called indirect since rst f is estimated by f n and then θn is taken to be any point for which f n ( θn ) = max x∈R d f n (x). In the i.i.d. case, various works such as [START_REF] Parzen | On estimation of a probability density function and mode[END_REF], [START_REF] Konakov | On asymptotic normality of the sample mode of multivariate distributions[END_REF], [START_REF] Romano | On weak convergence and optimality of kernel density estimates of the mode[END_REF] and [START_REF] Leclerc | Vitesse de convergence presque sre de l'estimateur b noyau du mode[END_REF] establish consistency properties of the approach under regularity assumptions. While yielding much insight into the problem, the estimator θn is hard to implement in the practice where the argmax is usually computed over a nite grid. We will deal with another estimate of the mode to overcome such problems. Denoting by S n = {X i , i ∈ I n } a set of observations of the spatial process (X i ) i∈Z N on the region I n , we let the estimate θ n of the mode θ is dened as

θ n ∈ arg max Sn f n .
This estimate is called direct (or simple mode) since it maximizes f n on data. This estimator has been rst considered by [START_REF] Devroye | Recursive estimation of the mode of a multivariate density[END_REF] in the independent case. More recently, Abraham et al. (2003Abraham et al. ( , 2004) ) have been established the strong consistency with the almost surely rate and the asymptotic normality for this estimate under mild conditions. In this paper, we will extend the strong consistency results of Abraham et al. (2003) to the spatial case. Before we establish the main results of this paper, we introduce some useful notations.

We consider for all > 0, the level sets

A( ) = {x ∈ R d : f (x) > f (θ) -},
and we denote by diam A( ) the diameter of A( ). Let V (δ) be the open ball centered at θ with radius δ > 0. We assume that there exists δ 0 > 0 small enough such that inf V f > 0 with V ≡ V (δ 0 ). For any function g, we denote by g ∞ (resp. g V ) the supremum norm of g on R d (resp. on V ). The Euclidean norm will be denoted by . . For any set B, its complement is denoted by B c . To avoid high density areas arbitrarily far from the exact mode θ, we suppose that the following property holds

diam A( ) → 0 as → 0. (2.1)
This assumption is used by Abraham et al. (2003) to establish the strong consistency of the estimate in the i.i.d. case. Furthermore, when the set of variables are i.i.d., Abraham et al. (2003) prove the following lemma.

Lemma 2.1. If (2.1) is fullled and X i , i ∈ I n are i.i.d., then, a.s., for any δ > 0, max

Sn∩V (δ) f → f (θ) as n → ∞.
As a consequence of this lemma, the density f asymptotically attains its maximum in an element of S n in a neighborhood of θ. Since the sample points are naturally concentrated in high density areas, the set S n can be regarded as the most natural (random) grid for approximating the mode. Lemma 2.1 is crucial in proving the consistency results so it is important to extend it to the dependent case where we will need it. Lemma 2.1 will be extended to the dependent case. To obtain asymptotic results, we will assume throughout the paper, the spatial process (X i ) i∈Z N is spatially dependent according to the following mixing condition: there exists a function ϕ : R → R + with ϕ(t)

0 as t → ∞, such that whenever E, E ⊂ Z N with nite cardinals, α B(E), B(E ) := sup{|P(A ∩ C) -P(A)P(C)|, A ∈ B(E), C ∈ B(E )} ≤ h(Card(E), Card(E ))ϕ(dist(E, E )), (2.2)
where B(E) (resp. B(E )) denotes the Borel σ-eld generated by (X i ) i∈E (resp.

(X i ) i∈E ), Card(E) (resp. Card(E )) the cardinality of E (E ), dist(E, E
) the Euclidean distance between E and E , and h : N 2 → R + is a symmetric positive function which is nondecreasing in each variable. Throughout the paper, it will be also assumed for simplicity that h satises

h(m 1 , m 2 ) ≤ L(m 1 + m 2 ) ξ , ∀m 1 , m 2 ∈ N * , (2.3)
for some L > 0 and ξ ≥ 0. If h ≡ 1, the random eld called strongly mixing. Conditions (2.2)-( 2.3) are satised by many spatial models. Examples can be found in [START_REF] Nedearhouser | Convergence of blocks spins dened by a random elds[END_REF], [START_REF] Rosenblatt | Stationary sequences and random elds[END_REF] and [START_REF] Guyon | Estimation d'un champ par pseudo-vraisemblance conditionnelle: Etude asymptotique et application au cas markovien[END_REF].

We suppose also that ϕ(t) tends to zero at a polynomial rate, i.e,

ϕ(t) = O(t -ζ ), (2.4)
The following proposition extends Lemma 2.1 to the dependent case.

Proposition 2.1. If (2.1) is fullled and X i , i ∈ I n are identically distributed and α-mixing, then, a.s., for any δ > 0, max

Sn∩V (δ) f → f (θ) as n → ∞.
We will also need the following lemma whose proof is the same as that of (Abraham et al. (2003), Lemma 1).

Lemma 2.2. If (2.1) is fullled, and X i , i ∈ I n are identically distributed and α-mixing, then, for any δ > 0, sup

V (δ) c f (x) < f (θ).
As a matter of fact, one can easily prove the equivalence between the two assertions of the lemma 2.2. Before we state the main results of this paper, we introduce some regularity assumptions.

H1. f (x) → 0 if x → ∞. H2. For each i = j, (X i , X j ) has a density f i,j such that sup u,v∈R d |f i,j (u, v) -f (u)f (v)| ≤ C, for some C > 0. H3. sup x∈R d x 2 f (x) < ∞.
H4. K satises the Lipschitz condition, i.e, there exists R > 0 such that for all

x, y ∈ R d , |K(x) -K(y)| < R x -y . H5. sup x∈R d x d+1 |K(x)| < ∞ and R d x 2 K(x)dx < ∞. H6.
support K is a compact set contained in the closed ball centered at the origin with radius a.

Remark 2.1. Note that H1 is satised, for example, if f is bounded and uniformly continuous. Assumption H2 is a standard technical conditions in nonparametric estimation used to replace the independence hypothesis. H3 is weaker than (2.30) in [START_REF] Bosq | Nonparametric Statistics for Stochastic Processes[END_REF]. Assumptions H4-H6 are classical conditions in nonparametric estimation. See [START_REF] Bosq | Nonparametric Statistics for Stochastic Processes[END_REF], page 39) for the use of H5.

Main results

In order to establish the strong consistency of the simple mode estimator, we need to show strong uniform consistency of f n on R d . Dene

ζ * 1 = d(ζ + 4N (d + 1)) ζ -N (2ξ + 3) and ζ * 2 = - ζ ζ -N (2ξ + 3) . Let g(n) = N i=1 (log n i )(log log n i ) 1+ for some > 0. It is well known that n∈(N * ) N 1/(ng(n)) < ∞. Suppose as n → ∞, nb d n /(log n) -→ ∞ (3.1)
and set

ψ n = nb d n / log n -1/2 .
Observe that (3.1) is classical to prove uniform consistency of the kernel density estimate in the i.i.d. case. and

nb ζ * 1 n (log n) ζ * 2 g(n) -2N/(ζ-N (2ξ+3)) -→ ∞, (3.3) then, sup x∈R d |f n (x) -Ef n (x)| = O(ψ n ) a.s.
Observe that (3.3) implies (3.1) as ζ → ∞ and that (3.2) is weaker than (2.6) in [START_REF] Carbon | Kernel density estimation for random elds[END_REF], Lemma 3.1). Condition (3.3) allows to determine the convergence of a series by comparing it with the convergent series

n∈(N * ) N 1/(ng(n))
in order to prove the almost sure uniform convergence of f n (x). The result of Theorem 3.1 is proved in the case of independent processes by techniques of empirical processes (see [START_REF] Pollard | Convergence of Stochastic Processes[END_REF]). These techniques have not yet developed for spatial processes. The consistency in Theorem 3.1 on compact is proved by [START_REF] Carbon | Kernel density estimation for random elds[END_REF].

Corollary 3.1. Under assumptions of Theorem 3.1, if furthermore f is twice dierentiable with bounded second partial derivatives. Then,

sup x∈R d |f n (x) -f (x)| = O(ψ n ) a.s.
Now we move to state consistency of the simple mode estimator. The following theorem extends the result of (Abraham et al. (2003), Theorem 1) to the spatially dependent case.

Theorem 3.2. Under assumptions of Corollary 3.1, if in addition (2.1) and H6 are satised, then, as n → ∞, θ n -→ θ, a.s.

To state the result about rate of convergence of the simple mode estimate, we need the following assumption. Suppose there exists L > 0 and β > 0 such that

diam A( ) ≤ L β (3.4)
for some ≤ 0 where 0 > 0 is small enough. (3.4) is used by Abraham et al. (2003Abraham et al. ( , 2004) ) to obtain strong consistency and asymptotic normality of the simple mode in the i.i.d. case. If (3.4) holds, we say that the density f admits a peak index β. This peak index measures the sharpness of the density around the mode. Roughly, the sharper the density around θ, the larger the peak index is. For example, β = 1/2 corresponds to the family of normal densities and β = 1 corresponds to the family of Laplace densities (see Abraham et al. (2003) for further details).

Theorem 3.3. Under assumptions of Theorem 3.2, if in addition (3.4) is satised, then,

|θ n -θ| = O(b n + ψ β n ) a.
s. The rate stated in the theorem is the same as that obtained in the i.i.d. case. Furthermore, if β = 1, we get

|θ n -θ| = O(ψ n ) a.s.
with ψ n = max(b n , ψ n ). The latter one is the same optimal rate as that of the density estimate obtained by [START_REF] Tran | Density estimation for time series by histograms[END_REF] in univariate temporal case.

Simulation studies

In this section, we conduct simulations using R program to evaluate our method. According to [START_REF] Biau | Optimal asymptotic quadratic errors of density estimators on random elds[END_REF]), Theorem 7), under some mild smoothness condi-tions, the mean integrated squared error (MISE) is given by

E R d (f n (x) -f (x)) 2 dx = b 4 n 4 Γ 2 d + R d K 2 (u)du nb d n + o b 4 n + 1 nb d n , with Γ 2 d = R d   1≤i,j≤d ∂ 2 f ∂x i ∂x j (x) R d z i z j K(z)dz   2 dx.
By calculus, one can show that the approximate MISE is minimized by choosing the asymptotic optimal bandwidth to be

b nopt = d R d K 2 (u)du Γ 2 d 1 d+4 n-1 d+4 .
Due to the need of a second-order dierentiable kernel to nd a solution for the problem of optimal bandwidth, we will suppose that K is the Gaussian kernel.

In this case, we have

R d K 2 (u)du = (2 d π d/2 ) -1 .
Therefore, we can write

b nopt = U (f )n -1 d+4 with U (f ) = d 2 d π d/2 Γ 2 d 1 d+4
.

(4.1)

Observe that b nopt depends on the unknown density f through the term Γ 2 d and so it is not calculable. There are two methods to get around the problem.

The rst method, called iterative plug-in rule, consists in estimating f by its kernel estimator f n (see [START_REF] Scott | Kernel density estimation revisited[END_REF]) and proceed as follows. Denote respectively Γ 2 d and U (f n ) the values of Γ 2 d and U (f ) when f is replaced by f n . Similarly to (4.1), to nd an approximate value for b nopt , say b nopt , we can write

b nopt = U (f n )n -1 d+4 with U (f n ) = d 2 d π d/2 Γ 2 d 1 d+4
. Let b n,i be an arbitrarily picked value of b n and denote f n,i (x) the kernel estimator constructed based on b n,i . Since U (f n,i ) is an explicit function of b n,i , the problem turns to use Newton's method to nd a zero for the following equation d+4) . Hence, we substitute each previous number for b n,i back into the equation (4.2) to get a closer and closer approximation to the exact solution. The second method, called normal reference rule, consists in replacing f by a Gaussian density with parameters estimated from the sample [START_REF] Scott | Multivariate Density Estimation: Theory, Practice, and Visualization[END_REF]). The problem we can have with the latter method is the biased estimates of the parameters due to the dependence condition. Therefore, in the rst two simulation studies which deals with the univariate case, we will use the rst method, and due to the diculty of calculations in the bivariate case, we will have to use the second method in the last two simulation studies.

b n,i+1 = Φ(b n,i ) for i ∈ N, (4.2) with Φ(b n,i ) = U (f n,i )n -1/(
Simulation 1. We assume that N = 2 and d = 1. We generate random variables X (i,j) distributed according to a normal distribution with mean µ = 0, standard deviation σ = 2 and covariance function c(t) = 2t -ζ on the grid I (n1,n2) for some ζ > 0. First of all, the target density f of X (i,j) is estimated by the kernel estimator f n based on bandwidth chosen by the iterative plug-in rule. To analyze the eect of the dependence degree on the density estimator, we consider three degrees of dependence corresponding to ζ = 2, 5, 8 and we take n 1 = n 2 = 25. For each value of ζ, three samples are generated to get three estimations for f based on optimal values for b n chosen as solutions of (4.2). Figure 1 show that as ζ gets large, the bias of the density estimate gets smaller which is interpreted by the fact that the larger the value of ζ, the closer the variables to the independence.

- As mentioned previously in the theoretical part, the target density mode can be estimated either by maximizing f n on the set of simulated observations {X (i,j) , (i, j) ∈ I (25,25) } to get the direct estimate θ n or by maximizing it on R d to get the indirect estimate θn . Although our focus will be on the direct estimate, we nd it useful to compare the paths of the two estimates. To this aim, it will be better to assign a number n varying from 1 to 625 for each site (s, t) ∈ I (25,25) based on lexicographical order relation. Hence, for each value of n associated with the site (s, t), the kernel estimate of the density is constructed using the set of observations

{X (i,j) , 1 ≤ i ≤ s, 1 ≤ j ≤ t}.
For each n = 1, ..., 625, the simulation scenario is repeated to generate 100 samples, and both the direct and indirect estimates are determined for each sample. Then, the average values for the direct and indirect estimates are calculated for each value of n. Figure 2 shows how similar the paths of the two estimates for ζ = 8. We clearly notice that the two paths are largely identical and that as the value of n gets large the two average estimates get closer and closer to the target mode. This is a motivating reason to focus on the simple estimator in this paper. Now, we study the eect of the dependence degree on the simple estimate for ζ = 2, 5, 8. For each value of ζ and each n, the above simulation scenario is repeated to get 100 samples. For each sample, the density f is estimated by f n and θ n is determined. The optimal bandwidth is obtained by solving (4.2). For each ζ = 2, 5, 8, the average trajectory of θ n is established. Figure 3 To analyze the eect of the bandwidth on the simple mode estimate, Figure 4 shows the average trajectories corresponding to three bandwidths: an optimal bandwidth bnopt obtained as solution of (4.2), a small bandwidth (0.3 bnopt ), and a large bandwidth (1.7 bnopt ). Note that too small values of the bandwidth produce an undersmoothing bath and too large values of the bandwidth produce an oversmoothing bath. In contrast to the indirect estimate, which is often not a sampling point, the simple estimate is an element of the sample and son site can be determined. In other words, it is easy to show in which site θ n is observed. To do that, for each n 1 = n 2 ∈ {17, ..., 25}, we generate a sample of size n 1 × n 2 to estimate f by f n and θ by θ n . Figure 5 shows the locations of θ n on the nine regions corresponding to n 1 = n 2 = 17, ..., 25. It also shows that the value of θ n remained in the same site for n 1 = n 2 = 17, 18, 19 while it began to move from a site to another for n 1 = n 2 = 20, ..., 25.

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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5 10 15 5 10 15 q q 5 10 15 5 10 15 q q 5 10 15 5 10 15 n1=n2=19 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5 10 15 20 5 10 15 20 q q 5 10 15 20 5 10 15 20 q q 5 10 15 20 5 10 15 20 n1=n2=20 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5 10 15 20 5 10 15 20 q q 5 10 15 20 5 10 15 20 q q 5 10 15 20 5 10 15 20 n1=n2=21 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5 10 15 20 5 10 15 20 q q 5 10 15 20 5 10 15 20 q q 5 10 15 20 5 10 15 20 n1=n2=22 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5 10 15 20 5 10 15 20 q q 5 10 15 20 5 10 15 20 q q 5 10 15 20 5 10 15 20 n1=n2=23 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5 10 15 20 5 10 15 20 q q 5 10 15 20 5 10 15 20 qq 5 10 15 20 5 10 15 20 n1=n2=24 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5 10 15 20 25 Simulation 2. For N = 2 and d = 1, we generate dependent variables X (i,j) on the grid I (25,25) from the mixture distribution 0.7N (-2, 1) + 0.3N (1, 0.4) with the same covariance function used above c(t) = 2t -ζ with ζ = 8. Figure 6 shows the curves of the density f and its estimate by f n . We retake 1 ≤ n ≤ 625 dened in the previous simulation. For each n = 1, ..., 625, we maximize f n on data to get the simple estimate θ n . This procedure is repeated 100 times for each n and the average value is calculated in each replication. Figure 7 shows the trajectory of the average values of the simple estimate in function of n.

Obviously, as the value of n gets large the average trajectory of θ n tends to be closer and closer to the target mode (θ = 1). Simulation 3. Using a single bandwidth is not suitable for data with dierent variability in the coordinate directions [START_REF] Wand | Comparison of smoothing parameterizations in bivariate kernel density estimation[END_REF]). To avoid applying the same amount of smoothing in each coordinate direction, it is important to have independent smoothing parameters for each of the coordinate directions. We turn our attention to the bivariate case (d = 2). In this case, there can be between one and three independent smoothing parameters, denoted b 1n , b 12n and b 2n , in the kernel estimator. In its most general bivariate form, the global bandwidth kernel estimator of f (x) is given by

f n (x; B) = 1 n|B| 1/2 i∈In K B -1/2 (x -X i ) , (4.3)
where the bandwidth matrix

B = B(n) = b 2 1n b 12n b 12n b 2 2n
is a 2 × 2 symmetric and positive denite matrix. The most employed bivariate kernel is the normal kernel

K(x) = (2π) -1 e -1 2 x x for x = (x 1 , x 2 ) ∈ R 2 . A simple choice is to consider B = diag(b 2 1n , b 2 2n ) with b in > 0 for i = 1, 2.
According to the normal reference rule in a general context (see [START_REF] Scott | Multivariate Density Estimation: Theory, Practice, and Visualization[END_REF]), if f is replaced by the centered bivariate normal density with variance-covariance matrix Σ = diag(σ 2 1 , σ 2 2 ), the optimal choice of B which minimizes MISE is given by

b in = σ i n-1/6 , i = 1, 2, (4.4)
where σ i can be replaced by its sample estimator in practical implementations.

In this simulation study, we consider the bivariate case where N = 2 and d = 2

with n 1 = n2 = 25. We generate random vectors (X (i,j) , Y (i,j) ) distributed according to the bivariate normal distribution with means data to get the simple estimate of the mode and we get θ n = (1.885, 2.055). We simulate a hundred samples and we obtain the simple estimate of the mode for each one. Then, a boxplot is constructed for each component of θ n = (θ 1n , θ 2n ) using its set of estimations. The scenario is repeated nine times to have nine boxplots as in Figure 9. From this gure, we can see that the medians are roughly at the same level in the middle of the box and the whiskers are about the same on both sides of the box. This shows the symmetry of distribution of θ n around the target mode (θ = 2).

µ X = µ Y = 2, standard deviations σ X = σ Y = 2,
q θ1n θ2n 1.4 1.8 2.2 2.6 q q θ1n θ2n 1.4 1.8 2.2 2.6 q q θ1n θ2n 1.4 1.8 2.2 2.6 q q q θ1n θ2n 1.4 1.8 2.2 2.6 q q q θ1n θ2n 1.4 1.8 2.2 2.6 θ1n θ2n 1.4 1.8 2.2 2.6 q q θ1n θ2n 1.4 1.8 2.2 2.6 θ1n θ2n 1.4 1.8 2.2 2.6 q θ1n θ2n 1.4 1.8 2.2 2.6
Figure 9: Bivariate normal distribution: boxplots for the components of θn using 100 samples of size 625.

Simulation 4. With the same covariance function of the latter simulation, we simulate random vectors (X (i,j) , Y (i,j) ) distributed according to the following Gaussian mixture model : 0.2N (µ 1 , Σ 2 ) + 0.8N (µ 2 , Σ 2 ) with µ 1 = (0, 0) and µ 2 = (3, 3), Σ 1 = Σ 2 = diag(1, 1) on the grid I (25,25) . The density of this distribution has a single global mode θ = (3, 3). With one simulated sample we have b 1n = 0.509 and b 2n = 0.517. Figure 10 displays the image plot and the perspective plot of the density estimate. As illustrated by the image plot, the two local maximas of the density permit to split the dataset into two clusters separated by a data-poor region, the rst cluster is centered at the rst maxima (the global mode) and the second cluster is centered at the second maxima. In the rst cluster, the colors range from yellow (the highest density) in the center to purple (lowest density), while the second cluster is represented by three colors corresponding to the lowest density in the rst cluster. For the simple estimate of the mode associated with this sample we obtain θ n = (2.976, 3.084). As in the previous simulation, we simulate 100 samples and we obtain the simple estimate of the mode for each one. Then, a boxplot is constructed for each component of θ n = (θ 1n , θ 2n ) using its set of estimations. Similarly to Figure 9, the medians are roughly at the same level in the middle of the box and the whiskers are about the same on both sides of the box. q q q q θ1n θ2n 2.7 2.9

3.1

θ1n θ2n

2.7 2.9

3.1

θ1n θ2n

2.7 2.9

3.1

3.3

q q θ1n θ2n 2.7 2.9

3.1

3.3

q q θ1n θ2n 2.7 2.9

3.1 3.3 q θ1n θ2n 2.7 2.9

3.1

θ1n θ2n

2.7 2.9

3.1 3.3 q q θ1n θ2n 2.7 2.9

3.1

θ1n θ2n

2.7 2.9

3.1

3.3

Figure 11: Bivariate normal mixture distribution: boxplots for the components of θn using 100 samples of size 625.

PROOFS

Simple mode of a density

Proofs

Proof of Proposition 2.1. Enumerate the r.v.'s X i in an arbitrary manner and refer to them respectively X 1 , X 2 ..., X n. Let X * 1 = X 1 . By [START_REF] Bradley | Approximation theorems for strongly mixing random variables[END_REF], Theorem 2) there exists a variable X * 2 independent of X * 1 such that X * 1 and X * 2 have the same distribution. In the same manner, there exists a variable X * 3 independent of X * 2 and X * 3 and has the same distribution as X * 1 . If we continue in this manner, we get X * 4 , ..., X * n which are mutually independent, independent of X * 1 , X f -→ f (θ) as n → ∞.

(5.1)

By contradiction; suppose (2.1) is fullled but the assertion of Proposition 2.1 is false, i.e., there are 0 > 0 and δ 0 > 0 such that for some n large enough:

P f (θ) -max Sn∩V (δ0) f ≥ 0 > 0.
Thus,

P f (θ) -max Sn∩V (δ0) f ≥ 0 = P max Sn∩V (δ0) f ≤ f (θ) -0 = P i∈In ((f (X i ) ≤ f (θ) -0 , X i ∈ V (δ 0 )) ∪ (X i ∈ V (δ 0 ) c ) > 0.
Consequently, for each i ∈ I n ,

P ((f (X i ) ≤ f (θ) -0 , X i ∈ V (δ 0 )) ∪ (X i ∈ V (δ 0 ) c ) > 0.
Then, since X * i has the same density f as X i , for each i ∈ I n ,

P ((f (X * i ) ≤ f (θ) -0 , X * i ∈ V (δ 0 )) ∪ (X * i ∈ V (δ 0 ) c ) > 0.
Since the elements of S * n are i.i.d., we can write

P i∈In ((f (X * i ) ≤ f (θ) -0 , X * i ∈ V (δ 0 )) ∪ (X * i ∈ V (δ 0 ) c ) > 0,
i.e., there are 0 > 0 and δ 0 > 0 such that for n large enough:

P f (θ) -max S * n ∩V (δ0) (f ) ≥ 0 > 0.

PROOFS

Simple mode of a density

The last inequality leads to contradiction with (5.1).

Proof of Theorem 3.1. Let a n = b -d n . Then,

sup x∈R d |f n (x)-Ef n (x) ≤ sup x ≤an |f n (x)-Ef n (x)|+ sup x >an |f n (x)-Ef n (x)|. (5.2)
Consequently, we prove the theorem if we show that each term in the right-hand side of (5.2) tends almost surely to zero as n → ∞. We rst show that

sup x ≤an |f n (x) -Ef n (x)| = O(ψ n ) a.s. (5.3) We denote h n = b d+3 n , s n = a n h -1
n and ν n = s d n where . stands for the integer part. Set B(x, r) the closed ball centered at x with radius r. Since B(0, a n ) = {x, x ≤ a n } is compact, it can be covered by ν n balls centered at y j , j = 1, ..., ν n with radius h n . Consequently,

sup x ≤an |f n (x) -Ef n (x)| ≤ max 1≤j≤νn sup x∈B(yj ,hn) |f n (x) -f n (y j )| + max 1≤j≤νn sup x∈B(yj ,hn) |Ef n (x) -Ef n (y j )| + max 1≤j≤νn |f n (y j ) -Ef n (y j )|.
We will bound each term in the right-hand side of the above inequality. One can easily verify that b 2 n = o(ψ n ) according to (3.2). First, we have

|f n (x) -f n (y j )| ≤ b -d n sup (x,z)∈B(yj ,hn)×R d K x -z b n -K y j -z b n < Rh n /b d+1 n = Rb 2 n .
Similarly, max 1≤j≤νn sup x∈B(yj ,hn)

|Ef n (x) -Ef n (y j )| = o(ψ n ).
(5.5)

To complete the proof of (5.3), it remains to show that (5.7)

Fortunately, the proof of (5.7), which depends mainly on (3.3), is exactly the same as that of (5.6) in [START_REF] Carbon | Kernel density estimation for random elds[END_REF]. Therefore, to complete the proof of the theorem, it remains, by (5.2), to prove that

sup x >an |f n (x) -Ef n (x)| = O(ψ n ) a.s.
(5.8)

We have

sup x >an |f n (x) -Ef n (x)| ≤ sup x >an |f n (x)| + sup x >an |Ef n (x)|.
(5.9)

Let us rst prove that sup

x >an E|f n (x)| = O(ψ n ). (5.10) Note that for each x ∈ R d , E|f n (x)| ≤ R d K(u)f (x -b n u)du = B(0,b -1 n ) K(u)f (x -b n u)du + B(0,b -1 n ) c K(u)f (x -b n u)du. (5.11) Thus, if x > a n and u ∈ B(0, b -1 n ), then x -b n u ≥ a n -1. Consequently, since b 2 = o(ψ n ), H3 yields, ψ -1 n sup x >an B(0,b -1 n ) K(u)f (x -b n u)du ≤ sup x >an-1 f (x) B(0,b -1 n ) K(u)du ≤ Cψ -1 n a -2 n sup x >an/2 x 2 f (x) ≤ Cψ -1 n b 2d n sup x∈R d x 2 f (x) = o(1).
(5.12)

In the other hand, since f is bounded by assumption, we have by H5,

ψ -1 n sup x >an B(0,b -1 n ) c K(u)f (x -b n u)du ≤ ψ -1 n b 2 n sup R d f u >b -1 n u 2 K(u)du ≤ Cψ -1 n b 2 n = o(1).
(5.13)

PROOFS

Simple mode of a density Combining (5.11)-(5.13), we get (5.10). Now, according to (5.9), to establish (5.8), it remains to show that

sup x >an |f n (x)| = O(ψ n ) a.s.
(5.14)

To do that, set n = ηψ n for some η > 0. We have

P sup x >an |f n (x)| ≥ n ≤ P sup x >an 1 nb d n x-X i >an/2 K x -X i b n ≥ n /2 + P sup x >an 1 nb d n x-X i ≤an/2 K x -X i b n ≥ n /2 .
However, by H5, for each

x ∈ R d , -1 n nb d n x-X i >an/2 K x -X i b n ≤ -1 n nb d n x-X i >an/2 x -X i b n d+1 K x -X i b n b n x -X i d+1 ≤ C -1 n nb d n x-X i >an/2 b n x -X i d+1 ≤ C -1 n a d+1 n ≤ Cψ -1 n b d(d+1) = o(1).
Hence, for n large enough,

P sup x >an |f n (x)| ≥ n ≤ P sup x >an 1 nb d n x-X i ≤an/2 K x -X i b n ≥ n /2 . Since for x > a n and x -X i ≤ a n /2, X i ≥ a n /2, if we set c = K ∞ , then, P sup x >an |f n (x)| ≥ n ≤ P c nb d n i∈In I { X i >an/2} ≥ n /2 ,
where I A denotes the indicator function of the set A. Markov's inequality yields

P X i ≥ a n /2 ≤ 2a -1 n E X i ≤ Cb d n → 0 as n → ∞.
Therefore, for n large enough, we can write (5.15) with, for each i ∈ I n ,

P sup x >an |f n (x)| ≥ n ≤ P i∈In ∆ i ≥ n /2 ,
∆ i = c(nb d n ) -1 I { X i ≥an/2} -P ( X i ≥ a n /2) .
We will apply the known block decomposition of [START_REF] Carbon | Kernel density estimation for random elds[END_REF] on the set of random variables ∆ i , i ∈ I n . Without loss of the generality, suppose that there exist positive integers p, q 1 , ..., q N such that n k = 2pq k , k = 1, ..., N . The variables ∆ i , i ∈ I n can be regrouped into 2 N q 1 × ... × q n cubic blocks of side p. Denote

U (1, n, j) = (2j k +1)p i k =2j k p+1 k=1,...,N ∆ i U (2, n, j) = (2j k +1)p i k =2j k p+1 k=1,...,N -1 2(jn+1)p in=(2jn+1)p+1 ∆ i U (3, n, j) = (2j k +1)p i k =2j k p+1 k=1,...,N -2 2(j N -1 +1)p i N -1 =(2j N -1 +1)p+1 (2jn+1)p in=2jnp+1 ∆ i ... etc ... U (2 N -1, n, j) = 2(j k +1)p i k =(2j k +1)p+1 k=1,...,N -1 (2jn+1)p in=2jnp+1 ∆ i U (2 N , n, j) = 2(j k +1)p i k =(2j k +1)p+1 k=1,...,N ∆ i .
For each integer i = 1, ..., 2 N , set

T (n, i) = q k -1 j k =1 k=1,...,N U (i, n, j). Thus, i∈In ∆ i = 2 N i=1 T (n, i).
(5.16) Furthermore, for each i = 1, ..., 2 N , T (n, i) is a sum of r = q 1 × ... × q N of the U (i, n, j)'s. If for example we let i = 1, then T (n, 1) is a sum of r = q 1 × ... × q N of the U (1, n, j)'s. The random term U (1, n, j) is measurable with respect to the σ-eld generated by ∆ i with i belonging to the set of sites

S j = {i : 2j k p + 1 ≤ i k ≤ (2j k + 1)p, k = 1, ..., N }.
(5.17)

For dierent values of j = (j 1 , ..., j N ), the sets of sites (5.17) are separated by a distance of at least p, i.e, dist(S j , S j ) ≥ p for any j = j .

(5.18) Hence, by (5.15)-(5.16), to show (5.14), it suces to prove that for each

i = 1, ..., 2 N , |T (n, i)| = O(ψ n ) a.s. (5.19)
Without loss of generality we will prove (5.19) for i = 1. Note that T (n, 1) is a sum of r = q 1 × ... × q n of the U (1, n, j)'s. Enumerate the r.v.'s U (1, n, j) and the corresponding sets of sites S j in an arbitrary manner and refer to them respectively as Z 1 , Z 2 , ..., Z r and S 1 , S 2 , ..., S r . Approximate Z 1 , Z 2 , ..., Z r by the r.v.'s Z * 1 , Z * 2 , ..., Z * r as was done in Lemma 4.5 of [START_REF] Carbon | Kernel density estimation for random elds[END_REF]. Clearly, since U (1, n, j) is a sum of p N random variables ∆ i , then for each k = 1, ..., r,

|Z k | ≤ Cp N (nb d n ) -1 := M n , (5.20)
for some generic constant C > 0. (5.32)

We can easily verify that nβ n g(n) → 0 as n → ∞ according to (3.3). Choosing η large enough to ensure that Cη 2 > 1, we get n∈(N * ) N P T (n, 1) ≥ n < ∞.

Finally, by Borel-Cantelli theorem, (5.19) is concluded and the proof is completed.

Proof of Corollary 3.1. By Taylor's formula in the neighborhood of x, we have As a consequence, we get sup

V (δ) c
Ef n ≤ sup

V (δ-abn) c f,
with a > 0 is the constant dened in H6. Now, since b n → 0 as n → ∞, we have for n large enough, sup

V (δ) c
Ef n ≤ sup

V (δ/2) c f.
Consequently, by Lemma 2.2, we have for any δ > 0, lim sup

n sup V (δ) c
Ef n < f (θ).

Since by Theorem 3.1, f n -Ef n ∞ -→ 0 a.s., it implies

lim sup n sup V (δ) c f n < f (θ).
Therefore, lim sup n max Sn∩V (δ) c f n < f (θ). By Corollary 3.1 (see also [START_REF] Carbon | Kernel density estimation for random elds[END_REF], Theorem 3.2), we have

f n -f V -→ 0 a.s.
Applying Proposition 2.1, we have a.s., for all δ ≤ δ 0 , lim sup

n max Sn∩V (δ) c f n < lim n max Sn∩V (δ) f n .
Finally, since δ is as small as desired, the last inequality shows that θ n -→ θ a.s.

Proof of Theorem 3.3. The proof is inspired from that of ((Abraham et al., 2003), Proposition 2). Set for n ∈ I n , k n = ab n + 8 β Lψ β n . For n large enough, to ensure that V (k n ) ⊂ V , we have

P (|θ n -θ| ≥ k n ) ≤ P max Sn∩V (kn) (f n ) ≤ max Sn∩V (kn) c (f n ) ≤ P -f n -Ef n ∞ + max Sn∩V (kn) (f n ) ≤ max Sn∩V (kn) c (f n ) + f n -Ef n ∞ = P max Sn∩V (kn) (f n ) ≤ max Sn∩V (kn) c (f n ) + 2 f n -Ef n ∞ ≤ P max Sn∩V (kn) (f ) ≤ sup V (kn) c (f n ) + 2 f n -Ef n ∞ + Ef n -f V
As a consequence, to prove (5.34), by (5.35), it suces to show that for n large enough,

P max

Sn∩V (kn)

(f ) ≤ f (θ) -ψ n = 0.

(5.38)

Let S * n = {X * 1 , ..., X * n } be a set of independent and identically distributed copies of S n obtained in the same manner as in the proof of Proposition 2.1. We can easily verify that

A(ψ n ) ⊂ V (k n ).
Hence,

P f (θ) -max S * n ∩V (ψn) f ≥ ψ n = P n i=1 ((f (X * i ) ≤ f (θ) -ψ n , X * i ∈ V (ψ n )) ∪ (X * i ∈ V (ψ n ) c ) ≤ n i=1 P (((f (X * i ) ≤ f (θ) -ψ n , X * i ∈ V (ψ n )) ∪ (X * i ∈ V (k n ) c )) = [1 -P {X ∈ A(ψ n ) ∩ V (k n )}] n = [1 -P {X ∈ A(ψ n )}] n .
(5.39)

Without loss of generality, suppose that 0 is small enough so that (5.40)

As was done in the proof of Proposition 2.1, if we assume that (5.38) is not true, then we will arrive to a contradiction with (5.40) and the proof is completed.

Figure 1 :

 1 Figure 1: Nine typical kernel estimates of the density f (x) corresponding to three degrees of dependence with ζ = 2, 5, 8.

Figure 2 :

 2 Figure 2: The average trajectories of the direct and indirect estimates of the mode with degree of dependence corresponding to ζ = 8.

Figure 3 :

 3 Figure 3: The average trajectories of θn corresponding to three degrees of dependence with ζ = 2, 5, 8.

Figure 4 :

 4 Figure4: The average trajectories of θn corresponding to too small (red), optimal (black) and too large (green) values of bn.

Figure 5 :

 5 Figure 5: Locations (in red color) of θn on the grid I (n 1 ,n 2 ) for n 1 = n 2 = 17, ..., 25.

Figure 6 :Figure 7 :

 67 Figure 6: Density of the mixed normal distribution and its kernel estimate for ζ = 8.

  and covariance function c( t ) = 2 t -ζ for ζ = 8. The target density f (x, y) is estimated by the kernel estimate (4.3) and the optimal smoothing parameters b 1n and b 2n are approximated by (4.4). We obtain b 1n = 0.608 and b 2n = 0.575. Figure 8 displays the image plot and the perspective plot of the density estimate. Colors on the image plot denote probability density at each point. The heights of the density estimate are represented by dierent colors varying from high (yellow region near to the mode) to low (purple region). The kernel estimate of the density is maximized on the simulated

Figure 8 :

 8 Figure 8: Bivariate normal distribution: image plot of density estimate (left) and perspective plot of the density estimate (right).

Figure 10 :

 10 Figure 10: Bivariate normal mixture distribution: image plot of density estimate (left) and perspective plot of the density estimate (right).

  |f n (x) -f n (y j )| = o(ψ n ) a.s.(5.4) since by H4, for each j = 1, ..., ν n , sup x∈B(yj ,hn)

  max 1≤j≤νn |f n (y j ) -Ef n (y j )| = O(ψ n ) a.s.(5.6) Let S n (y) = f n (y) -Ef n (y).

  is equivalent to show that max 1≤j≤νn |S n (y j )| = O(ψ n ) a.s.

  20) together with Lemma 4.5 of[START_REF] Carbon | Kernel density estimation for random elds[END_REF],r i=1 E|Z i -Z * i | ≤ 2rM n h((r -1)p N , p N )ϕ(p).(5.22) By (2.3), (5.20) and (5.22), Markov's inequality yieldsP r i=1 |Z i -Z * i | ≥ n ≤ CrM n nξ ϕ(p)25)-(5.26), (5.28)-(5.29), we havevarZ 1 ≤ Cp N (n 2 b d n ) -1 .(5.30)According to (5.24) with taking into account (5.20), (5.30) and that n = ηψ n , if we set p ∼ ψ ≥ n ) ≤ 2 exp -Cη 2 log n .(5.31) By (5.21), (5.23) and (5.31), we have P T (n, 1) ≥ n ≤ 2 exp -Cη 2 log n + β n .

  f jk (x -τ b n z)z i z j K(z)dz, with τ = τ (f, x, b n , z) ∈]0, 1[. Consequently, since f jk , for j, k = 1, ..., d, are bounded by assumption, by H5, we get the following inequalitysup x∈R d |Ef n (x) -f (x)| ≤ Cb 2 n where C > 0 is a generic constant. Since b 2 n = o(ψ n ), then, sup x∈R d |Ef n (x) -f (x)| = o(ψ n ).(5.33) By (5.33) together with Theorem 3.1, the proof is completed. . Proof of Theorem 3.2. We have, for any x ∈ R d and n ∈ (N * ) N , Ef n (x) = R d K(u)f (x -b n u)du.

  > 0 and ∃l > 0 : λ(A( )) ≥ l β , ∀ ≤ 0 , with λ denotes the Lebesgue measure. Thus, by (5.39),P f (θ) -max S * n ∩V (ψn) f ≥ ψ n ≤ 1 -lψ β n infCantelli Lemma, we have for n large enoughP f (θ) -max S * n ∩V (kn) (f ) ≥ ψ n = 0.

  * 2 , X * 3 and have the same distribution as X * 1 . Now, we get back to the original numbering method and we denote S * n = {X * i , i ∈ I n }. The set S *

	n
	consists of i.i.d. copies of S n with common density function f . Since (2.1) is
	fullled, Lemma 2.1, yields, a.s., for any δ > 0,
	max S * n ∩V (δ)

SIMULATION STUDIES

Acknowledgments

The author would like to thank the anonymous referees whose valuable comments led to an improved version of the paper. Abraham, C., Biau, G., Cadre, B., 2003. Simple estimation of the mode of a multivariate density. The Canadian Journal of Statistics 31, 2334. Abraham, C., Biau, G., Cadre, B., 2004. On the asymptotic properties of a simple estimate of the mode. ESAIM: PS 8, 111.

Simple mode of a density Furthermore, by Bernstein's inequality, we have

(5.25)

We will nd an upper bound for each term in the right-hand side of (5.25). By Markov's inequality,

(5.26)

In the other hand, using Markov's inequality and H3, we have for any i = j,

. By (5.27), since

(5.29)

Theorem 3.1 and (5.33) imply, for n large enough,

Hence, to establish the proof, it remains to prove that, for n large enough,

(5.34)

According to H6, we can write for all

Choosing n large enough, we may assume that 8ψ n ≤ 0 , with 0 > 0 is the constant dened in (3.4), that is

Furthermore, θ ∈ A(2(f (θ) -f (t))). Hence, we have

The last ineqality is due to (3.4). As a consequence,

(5.37)

In the other hand, suppose that t meets the condition f (θ) -f (t) > 0 /2. Obviously, the inequality (5.37) is still valid in this case due to (5.36). Then, sup