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Register a new userSemiparametric estimation of mutual information and related criteria : optimal test of independence
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Added by
Amor Keziou
Publication date
2015
fields
Mathematical Statistics
and research's language is
English
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We derive independence tests by means of dependence measures thresholding in a semiparametric context. Precisely, estimates of phi-mutual informations, associated to phi-divergences between a joint distribution and the product distribution of its margins, are derived through the dual representation of phi-divergences. The asymptotic properties of the proposed estimates are established, including consistency, asymptotic distributions and large deviations principle. The obtained tests of independence are compared via their relative asymptotic Bahadur efficiency and numerical simulations. It follows that the proposed semiparametric Kullback-Leibler Mutual information test is the optimal one. On the other hand, the proposed approach provides a new method for estimating the Kullback-Leibler mutual information in a semiparametric setting, as well as a model selection procedure in large class of dependency models including semiparametric copulas.
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In this paper we study the problem of semiparametric estimation for a class of McKean-Vlasov stochastic differential equations. Our aim is to estimate the drift coefficient of a MV-SDE based on observations of the corresponding particle system. We propose a semiparametric estimation procedure and derive the rates of convergence for the resulting estimator. We further prove that the obtained rates are essentially optimal in the minimax sense.
Mutual information is a well-known tool to measure the mutual dependence between variables. In this paper, a Bayesian nonparametric estimation of mutual information is established by means of the Dirichlet process and the $k$-nearest neighbor distance. As a direct outcome of the estimation, an easy-to-implement test of independence is introduced through the relative belief ratio. Several theoretical properties of the approach are presented. The procedure is investigated through various examples where the results are compared to its frequentist counterpart and demonstrate a good performance.
We consider the problem of conditional independence testing of $X$ and $Y$ given $Z$ where $X,Y$ and $Z$ are three real random variables and $Z$ is continuous. We focus on two main cases - when $X$ and $Y$ are both discrete, and when $X$ and $Y$ are both continuous. In view of recent results on conditional independence testing (Shah and Peters, 2018), one cannot hope to design non-trivial tests, which control the type I error for all absolutely continuous conditionally independent distributions, while still ensuring power against interesting alternatives. Consequently, we identify various, natural smoothness assumptions on the conditional distributions of $X,Y|Z=z$ as $z$ varies in the support of $Z$, and study the hardness of conditional independence testing under these smoothness assumptions. We derive matching lower and upper bounds on the critical radius of separation between the null and alternative hypotheses in the total variation metric. The tests we consider are easily implementable and rely on binning the support of the continuous variable $Z$. To complement these results, we provide a new proof of the hardness result of Shah and Peters.
Let ${bf R}$ be the Pearson correlation matrix of $m$ normal random variables. The Raos score test for the independence hypothesis $H_0 : {bf R} = {bf I}_m$, where ${bf I}_m$ is the identity matrix of dimension $m$, was first considered by Schott (2005) in the high dimensional setting. In this paper, we study the asymptotic minimax power function of this test, under an asymptotic regime in which both $m$ and the sample size $n$ tend to infinity with the ratio $m/n$ upper bounded by a constant. In particular, our result implies that the Raos score test is rate-optimal for detecting the dependency signal $|{bf R} - {bf I}_m|_F$ of order $sqrt{m/n}$, where $|cdot|_F$ is the matrix Frobenius norm.
We introduce a new test procedure of independence in the framework of parametric copulas with unknown marginals. The method is based essentially on the dual representation of $chi^2$-divergence on signed finite measures. The asymptotic properties of the proposed estimate and the test statistic are studied under the null and alternative hypotheses, with simple and standard limit distributions both when the parameter is an interior point or not.
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