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Register a new userFunctional estimation in log-concave location families
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Added by
Vladimir Koltchinskii
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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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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We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form $f_0=expvarphi_0$ 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.
The log-concave projection is an operator that maps a d-dimensional distribution P to an approximating log-concave density. Prior work by D{u}mbgen et al. (2011) establishes that, with suitable metrics on the underlying spaces, this projection is continuous, but not uniformly continuous. In this work we prove a local uniform continuity result for log-concave projection -- in particular, establishing that this map is locally H{o}lder-(1/4) continuous. A matching lower bound verifies that this exponent cannot be improved. We also examine the implications of this continuity result for the empirical setting -- given a sample drawn from a distribution P, we bound the squared Hellinger distance between the log-concave projection of the empirical distribution of the sample, and the log-concave projection of P. In particular, this yields interesting statistical results for the misspecified setting, where P is not itself log-concave.
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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.
We are interested in estimating the location of what we call smooth change-point from $n$ independent observations of an inhomogeneous Poisson process. The smooth change-point is a transition of the intensity function of the process from one level to another which happens smoothly, but over such a small interval, that its length $delta_n$ is considered to be decreasing to $0$ as $nto+infty$. We show that if $delta_n$ goes to zero slower than $1/n$, our model is locally asymptotically normal (with a rather unusual rate $sqrt{delta_n/n}$), and the maximum likelihood and Bayesian estimators are consistent, asymptotically normal and asymptotically efficient. If, on the contrary, $delta_n$ goes to zero faster than $1/n$, our model is non-regular and behaves like a change-point model. More precisely, in this case we show that the Bayesian estimators are consistent, converge at rate $1/n$, have non-Gaussian limit distributions and are asymptotically efficient. All these results are obtained using the likelihood ratio analysis method of Ibragimov and Khasminskii, which equally yields the convergence of polynomial moments of the considered estimators. However, in order to study the maximum likelihood estimator in the case where $delta_n$ goes to zero faster than $1/n$, this method cannot be applied using the usual topologies of convergence in functional spaces. So, this study should go through the use of an alternative topology and will be considered in a future work.
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