Showing papers in "Esaim: Probability and Statistics in 2013"

Local asymptotic normality for normal inverse Gaussian Lévy processes with high-frequency sampling∗

Abstract: We prove the local asymptotic normality for the full parameters of the normal inverse Gaussian Levy process X, when we observe high-frequency data XΔn,X2Δn,...,XnΔn with sampling mesh Δn → 0 and the terminal sampling time nΔn → ∞. The rate of convergence turns out to be (√nΔn, √nΔn, √n, √n) for the dominating parameter (α,β,δ,μ), where α stands for the heaviness of the tails, β the degree of skewness, δ the scale, and μ the location. The essential feature in our study is that the suitably normalized increments of X in small time is approximately Cauchy-distributed, which specifically comes out in the form of the asymptotic Fisher information matrix.

How the result of graph clustering methods depends on the construction of the graph

Abstract: We study the scenario of graph-based clustering algorithms such as spectral clustering. Given a set of data points, one first has to construct a graph on the data points and then apply a graph clustering algorithm to find a suitable partition of the graph. Our main question is if and how the construction of the graph (choice of the graph, choice of parameters, choice of weights) influences the outcome of the final clustering result. To this end we study the convergence of cluster quality measures such as the normalized cut or the Cheeger cut on various kinds of random geometric graphs as the sample size tends to infinity. It turns out that the limit values of the same objective function are systematically different on different types of graphs. This implies that clustering results systematically depend on the graph and can be very different for different types of graph. We provide examples to illustrate the implications on spectral clustering.

Mixing conditions for multivariate infinitely divisible processes with an application to mixed moving averages and the supOU stochastic volatility model

Abstract: We consider strictly stationary infinitely divisible processes and first extend the mixing conditions given in Maruyama [Theory Probab. Appl. 15 (1970) 1–22] and Rosinski and Żak [Stoc. Proc. Appl. 61 (1996) 277–288] from the univariate to the d -dimensional case. Thereafter, we show that multivariate Levy-driven mixed moving average processes satisfy these conditions and hence a wide range of well-known processes such as superpositions of Ornstein − Uhlenbeck (supOU) processes or (fractionally integrated) continuous time autoregressive moving average (CARMA) processes are always mixing. Finally, mixing of the log-returns and the integrated volatility process of a multivariate supOU type stochastic volatility model, recently introduced in Barndorff − Nielsen and Stelzer [Math. Finance 23 (2013) 275–296], is established.

A central limit theorem for triangular arrays of weakly dependent random variables, with applications in statistics

Abstract: We derive a central limit theorem for triangular arrays of possibly nonstationary random variables satisfying a condition of weak dependence in the sense of Doukhan and Louhichi [Stoch. Proc. Appl. 84 (1999) 313–342]. The proof uses a new variant of the Lindeberg method: the behavior of the partial sums is compared to that of partial sums of dependent Gaussian random variables. We also discuss a few applications in statistics which show that our central limit theorem is tailor-made for statistics of different type.

Carthaginian enlargement of filtrations

Abstract: This work is concerned with the theory of initial and progressive enlargements of a reference filtration with a random time τ . We provide, under an equivalence assumption, slightly stronger than the absolute continuity assumption of Jacod, alternative proofs to results concerning canonical decomposition of an -martingale in the enlarged filtrations. Also, we address martingales’ characterization in the enlarged filtrations in terms of martingales in the reference filtration, as well as predictable representation theorems in the enlarged filtrations.

A sharp analysis on the asymptotic behavior of the Durbin-Watson statistic for the first-order autoregressive process

Abstract: The purpose of this paper is to provide a sharp analysis on the asymptotic behavior of the Durbin-Watson statistic. We focus our attention on the first-order autoregressive process where the driven noise is also given by a first-order autoregressive process. We establish the almost sure convergence and the asymptotic normality for both the least squares estimator of the unknown parameter of the autoregressive process as well as for the serial correlation estimator associated to the driven noise. In addition, the almost sure rates of convergence of our estimates are also provided. It allows us to establish the almost sure convergence and the asymptotic normality for the Durbin-Watson statistic. Finally, we propose a new bilateral statistical test for residual autocorrelation.

Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory

Abstract: This paper deals with the problem of estimating the level sets of an unknown distribution function $F$. A plug-in approach is followed. That is, given a consistent estimator $F_n$ of $F$, we estimate the level sets of $F$ by the level sets of $F_n$. In our setting no compactness property is a priori required for the level sets to estimate. We state consistency results with respect to the Hausdorff distance and the volume of the symmetric difference. Our results are motivated by applications in multivariate risk theory. In this sense we also present simulated and real examples which illustrate our theoretical results.

Penalization versus Goldenshluger-Lepski strategies in warped bases regression

Abstract: This paper deals with the problem of estimating a regression function f , in a random design framework We build and study two adaptive estimators based on model selection, applied with warped bases We start with a collection of finite dimensional linear spaces, spanned by orthonormal bases Instead of expanding directly the target function f on these bases, we rather consider the expansion of h = f ∘ G -1 , where G is the cumulative distribution function of the design, following Kerkyacharian and Picard [Bernoulli 10 (2004) 1053–1105] The data-driven selection of the (best) space is done with two strategies: we use both a penalization version of a “warped contrast”, and a model selection device in the spirit of Goldenshluger and Lepski [Ann Stat 39 (2011) 1608–1632] We propose by these methods two functions, ĥ l (l = 1, 2), easier to compute than least-squares estimators We establish nonasymptotic mean-squared integrated risk bounds for the resulting estimators, if G is known, or (l = 1,2) otherwise, where Ĝ is the empirical distribution function We study also adaptive properties, in case the regression function belongs to a Besov or Sobolev space, and compare the theoretical and practical performances of the two selection rules

Wavelet analysis of the multivariate fractional Brownian motion

Abstract: The work developed in the paper concerns the multivariate fractional Brownian motion (mfBm) viewed through the lens of the wavelet transform. After recalling some basic properties on the mfBm, we calculate the correlation structure of its wavelet transform. We particularly study the asymptotic behavior of the correlation, showing that if the analyzing wavelet has a sufficient number of null first order moments, the decomposition eliminates any possible long-range (inter)dependence. The cross-spectral density is also considered in a second part. Its existence is proved and its evaluation is performed using a von Bahr-Essen like representation of the function $\sign(t) |t|^\alpha$. The behavior of the cross-spectral density of the wavelet field at the zero frequency is also developed and confirms the results provided by the asymptotic analysis of the correlation.

Necessary and sufficient condition for the existence of a Fréchet mean on the circle

Abstract: Let ( \hbox{}S1,dS1 ) be the unit circle in ℝ2 endowed with the arclength distance. We give a sufficient and necessary condition for a general probability measure μ to admit a well defined Frechet mean on ( \hbox{}S1,dS1 ). We derive a new sufficient condition of existence P(α, ϕ) with no restriction on the support of the measure. Then, we study the convergence of the empirical Frechet mean to the Frechet mean and we give an algorithm to compute it.

Incremental moments and Hölder exponents of multifractional multistable processes

Abstract: Multistable processes, that is, processes which are, at each “time”, tangent to a stable process, but where the index of stability varies along the path, have been recently introduced as models for phenomena where the intensity of jumps is non constant. In this work, we give further results on (multifractional) multistable processes related to their local structure. We show that, under certain conditions, the incremental moments display a scaling behaviour, and that the pointwise Holder exponent is, as expected, related to the local stability index. We compute the precise value of the almost sure Holder exponent in the case of the multistable Levy motion, which turns out to reveal an interesting phenomenon.

Adaptive density estimation for clustering with gaussian mixtures

Abstract: Gaussian mixture models are widely used to study clustering problems. These model-based clustering methods require an accurate estimation of the unknown data density by Gaussian mixtures. In Maugis and Michel (2009), a penalized maximum likelihood estimator is proposed for automatically selecting the number of mixture components. In the present paper, a collection of univariate densities whose logarithm is locally β -Holder with moment and tail conditions are considered. We show that this penalized estimator is minimax adaptive to the β regularity of such densities in the Hellinger sense.

Partition-based conditional density estimation

Abstract: We propose a general partition-based strategy to estimate conditional density with candidate densities that are piecewise constant with respect to the covariate. Capitalizing on a general penalized maximum likelihood model selection result, we prove, on two specific examples, that the penalty of each model can be chosen roughly proportional to its dimension. We first study a classical strategy in which the densities are chosen piecewise conditional according to the variable. We then consider Gaussian mixture models with mixing proportion that vary according to the covariate but with common mixture components. This model proves to be interesting for an unsupervised segmentation application that was our original motivation for this work.

Testing randomness of spatial point patterns with the Ripley statistic

Abstract: Aggregation patterns are often visually detected in sets of location data. These clusters may be the result of interesting dynamics or the effect of pure randomness. We build an asymptotically Gaussian test for the hypothesis of randomness corresponding to a homogeneous Poisson point process. We first compute the exact first and second moment of the Ripley K-statistic under the homogeneous Poisson point process model. Then we prove the asymptotic normality of a vector of such statistics for different scales and compute its covariance matrix. From these results, we derive a test statistic that is chi-square distributed. By a Monte-Carlo study, we check that the test is numerically tractable even for large data sets and also correct when only a hundred of points are observed

An l1-oracle inequality for the Lasso in finite mixture Gaussian regression models

Abstract: We consider a finite mixture of Gaussian regression models for high-dimensional heterogeneous data where the number of covariates may be much larger than the sample size. We propose to estimate the unknown conditional mixture density by an l 1 -penalized maximum likelihood estimator. We shall provide an l 1 -oracle inequality satisfied by this Lasso estimator with the Kullback–Leibler loss. In particular, we give a condition on the regularization parameter of the Lasso to obtain such an oracle inequality. Our aim is twofold: to extend the l 1 -oracle inequality established by Massart and Meynet [12] in the homogeneous Gaussian linear regression case, and to present a complementary result to Stadler et al. [18], by studying the Lasso for its l 1 -regularization properties rather than considering it as a variable selection procedure. Our oracle inequality shall be deduced from a finite mixture Gaussian regression model selection theorem for l 1 -penalized maximum likelihood conditional density estimation, which is inspired from Vapnik’s method of structural risk minimization [23] and from the theory on model selection for maximum likelihood estimators developed by Massart in [11].

Towards a universally consistent estimator of the Minkowski content

Abstract: We deal with a subject in the interplay between nonparametric statistics and geometric measure theory. The measure L 0 (G ) of the boundary of a set G ⊂ ℝd (with d ≥ 2) can be formally defined, via a simple limit, by the so-called Minkowski content. We study the estimation of L 0 (G ) from a sample of random points inside and outside G . The sample design assumes that, for each sample point, we know (without error) whether or not that point belongs to G . Under this design we suggest a simple nonparametric estimator and investigate its consistency properties. The main emphasis in this paper is on generality. So we are especially concerned with proving the consistency of our estimator under minimal assumptions on the set G . In particular, we establish a mild shape condition on G under which the proposed estimator is consistent in L 2 . Roughly speaking, such condition establishes that the set of “very spiky” points at the boundary of G must be “small”. This is formalized in terms of the Minkowski content of such set. Several examples are discussed.

Local estimation of the Hurst index of multifractional brownian motion by increment ratio statistic method

Abstract: We investigate here the central limit theorem of the increment ratio statistic of a multifractional Brownian motion, leading to a CLT for the time varying Hurst index. The proofs are quite simple relying on Breuer–Major theorems and an original freezing of time strategy. A simulation study shows the goodness of fit of this estimator.

Nonparametric estimation of the derivatives of the stationary density for stationary processes

Abstract: In this article, our aim is to estimate the successive derivatives of the stationary density f of a strictly stationary and β -mixing process (Xt )t≥0 . This process is observed at discrete times t = 0,Δ, ... ,nΔ . The sampling interval Δ can be fixed or small. We use a penalized least-square approach to compute adaptive estimators. If the derivative f (j ) belongs to the Besov space , then our estimator converges at rate (nΔ )−α /(2α +2j +1) . Then we consider a diffusion with known diffusion coefficient. We use the particular form of the stationary density to compute an adaptive estimator of its first derivative f ′. When the sampling interval Δ tends to 0, and when the diffusion coefficient is known, the convergence rate of our estimator is (nΔ )−α /(2α +1) . When the diffusion coefficient is known, we also construct a quotient estimator of the drift for low-frequency data.

On ℝd-valued peacocks

Abstract: In this paper, we consider ℝd -valued integrable processes which are increasing in the convex order, i.e. ℝd -valued peacocks in our terminology. After the presentation of some examples, we show that an ℝd -valued process is a peacock if and only if it has the same one-dimensional marginals as an ℝd -valued martingale. This extends former results, obtained notably by Strassen [Ann. Math. Stat. 36 (1965) 423–439], Doob [J. Funct. Anal. 2 (1968) 207–225] and Kellerer [Math. Ann. 198 (1972) 99–122].

Asymptotic normality of randomly truncated stochastic algorithms

Abstract: We study the convergence rate of randomly truncated stochastic algorithms, which consist in the truncation of the standard Robbins-Monro procedure on an increasing sequence of compact sets. Such a truncation is often required in practice to ensure convergence when standard algorithms fail because the expected-value function grows too fast. In this work, we give a self contained proof of a central limit theorem for this algorithm under local assumptions on the expected-value function, which are fairly easy to check in practice.

On the convergence of moments in the almost sure central limit theorem for stochastic approximation algorithms

Abstract: We study the almost sure asymptotic behaviour of stochastic approximation algorithms for the search of zero of a real function. The quadratic strong law of large numbers is extended to the powers greater than one. In other words, the convergence of moments in the almost sure central limit theorem (ASCLT) is established. As a by-product of this convergence, one gets another proof of ASCLT for stochastic approximation algorithms. The convergence result is applied to several examples as estimation of quantiles and recursive estimation of the mean.

Polynomial deviation bounds for recurrent Harris processes having general state space

Abstract: Consider a strong Markov process in continuous time, taking values in some Polish state space. Recently, Douc, Fort and Guillin (2009) introduced verifiable conditions in terms of a supermartingale property implying an explicit control of modulated moments of hitting times. We show how this control can be translated into a control of polynomial moments of abstract regeneration times which are obtained by using the coupling method of Nummelin, extended to the time-continuous context. As a consequence, if a p th moment of the regeneration times exists, then we obtain non asymptotic deviation bounds of the form

Smoothness of Metropolis-Hastings algorithm and application to entropy estimation

Abstract: The transition kernel of the well-known Metropolis-Hastings (MH) algorithm has a point mass at the chain’s current position, which prevent direct smoothness properties to be derived for the successive densities of marginals issued from this algorithm We show here that under mild smoothness assumption on the MH algorithm “input” densities (the initial, proposal and target distributions), propagation of a Lipschitz condition for the iterative densities can be proved This allows us to build a consistent nonparametric estimate of the entropy for these iterative densities This theoretical study can be viewed as a building block for a more general MCMC evaluation tool grounded on such estimates

Meeting time of independent random walks in random environment

Abstract: We consider, in the continuous time version, γ independent random walks on Z+ in random environment in Sinai’s regime. Let T γ be the first meeting time of one pair of the γ random walks starting at different positions. We first show that the tail of the quenched distribution of T γ , after a suitable rescaling, converges in probability, to some functional of the Brownian motion. Then we compute the law of this functional. Eventually, we obtain results about the moments of this meeting time. Being Eω the quenched expectation, we show that, for almost all environments ω , Eω [Tγ c ] is finite for c (γ − 1) / 2 and infinite for c > γ (γ − 1) / 2.

Semimartingale decomposition of convex functions of continuous semimartingales by Brownian perturbation

Abstract: In this note we prove that the local martingale part of a convex function f of a d -dimensional semimartingale X = M + A can be written in terms of an Ito stochastic integral ∫ H ( X )d M , where H ( x ) is some particular measurable choice of subgradient of f at x , and M is the martingale part of X . This result was first proved by Bouleau in [N. Bouleau, C. R. Acad. Sci. Paris Ser. I Math. 292 (1981) 87–90]. Here we present a new treatment of the problem. We first prove the result for , ϵ > 0, where B is a standard Brownian motion, and then pass to the limit as ϵ → 0, using results in [M.T. Barlow and P. Protter, On convergence of semimartingales. In Seminaire de Probabilites, XXIV, 1988/89 , Lect. Notes Math., vol. 1426. Springer, Berlin (1990) 188–193; E. Carlen and P. Protter, Illinois J. Math. 36 (1992) 420–427]. The former paper concerns convergence of semimartingale decompositions of semimartingales, while the latter studies a special case of converging convex functions of semimartingales.

Risk bounds for new M-estimation problems

Abstract: In this paper, we develop new algorithms for parameter estimation in the case of models type Input/Output in order to represent and to characterize a phenomenon Y. From experimental data Y_{1},...,Y_{n} supposed to be i.i.d from Y, we prove risk bounds qualifying the proposed procedures in terms of the number of experimental data n, computing budget m and model complexity. The methods we present are general enough which should cover a wide range of applications.

Adaptive hard-thresholding for linear inverse problems

Abstract: A number of regularization methods for discrete inverse problems consist in considering weighted versions of the usual least square solution. These filter methods are generally restricted to monotonic transformations, e.g. the Tikhonov regularization or the spectral cut-off. However, in several cases, non-monotonic sequences of filters may appear more appropriate. In this paper, we study a hard-thresholding regularization method that extends the spectral cut-off procedure to non-monotonic sequences. We provide several oracle inequalities, showing the method to be nearly optimal under mild assumptions. Contrary to similar methods discussed in the literature, we use here a non-linear threshold that appears to be adaptive to all degrees of irregularity, whether the problem is mildly or severely ill-posed. Finally, we extend the method to inverse problems with noisy operator and provide efficiency results in a conditional framework.

Convolution property and exponential bounds for symmetric monotone densities

Abstract: Our first theorem states that the convolution of two symmetric densities which are k -monotone on (0, ∞) is again (symmetric) k -monotone provided 0 k ≤ 1. We then apply this result, together with an extremality approach, to derive sharp moment and exponential bounds for distributions having such shape constrained densities.

Moment measures of heavy-tailed renewal point processes: asymptotics and applications

Abstract: We study higher-order moment measures of heavy-tailed renewal models, including a renewal point process with heavy-tailed inter-renewal distribution and its continuous analog, the occupation measure of a heavy-tailed Levy subordinator. Our results reveal that the asymptotic structure of such moment measures are given by explicit power-law density functions. The same power-law densities appear naturally as cumulant measures of certain Poisson and Gaussian stochastic integrals. This correspondence provides new and extended results regarding the asymptotic fluctuations of heavy-tailed sources under aggregation, and clarifies existing links between renewal models and fractional random processes.

Asymptotics of counts of small components in random structures and models of coagulation-fragmentation

Abstract: We establish necessary and sufficient conditions for the convergence (in the sense of finite dimensional distributions) of multiplicative measures on the set of partitions. The multiplicative measures depict distributions of component spectra of random structures and also the equilibria of classic models of statistical mechanics and stochastic processes of coagulation-fragmentation. We show that the convergence of multiplicative measures is equivalent to the asymptotic independence of counts of components of fixed sizes in random structures. We then apply Schur’s tauberian lemma and some results from additive number theory and enumerative combinatorics in order to derive plausible sufficient conditions of convergence. Our results demonstrate that the common belief, that counts of components of fixed sizes in random structures become independent as the number of particles goes to infinity, is not true in general.