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Register a new userAdaptive pointwise estimation of conditional density function
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Added by
Vincent Rivoirard
Publication date
2013
fields
Mathematical Statistics
and research's language is
English
Authors
Karine Bertin
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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.
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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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We address the problem of adaptive minimax density estimation on $bR^d$ with $bL_p$--loss on the anisotropic Nikolskii classes. We fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm indexes in definitions of the functional class and of the risk. In particular, we show that there are four different regimes with respect to the behavior of the minimax risk. We develop a single estimator which is (nearly) optimal in orderover the complete scale of the anisotropic Nikolskii classes. Our estimation procedure is based on a data-driven selection of an estimator from a fixed family of kernel estimators.
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