ترغب بنشر مسار تعليمي؟ اضغط هنا

Optimal cross-validation in density estimation with the $L^2$-loss

141   0   0.0 ( 0 )
 نشر من قبل Alain Celisse
 تاريخ النشر 2014
  مجال البحث الاحصاء الرياضي
والبحث باللغة English
 تأليف Alain Celisse




اسأل ChatGPT حول البحث

We analyze the performance of cross-validation (CV) in the density estimation framework with two purposes: (i) risk estimation and (ii) model selection. The main focus is given to the so-called leave-$p$-out CV procedure (Lpo), where $p$ denotes the cardinality of the test set. Closed-form expressions are settled for the Lpo estimator of the risk of projection estimators. These expressions provide a great improvement upon $V$-fold cross-validation in terms of variability and computational complexity. From a theoretical point of view, closed-form expressions also enable to study the Lpo performance in terms of risk estimation. The optimality of leave-one-out (Loo), that is Lpo with $p=1$, is proved among CV procedures used for risk estimation. Two model selection frameworks are also considered: estimation, as opposed to identification. For estimation with finite sample size $n$, optimality is achieved for $p$ large enough [with $p/n=o(1)$] to balance the overfitting resulting from the structure of the model collection. For identification, model selection consistency is settled for Lpo as long as $p/n$ is conveniently related to the rate of convergence of the best estimator in the collection: (i) $p/nto1$ as $nto+infty$ with a parametric rate, and (ii) $p/n=o(1)$ with some nonparametric estimators. These theoretical results are validated by simulation experiments.



قيم البحث

اقرأ أيضاً

In model selection, several types of cross-validation are commonly used and many variants have been introduced. While consistency of some of these methods has been proven, their rate of convergence to the oracle is generally still unknown. Until now, an asymptotic analysis of crossvalidation able to answer this question has been lacking. Existing results focus on the pointwise estimation of the risk of a single estimator, whereas analysing model selection requires understanding how the CV risk varies with the model. In this article, we investigate the asymptotics of the CV risk in the neighbourhood of the optimal model, for trigonometric series estimators in density estimation. Asymptotically, simple validation and incomplete V --fold CV behave like the sum of a convex function fn and a symmetrized Brownian changed in time W gn/V. We argue that this is the right asymptotic framework for studying model selection.
We investigate predictive density estimation under the $L^2$ Wasserstein loss for location families and location-scale families. We show that plug-in densities form a complete class and that the Bayesian predictive density is given by the plug-in den sity with the posterior mean of the location and scale parameters. We provide Bayesian predictive densities that dominate the best equivariant one in normal models.
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 t hat $p_{X|Z}$ 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.
Consider estimating the n by p matrix of means of an n by p matrix of independent normally distributed observations with constant variance, where the performance of an estimator is judged using a p by p matrix quadratic error loss function. A matrix version of the James-Stein estimator is proposed, depending on a tuning constant. It is shown to dominate the usual maximum likelihood estimator for some choices of of the tuning constant when n is greater than or equal to 3. This result also extends to other shrinkage estimators and settings.
We aim at estimating the invariant density associated to a stochastic differential equation with jumps in low dimension, which is for $d=1$ and $d=2$. We consider a class of jump diffusion processes whose invariant density belongs to some Holder spac e. Firstly, in dimension one, we show that the kernel density estimator achieves the convergence rate $frac{1}{T}$, which is the optimal rate in the absence of jumps. This improves the convergence rate obtained in [Amorino, Gloter (2021)], which depends on the Blumenthal-Getoor index for $d=1$ and is equal to $frac{log T}{T}$ for $d=2$. Secondly, we show that is not possible to find an estimator with faster rates of estimation. Indeed, we get some lower bounds with the same rates ${frac{1}{T},frac{log T}{T}}$ in the mono and bi-dimensional cases, respectively. Finally, we obtain the asymptotic normality of the estimator in the one-dimensional case.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا