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Register a new userOn maxima of periodograms of stationary processes
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We consider the limit distribution of maxima of periodograms for stationary processes. Our method is based on $m$-dependent approximation for stationary processes and a moderate deviation result.
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Suppose that particles are randomly distributed in $bR^d$, and they are subject to identical stochastic motion independently of each other. The Smoluchowski process describes fluctuations of the number of particles in an observation region over time. This paper studies properties of the Smoluchowski processes and considers related statistical problems. In the first part of the paper we revisit probabilistic properties of the Smoluchowski process in a unified and principled way: explicit formulas for generating functionals and moments are derived, conditions for stationarity and Gaussian approximation are discussed, and relations to other stochastic models are highlighted. The second part deals with statistics of the Smoluchowki processes. We consider two different models of the particle displacement process: the undeviated uniform motion (when a particle moves with random constant velocity along a straight line) and the Brownian motion displacement. In the setting of the undeviated uniform motion we study the problems of estimating the mean speed and the speed distribution, while for the Brownian displacement model the problem of estimating the diffusion coefficient is considered. In all these settings we develop estimators with provable accuracy guarantees.
The theory for multiplier empirical processes has been one of the central topics in the development of the classical theory of empirical processes, due to its wide applicability to various statistical problems. In this paper, we develop theory and tools for studying multiplier $U$-processes, a natural higher-order generalization of the multiplier empirical processes. To this end, we develop a multiplier inequality that quantifies the moduli of continuity of the multiplier $U$-process in terms of that of the (decoupled) symmetrized $U$-process. The new inequality finds a variety of applications including (i) multiplier and bootstrap central limit theorems for $U$-processes, (ii) general theory for bootstrap $M$-estimators based on $U$-statistics, and (iii) theory for $M$-estimation under general complex sampling designs, again based on $U$-statistics.
Bayesian nonparametric hierarchical priors are highly effective in providing flexible models for latent data structures exhibiting sharing of information between and across groups. Most prominent is the Hierarchical Dirichlet Process (HDP), and its subsequent variants, which model latent clustering between and across groups. The HDP, may be viewed as a more flexible extension of Latent Dirichlet Allocation models (LDA), and has been applied to, for example, topic modelling, natural language processing, and datasets arising in health-care. We focus on analogous latent feature allocation models, where the data structures correspond to multisets or unbounded sparse matrices. The fundamental development in this regard is the Hierarchical Indian Buffet process (HIBP), which utilizes a hierarchy of Beta processes over J groups, where each group generates binary random matrices, reflecting within group sharing of features, according to beta-Bernoulli IBP priors. To encompass HI
Dependence measures based on reproducing kernel Hilbert spaces, also known as Hilbert-Schmidt Independence Criterion and denoted HSIC, are widely used to statistically decide whether or not two random vectors are dependent. Recently, non-parametric HSIC-based statistical tests of independence have been performed. However, these tests lead to the question of the choice of the kernels associated to the HSIC. In particular, there is as yet no method to objectively select specific kernels with theoretical guarantees in terms of first and second kind errors. One of the main contributions of this work is to develop a new HSIC-based aggregated procedure which avoids such a kernel choice, and to provide theoretical guarantees for this procedure. To achieve this, we first introduce non-asymptotic single tests based on Gaussian kernels with a given bandwidth, which are of prescribed level $alpha in (0,1)$. From a theoretical point of view, we upper-bound their uniform separation rate of testing over Sobolev and Nikolskii balls. Then, we aggregate several single tests, and obtain similar upper-bounds for the uniform separation rate of the aggregated procedure over the same regularity spaces. Another main contribution is that we provide a lower-bound for the non-asymptotic minimax separation rate of testing over Sobolev balls, and deduce that the aggregated procedure is adaptive in the minimax sense over such regularity spaces. Finally, from a practical point of view, we perform numerical studies in order to assess the efficiency of our aggregated procedure and compare it to existing independence tests in the literature.
We obtain the optimal proxy variance for the sub-Gaussianity of Beta distribution, thus proving upper bounds recently conjectured by Elder (2016). We provide different proof techniques for the symmetrical (around its mean) case and the non-symmetrical case. The technique in the latter case relies on studying the ordinary differential equation satisfied by the Beta moment-generating function known as the confluent hypergeometric function. As a consequence, we derive the optimal proxy variance for the Dirichlet distribution, which is apparently a novel result. We also provide a new proof of the optimal proxy variance for the Bernoulli distribution, and discuss in this context the proxy variance relation to log-Sobolev inequalities and transport inequalities.
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