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Register a new userAdaptive greedy algorithm for moderately large dimensions in kernel conditional density estimation
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Added by
Vincent Rivoirard
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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This paper studies the estimation of the conditional density f (x, $times$) of Y i given X i = x, from the observation of an i.i.d. sample (X i , Y i) $in$ R d , i = 1,. .. , n. We assume that f depends only on r unknown components with typically r d. We provide an adaptive fully-nonparametric strategy based on kernel rules to estimate f. To select the bandwidth of our kernel rule, we propose a new fast iterative algorithm inspired by the Rodeo algorithm (Wasserman and Lafferty (2006)) to detect the sparsity structure of f. More precisely, in the minimax setting, our pointwise estimator, which is adaptive to both the regularity and the sparsity, achieves the quasi-optimal rate of convergence. Its computational complexity is only O(dn log n).
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In this paper we consider the problem of estimating $f$, the conditional density of $Y$ given $X$, by using an independent sample distributed as $(X,Y)$ in the multivariate setting. We consider the estimation of $f(x,.)$ where $x$ is a fixed point. We define two different procedures of estimation, the first one using kernel rules, the second one inspired from projection methods. Both adapted estimators are tuned by using the Goldenshluger and Lepski methodology. After deriving lower bounds, we show that these procedures satisfy oracle inequalities and are optimal from the minimax point of view on anisotropic H{o}lder balls. Furthermore, our results allow us to measure precisely the influence of $mathrm{f}_X(x)$ on rates of convergence, where $mathrm{f}_X$ is the density of $X$. Finally, some simulations illustrate the good behavior of our tuned estimates in practice.
We present a new adaptive kernel density estimator based on linear diffusion processes. The proposed estimator builds on existing ideas for adaptive smoothing by incorporating information from a pilot density estimate. In addition, we propose a new plug-in bandwidth selection method that is free from the arbitrary normal reference rules used by existing methods. We present simulation examples in which the proposed approach outperforms existing methods in terms of accuracy and reliability.
We study the estimation, in Lp-norm, of density functions defined on [0,1]^d. We construct a new family of kernel density estimators that do not suffer from the so-called boundary bias problem and we propose a data-driven procedure based on the Goldenshluger and Lepski approach that jointly selects a kernel and a bandwidth. We derive two estimators that satisfy oracle-type inequalities. They are also proved to be adaptive over a scale of anisotropic or isotropic Sobolev-Slobodetskii classes (which are particular cases of Besov or Sobolev classical classes). The main interest of the isotropic procedure is to obtain adaptive results without any restriction on the smoothness parameter.
This paper studies the minimax rate of nonparametric conditional density estimation under a weighted absolute value loss function in a multivariate setting. We first demonstrate that conditional density estimation is impossible if one only requires that $p_{X|Z}$ is smooth in $x$ for all values of $z$. This motivates us to consider a sub-class of absolutely continuous distributions, restricting the conditional density $p_{X|Z}(x|z)$ to not only be Holder smooth in $x$, but also be total variation smooth in $z$. We propose a corresponding kernel-based estimator and prove that it achieves the minimax rate. We give some simple examples of densities satisfying our assumptions which imply that our results are not vacuous. Finally, we propose an estimator which achieves the minimax optimal rate adaptively, i.e., without the need to know the smoothness parameter values in advance. Crucially, both of our estimators (the adaptive and non-adaptive ones) impose no assumptions on the marginal density $p_Z$, and are not obtained as a ratio between two kernel smoothing estimators which may sound like a go to approach in this problem.
This article is concerned with the spectral behavior of $p$-dimensional linear processes in the moderately high-dimensional case when both dimensionality $p$ and sample size $n$ tend to infinity so that $p/nto0$. It is shown that, under an appropriate set of assumptions, the empirical spectral distributions of the renormalized and symmetrized sample autocovariance matrices converge almost surely to a nonrandom limit distribution supported on the real line. The key assumption is that the linear process is driven by a sequence of $p$-dimensional real or complex random vectors with i.i.d. entries possessing zero mean, unit variance and finite fourth moments, and that the $ptimes p$ linear process coefficient matrices are Hermitian and simultaneously diagonalizable. Several relaxations of these assumptions are discussed. The results put forth in this paper can help facilitate inference on model parameters, model diagnostics and prediction of future values of the linear process.
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