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Register a new userAffine equivariant rank-weighted L-estimation of multivariate location
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Jana Jure\\v{c}kov\\'a
Publication date
2015
fields
Mathematical Statistics
and research's language is
English
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In the multivariate one-sample location model, we propose a class of flexible robust, affine-equivariant L-estimators of location, for distributions invoking affine-invariance of Mahalanobis distances of individual observations. An involved iteration process for their computation is numerically illustrated.
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Bayesian methods are developed for the multivariate nonparametric regression problem where the domain is taken to be a compact Riemannian manifold. In terms of the latter, the underlying geometry of the manifold induces certain symmetries on the multivariate nonparametric regression function. The Bayesian approach then allows one to incorporate hierarchical Bayesian methods directly into the spectral structure, thus providing a symmetry-adaptive multivariate Bayesian function estimator. One can also diffuse away some prior information in which the limiting case is a smoothing spline on the manifold. This, together with the result that the smoothing spline solution obtains the minimax rate of convergence in the multivariate nonparametric regression problem, provides good frequentist properties for the Bayes estimators. An application to astronomy is included.
In this paper we propose a class of weighted rank correlation coefficients extending the Spearmans rho. The proposed class constructed by giving suitable weights to the distance between two sets of ranks to place more emphasis on items having low rankings than those have high rankings or vice versa. The asymptotic distribution of the proposed measures and properties of the parameters estimated by them are studied through the associated copula. A simulation study is performed to compare the performance of the proposed statistics for testing independence using asymptotic relative efficiency calculations.
This paper studies nonparametric estimation of parameters of multivariate Hawkes processes. We consider the Bayesian setting and derive posterior concentration rates. First rates are derived for L1-metrics for stochastic intensities of the Hawkes process. We then deduce rates for the L1-norm of interactions functions of the process. Our results are exemplified by using priors based on piecewise constant functions, with regular or random partitions and priors based on mixtures of Betas distributions. Numerical illustrations are then proposed with in mind applications for inferring functional connec-tivity graphs of neurons.
We consider the problem of estimating a low rank covariance function $K(t,u)$ of a Gaussian process $S(t), tin [0,1]$ based on $n$ i.i.d. copies of $S$ observed in a white noise. We suggest a new estimation procedure adapting simultaneously to the low rank structure and the smoothness of the covariance function. The new procedure is based on nuclear norm penalization and exhibits superior performances as compared to the sample covariance function by a polynomial factor in the sample size $n$. Other results include a minimax lower bound for estimation of low-rank covariance functions showing that our procedure is optimal as well as a scheme to estimate the unknown noise variance of the Gaussian process.
Let ${P_{theta}:theta in {mathbb R}^d}$ be a log-concave location family with $P_{theta}(dx)=e^{-V(x-theta)}dx,$ where $V:{mathbb R}^dmapsto {mathbb R}$ is a known convex function and let $X_1,dots, X_n$ be i.i.d. r.v. sampled from distribution $P_{theta}$ with an unknown location parameter $theta.$ The goal is to estimate the value $f(theta)$ of a smooth functional $f:{mathbb R}^dmapsto {mathbb R}$ based on observations $X_1,dots, X_n.$ In the case when $V$ is sufficiently smooth and $f$ is a functional from a ball in a Holder space $C^s,$ we develop estimators of $f(theta)$ with minimax optimal error rates measured by the $L_2({mathbb P}_{theta})$-distance as well as by more general Orlicz norm distances. Moreover, we show that if $dleq n^{alpha}$ and $s>frac{1}{1-alpha},$ then the resulting estimators are asymptotically efficient in Hajek-LeCam sense with the convergence rate $sqrt{n}.$ This generalizes earlier results on estimation of smooth functionals in Gaussian shift models. The estimators have the form $f_k(hat theta),$ where $hat theta$ is the maximum likelihood estimator and $f_k: {mathbb R}^dmapsto {mathbb R}$ (with $k$ depending on $s$) are functionals defined in terms of $f$ and designed to provide a higher order bias reduction in functional estimation problem. The method of bias reduction is based on iterative parametric bootstrap and it has been successfully used before in the case of Gaussian models.
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