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The FANOVA (or "Sobol'-Hoeffding") decomposition of multivariate functions has been used for high-dimensional model representation and global sensitivity analysis. When the objective function f has no simple analytic form and is costly to evaluate, a practical limitation is that computing FANOVA terms may be unaffordable due to numerical integration costs. Several approximate approaches relying on random field models have been proposed to alleviate these costs, where f is substituted by a (kriging) predictor or by conditional simulations. In the present work, we focus on FANOVA decompositions of Gaussian random field sample paths, and we notably introduce an associated kernel decomposition (into 2^{2d} terms) called KANOVA. An interpretation in terms of tensor product projections is obtained, and it is shown that projected kernels control both the sparsity of Gaussian random field sample paths and the dependence structure between FANOVA effects. Applications on simulated data show the relevance of the approach for designing new classes of covariance kernels dedicated to high-dimensional kriging.

International audience; In this article we consider Lévy driven continuous time moving average processes observed on a lattice, which are stationary time series. We show asymptotic normality of the sample mean, the sample autocovariances and the sample autocorrelations. A comparison with the classical setting of discrete moving average time series shows that in the last case a correction term should be added to the classical Bartlett formula that yields the asymptotic variance. An application to the asymptotic normality of the estimator of the Hurst exponent of fractional Lévy processes is also deduced from these results.

It is common practice to treat small jumps of L\'evy processes as Wiener noise and thus to approximate its marginals by a Gaussian distribution. However, results that allow to quantify the goodness of this approximation according to a given metric are rare. In this paper, we clarify what happens when the chosen metric is the total variation distance. Such a choice is motivated by its statistical interpretation. If the total variation distance between two statistical models converges to zero, then no tests can be constructed to distinguish the two models which are therefore equivalent, statistically speaking. We elaborate a fine analysis of a Gaussian approximation for the small jumps of L\'evy processes with infinite L\'evy measure in total variation distance. Non asymptotic bounds for the total variation distance between $n$ discrete observations of small jumps of a L\'evy process and the corresponding Gaussian distribution is presented and extensively discussed. As a byproduct, new upper bounds for the total variation distance between discrete observations of L\'evy processes are provided. The theory is illustrated by concrete examples., Comment: Important and necessary changes have been made in this new version, this version supersedes version 1

We revisit conservative/dissipative and positive/null decompositions of stationary max-stable processes. Originally, both decompositions were defined in an abstract way based on the underlying non-singular flow representation. We provide simple criteria which allow to tell whether a given spectral function belongs to the conservative/dissipative or positive/null part of the de Haan spectral representation. Specifically, we prove that a spectral function is null-recurrent iff it converges to $0$ in the Ces\`{a}ro sense. For processes with locally bounded sample paths we show that a spectral function is dissipative iff it converges to $0$. Surprisingly, for such processes a spectral function is integrable a.s. iff it converges to $0$ a.s. Based on these results, we provide new criteria for ergodicity, mixing, and existence of a mixed moving maximum representation of a stationary max-stable process in terms of its spectral functions. In particular, we study a decomposition of max-stable processes which characterizes the mixing property., Comment: 21 pages, no figures

We study the monotone single index model where a real response variable $Y $ is linked to a $d$-dimensional covariate $X$ through the relationship $E[Y | X] = \Psi_0(\alpha^T_0 X)$ almost surely. Both the ridge function, $\Psi_0$, and the index parameter, $\alpha_0$, are unknown and the ridge function is assumed to be monotone on its interval of support. Under some regularity conditions, without imposing a particular distribution on the regression error, we show the $n^{-1/3}$ rate of convergence in the $\ell_2$-norm for the least squares estimator of the bundled function $\psi_0({\alpha}^T_0 \cdot),$ and also that of the ridge function and the index separately. Furthermore, we show that the least squares estimator is nearly parametrically rate-adaptive to piecewise constant ridge functions., Comment: 54 pages

31 pages, 7 figures; International audience; In this paper, the extinction problem for a class of distylous plant populations is considered within the framework of certain nonhomogeneous nearest-neighbor random walks in the positive quadrant. For the latter, extinction means absorption at one of the axes. Despite connections with some classical probabilistic models (standard two-type Galton-Watson process, two-urn model), exact formulae for the probabilities of absorption seem to be difficult to come by and one must therefore resort to good approximations. In order to meet this task, we develop potential-theoretic tools and provide various sub- and super-harmonic functions which, for large initial populations, provide bounds which in particular improve those that have appeared earlier in the literature.

We consider composite-composite testing problems for the expectation in the Gaussian sequence model where the null hypothesis corresponds to a convex subset $\mathcal{C}$ of $\mathbb{R}^d$. We adopt a minimax point of view and our primary objective is to describe the smallest Euclidean distance between the null and alternative hypotheses such that there is a test with small total error probability. In particular, we focus on the dependence of this distance on the dimension $d$ and the sample size/variance parameter $n$ giving rise to the minimax separation rate. In this paper we discuss lower and upper bounds on this rate for different smooth and non- smooth choices for $\mathcal{C}$.

In this paper, we propose a model-free bootstrap method for the empirical process under absolute regularity. More precisely, consistency of an adapted version of the so-called dependent wild bootstrap, which was introduced by Shao () and is very easy to implement, is proved under minimal conditions on the tuning parameter of the procedure. We show how our results can be applied to construct confidence intervals for unknown parameters and to approximate critical values for statistical tests. In a simulation study, we investigate the size properties of a bootstrap-aided Kolmogorov-Smirnov test and show that our method is competitive to standard block bootstrap methods in finite samples.

We propose a new testing procedure for detecting localized departures from monotonicity of a signal embedded in white noise. In fact, we perform simultaneously several tests that aim at detecting departures from concavity for the integrated signal over various intervals of different sizes and localizations. Each of these local tests relies on estimating the distance between the restriction of the integrated signal to some interval and its least concave majorant. Our test can be easily implemented and is proved to achieve the optimal uniform separation rate simultaneously for a wide range of H\"{o}lderian alternatives. Moreover, we show how this test can be extended to a Gaussian regression framework with unknown variance. A simulation study confirms the good performance of our procedure in practice., Comment: Published in at http://dx.doi.org/10.3150/12-BEJ496 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

We generalize a theorem of Shao [Proc. Amer. Math. Soc. 123 (1995) 575–582] on the almost-sure limiting behavior of the maximum of standardized random walk increments to multidimensional arrays of i.i.d. random variables. The main difficulty is the absence of an appropriate strong approximation result in the multidimensional setting. The multiscale statistic under consideration was used recently for the selection of the regularization parameter in a number of statistical algorithms as well as for the multiscale signal detection. peerReviewed

We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form $f_0=\exp\varphi_0$ where $\varphi_0$ is a concave function on $\mathbb{R}$. 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 $\varphi_0=\log f_0$ 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., Comment: Published in at http://dx.doi.org/10.1214/08-AOS609 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We prove a second Marshall inequality for adaptive convex density estimation via least squares. The result completes the first inequality proved recently by Dumbgen et al. [2007. Marshall's lemma for convex density estimation. IMS Lecture Notes—Monograph Series, submitted for publication. Preprint available at 〈 http://arxiv.org/abs/math.ST/0609277 〉 ], and is very similar to the original Marshall inequality in monotone estimation.

We study the asymptotic behavior of the Maximum Likelihood and Least Squares estimators of a $k-$monotone density $g_0$ at a fixed point $x_0$ when $k > 2$. In \mycite{balabwell:04a}, it was proved that both estimators exist and are splines of degree $k-1$ with simple knots. These knots, which are also the jump points of the $(k-1)-$st derivative of the estimators, cluster around a point $x_0 > 0$ under the assumption that $g_0$ has a continuous $k$-th derivative in a neighborhood of $x_0$ and $(-1)^k g^{(k)}_0(x_0) > 0$. If $\tau^{-}_n$ and $\tau^{+}_n$ are two successive knots, we prove that the random ``gap'' \ $\tau^{+}_n - \tau^{-}_n $ is $O_p(n^{-1/(2k+1)})$ for any $k > 2$ if a conjecture about the upper bound on the error in a particular Hermite interpolation via odd-degree splines holds. Based on the order of the gap, the asymptotic distribution of the Maximum Likelihood and Least Squares estimators can be established. We find that the $j-$th derivative of the estimators at $x_0$ converges at the rate $n^{-(k-j)/(2k+1)}$ for $j=0, \ldots, k-1$. The limiting distribution depends on an almost surely uniquely defined stochastic process $H_k$ that stays above (below) the $k$-fold integral of Brownian motion plus a deterministic drift, when $k$ is even (odd).

Motivated by Hampel’s birds migration problem, Groeneboom, Jongbloed, and Wellner [7] established the asymptotic distribution theory for the nonparametric Least Squares and Maximum Likelihood estimators of a convex and decreasing density, g 0, at a fixed point t 0 > 0. However, estimation of the distribution function of the birds’ resting times involves estimation of g′0 at 0, a boundary point at which the estimators are not consistent. In this paper, we focus on the Least Squares estimator, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaGafm4zaCMbaGaadaWgaaWcbaGaemOBa4gabeaaaaa!3DC2! $$\tilde g_n $$ . Our goal is to show that consistent estimators of both g 0(0) and g′0(0) can be based solely on % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaGafm4zaCMbaGaadaWgaaWcbaGaemOBa4gabeaaaaa!3DC2! $$\tilde g_n $$ . Following the idea of Kulikov and Lopuhaa [14] in monotone estimation, we show that it suffices to take % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaGafm4zaCMbaGaadaWgaaWcbaGaemOBa4gabeaaaaa!3DC2! $$\tilde g_n $$ (n −α ) and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaGafm4zaCMbaGGbauaadaWgaaWcbaGaemOBa4gabeaa% aaa!3DCD! $$\tilde g'_n $$ (n −α ), with α ∈ (0, 1/3). We establish their joint asymptotic distributions and show that α = 1/5 should be taken as it yields the fastest rates of convergence.

Probabilistic forecasts of a continuous variable take the form of predictive densities or predictive cumulative distribution functions. We propose a diagnostic approach to the evaluation of predictive performance that is based on the paradigm of {\sl maximizing the sharpness of the predictive distributions subject to calibration}. Calibration refers to the statistical consistency between the distributional forecasts and the observations and is a joint property of the predictions and the events that materialize. Sharpness refers to the concentration of the predictive distributions and is a property of the forecasts only. A simple theoretical framework phrased in terms of a game between nature and forecaster allows us to distinguish probabilistic calibration, exceedance calibration and marginal calibration. We propose and study tools for checking calibration and sharpness, among them the probability integral transform (PIT) histogram, marginal calibration plots, the sharpness diagram and proper scoring rules. The diagnostic approach is illustrated by an assessment and ranking of probabilistic forecasts of wind speed at the Stateline wind energy center in the U.S.~Pacific Northwest. In combination with cross-validation or in the time series context, our proposal provides very general, nonparametric alternatives to the use of information criteria for model diagnostics and model selection.

In this dissertation, we consider the problem of nonparametric estimation of a k-monotone density on (0,∞) for a fixed integer k > 0 via the methods of Maximum Likelihood (ML) and Least Squares (LS). In the introduction, we present the original question that motivated us to look into this problem and also put other existing results in our general framework. In Chapter 2, we study the MLE and LSE of a k-monotone density based on n i.i.d. observations. Here, our study of the estimation problem is local in the sense that we only study the estimator and its derivatives at a fixed point x_0. Under some specific working assumptions, asymptotic minimax lower bounds for estimating g^{(j)}_0 (x_0), j=1,...,k-1, are derived. These bounds show that the rates of convergence of any estimator of g^{(j)}_0 (x_0) can be at most n^{-(k-j)/(2k+1)}. Furthermore, under the same working assumptions we prove that this rate is achieved by the j-th derivative of either the MLE or LSE if a certain conjecture concerning the error in a particular Hermite interpolation problem holds. To make the asymptotic distribution theory complete, the limiting distribution needs to be determined. This distribution depends on a very special stochastic process H_k which is almost surely uniquely defined on R. Chapter 3 is essentially devoted to an effort to prove the existence of such a process and to establish conditions characterizing it. It turns out that we can establish the existence and uniqueness of the process Hk if the same conjecture mentioned above with the finite sample problem holds. If Y_k is the (k-1) -fold integral of two-sided Brownian motion + k!/(2k)! t2k, then H_k is a random spline of degree 2k-1 that stays above Y_k if k is even and below it if k is odd. By applying a change of scale, our results include the special cases of estimation of monotone densities (k =1), and monotone and convex densities (k =2) for which an asymptotic distribution theory is available.Iterative spline algorithms developed to calculate the estimators and approximate the process H_k on finite intervals are described in Chapter 4. These algorithms exploit both the spline structure of the estimators and the process H_k as well as their characterizations and are based on iterative addition and deletion of the knot points.; Nous considérons l'estimation non-paramétrique d'une densité k-monotone définie sur (0,∞), pour un entier k > 0 donné, via les méthodes de maximum de vraisemblance et des moindres carrés qu'on note respectivement par MLE et LSE.Dans l'introduction, nous présentons tout d'abord la motivation principale derrière ce problème et nous faisons l'effort d'inclure dans le cadre général de notre travail les résultats asymptotiques qui étaient déjà établis pour les cas spéciaux k=1 et k=2. Ensuite, nous nous penchons sur l'étude des propriétés des MLE et LSE d'une densité k-monotone g_0 dans le cas où on dispose de n observations indépendantes générées de g_0. Notre étude asymptotique est locale, c'est-à-dire que nous nous intéressons uniquement aux propriétés asymptotiques des estimateurs et de leur dérivées à un point fixe, x_0. Sous certaines hypothèses que nous précisons, nous établissons d'abord les bornes inférieures minimax pour l'estimation des dérivées g^{(j)}_0(x_0), j=0,...,k-1. Les bornes obtenues indiquent que n^{-(k-j)/(2k+1)} est la vitesse de convergence optimale de n'importe quel estimateur non-paramétrique de g^{(j)}_0(x_0). Sous les mêmes hypothèses et si une certaine conjecture est vraie, nous démontrons que cette vitesse optimale est atteinte dans le cas des MLE et LSE.Pour compléter la théorie asymptotique des estimateurs et de leur dérivées au point x_0, nous passons à la dérivation de leurs distributions limites lorsque la taille de l'échantillon n tend vers l'infini. Il s'avère que ces distributions dépendent d'un processus stochastique bien particulier défini sur l'ensemble des réels R. On note ce processus par H_k Le 3ème chapitre est consacré essentiellement à l'existence et à l'unicité de H_k, ainsi qu'à sa caractérisation. Nous démontrons que si Y_k est la primitive (k-1)-ème d'un mouvement Brownien + k!/(2k)! t^{2k}, alors H_k reste au-dessus (au-dessous) de Y_k lorsque k est pair (impair). Un simple changement de variable suffit pour reconnaître que nos résultats comprennent les cas spéciaux k=1 et k=2 où le problème se réduit à l'estimation d'une densité décroissante et d'une densité décroissante et convexe respectivement. Pour ces cas-là, la théorie asymptotique des MLE et LES a été déjà établie.L'aspect algorithmique fait l'objet du 4ème chapitre. Les algorithmes de Splines itératifs (Iterative Spline algorithms) sont développés et implémentés afin de calculer les estimateurs et aussi pour obtenir une approximation du processus limite sur n'importe quel compact dans R. Ces algorithmes exploitent essentiellement la structure 'splineuse' des MLE, LSE et H_k, et se basent ainsi sur la suppression et l'addition itératives des noeuds de certains Splines aléatoires.

In this dissertation, we consider the problem of nonparametric estimation of a k-monotone density on (0,∞) for a fixed integer k > 0 via the methods of Maximum Likelihood (ML) and Least Squares (LS). In the introduction, we present the original question that motivated us to look into this problem and also put other existing results in our general framework. In Chapter 2, we study the MLE and LSE of a k-monotone density based on n i.i.d. observations. Here, our study of the estimation problem is local in the sense that we only study the estimator and its derivatives at a fixed point x_0. Under some specific working assumptions, asymptotic minimax lower bounds for estimating g^{(j)}_0 (x_0), j=1,...,k-1, are derived. These bounds show that the rates of convergence of any estimator of g^{(j)}_0 (x_0) can be at most n^{-(k-j)/(2k+1)}. Furthermore, under the same working assumptions we prove that this rate is achieved by the j-th derivative of either the MLE or LSE if a certain conjecture concerning the error in a particular Hermite interpolation problem holds. To make the asymptotic distribution theory complete, the limiting distribution needs to be determined. This distribution depends on a very special stochastic process H_k which is almost surely uniquely defined on R. Chapter 3 is essentially devoted to an effort to prove the existence of such a process and to establish conditions characterizing it. It turns out that we can establish the existence and uniqueness of the process Hk if the same conjecture mentioned above with the finite sample problem holds. If Y_k is the (k-1) -fold integral of two-sided Brownian motion + k!/(2k)! t2k, then H_k is a random spline of degree 2k-1 that stays above Y_k if k is even and below it if k is odd. By applying a change of scale, our results include the special cases of estimation of monotone densities (k =1), and monotone and convex densities (k =2) for which an asymptotic distribution theory is available.Iterative spline algorithms developed to calculate the estimators and approximate the process H_k on finite intervals are described in Chapter 4. These algorithms exploit both the spline structure of the estimators and the process H_k as well as their characterizations and are based on iterative addition and deletion of the knot points.; Nous considérons l'estimation non-paramétrique d'une densité k-monotone définie sur (0,∞), pour un entier k > 0 donné, via les méthodes de maximum de vraisemblance et des moindres carrés qu'on note respectivement par MLE et LSE.Dans l'introduction, nous présentons tout d'abord la motivation principale derrière ce problème et nous faisons l'effort d'inclure dans le cadre général de notre travail les résultats asymptotiques qui étaient déjà établis pour les cas spéciaux k=1 et k=2. Ensuite, nous nous penchons sur l'étude des propriétés des MLE et LSE d'une densité k-monotone g_0 dans le cas où on dispose de n observations indépendantes générées de g_0. Notre étude asymptotique est locale, c'est-à-dire que nous nous intéressons uniquement aux propriétés asymptotiques des estimateurs et de leur dérivées à un point fixe, x_0. Sous certaines hypothèses que nous précisons, nous établissons d'abord les bornes inférieures minimax pour l'estimation des dérivées g^{(j)}_0(x_0), j=0,...,k-1. Les bornes obtenues indiquent que n^{-(k-j)/(2k+1)} est la vitesse de convergence optimale de n'importe quel estimateur non-paramétrique de g^{(j)}_0(x_0). Sous les mêmes hypothèses et si une certaine conjecture est vraie, nous démontrons que cette vitesse optimale est atteinte dans le cas des MLE et LSE.Pour compléter la théorie asymptotique des estimateurs et de leur dérivées au point x_0, nous passons à la dérivation de leurs distributions limites lorsque la taille de l'échantillon n tend vers l'infini. Il s'avère que ces distributions dépendent d'un processus stochastique bien particulier défini sur l'ensemble des réels R. On note ce processus par H_k Le 3ème chapitre est consacré essentiellement à l'existence et à l'unicité de H_k, ainsi qu'à sa caractérisation. Nous démontrons que si Y_k est la primitive (k-1)-ème d'un mouvement Brownien + k!/(2k)! t^{2k}, alors H_k reste au-dessus (au-dessous) de Y_k lorsque k est pair (impair). Un simple changement de variable suffit pour reconnaître que nos résultats comprennent les cas spéciaux k=1 et k=2 où le problème se réduit à l'estimation d'une densité décroissante et d'une densité décroissante et convexe respectivement. Pour ces cas-là, la théorie asymptotique des MLE et LES a été déjà établie.L'aspect algorithmique fait l'objet du 4ème chapitre. Les algorithmes de Splines itératifs (Iterative Spline algorithms) sont développés et implémentés afin de calculer les estimateurs et aussi pour obtenir une approximation du processus limite sur n'importe quel compact dans R. Ces algorithmes exploitent essentiellement la structure 'splineuse' des MLE, LSE et H_k, et se basent ainsi sur la suppression et l'addition itératives des noeuds de certains Splines aléatoires.