arXiv subjects

Thi Mong Ngoc Nguyen

Publications and source records attributed to Thi Mong Ngoc Nguyen.

6 recordsLinked to original sources

arXiv2026

Adaptive Density Estimation Using Projection Kernels and Penalized Comparison to Overfitting

In this work, we study wavelet projection estimators for density estimation, based on compactly supported $\mathcal S$-regular scaling functions. The main issue is the choice of the resolution level, which determines the bias--variance trade-off. We select this level by a Penalized Comparison to Overfitting (PCO) criterion: each candidate estimator is compared in $Ł^2(\R)$ with a single overfitting reference, and the additional stochastic fluctuation is corrected by an explicit penalty. For the selected estimator, we prove a high-probability oracle inequality and a risk oracle inequality in expectation. Over Besov balls of densities satisfying a common $Ł^\infty$ bound, the risk inequality yields the rate $n^{-2r/(2r+1)}$ uniformly over the class, with a sufficient penalty threshold that can be chosen uniformly over the class. This rate has the classical minimax order, and the procedure adapts to the unknown Besov regularity. Numerical experiments on several density shapes confirm that the data-driven level is close to the oracle one and adapts to the structure of the target.

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arXiv2025

Asymptotic efficiency for Sobol' and Cram{é}r-von Mises indices under two designs of experiments

A variety of indices aim to quantify the impact of input variables on a response, typically the output from a complex computer code or black-box model. Most commonly used, the Sobol' index typically measures the influence of some inputs from an explained variance perspective. However, some situations may require a more targeted analysis of some inputs influence. With no prior information, distribution-based measures appear to be appealing. In this purpose, so-called Cram{é}r-von Mises indices (and their generalization) have been proposed in the literature, defined as an excess probability integrated over the output distribution that aim to reflect influence on the whole distribution of the output rather than on the variance solely. Inference of these various indices has remained a challenging topic especially in presence of many inputs. While several Sobol' indices estimators are known to be optimal under regularity conditions, the issue of asymptotic efficiency for Cram{é}r-von Mises indices has been unaddressed in the literature so far. For these indices, we derive in this paper the efficiency bounds and discuss the known methods to achieve such optimal bounds. Two estimation contexts are considered: the so-called Pick-Freeze scheme and the Given-Data setting, for which the estimation is produced from a unique input-output sample.

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arXiv2020

Semi-parametric estimation of the variogram of a Gaussian process with stationary increments

We consider the semi-parametric estimation of a scale parameter of a one-dimensional Gaussian process with known smoothness. We suggest an estimator based on quadratic variations and on the moment method. We provide asymptotic approximations of the mean and variance of this estimator, together with asymptotic normality results, for a large class of Gaussian processes. We allow for general mean functions and study the aggregation of several estimators based on various variation sequences. In extensive simulation studies, we show that the asymptotic results accurately depict thefinite-sample situations already for small to moderate sample sizes. We also compare various variation sequences and highlight the efficiency of the aggregation procedure.

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arXiv2018

On the Bickel-Rosenblatt test of goodness-of-fit for the residuals of autoregressive processes

We investigate in this paper a Bickel-Rosenblatt test of goodness-of-fit for the density of the noise in an autoregressive model. Since the seminal work of Bickel and Rosenblatt, it is well-known that the integrated squared error of the Parzen-Rosenblatt density estimator, once correctly renormalized, is asymptotically Gaussian for independent and identically distributed (i.i.d.) sequences. We show that the result still holds when the statistic is built from the residuals of general stable and explosive autoregressive processes. In the univariate unstable case, we prove that the result holds when the unit root is located at $-1$ whereas we give further results when the unit root is located at $1$. In particular, we establish that except for some particular asymmetric kernels leading to a non-Gaussian limiting distribution and a slower convergence, the statistic has the same order of magnitude. We also study some common unstable cases, like the integrated seasonal process. Finally we build a goodness-of-fit Bickel-Rosenblatt test for the true density of the noise together with its empirical properties on the basis of a simulation study.

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arXiv2017

Cross-validation estimation of covariance parameters under fixed-domain asymptotics

We consider a one-dimensional Gaussian process having exponential covariance function. Under fixed-domain asymptotics, we prove the strong consistency and asymptotic normality of a cross validation estimator of the microergodic covariance parameter. In this setting, Ying [40] proved the same asymptotic properties for the maximum likelihood estimator. Our proof includes several original or more involved components, compared to that of Ying. Also, while the asymptotic variance of maximum likelihood does not depend on the triangular array of observation points under consideration, that of cross validation does, and is shown to be lower and upper bounded. The lower bound coincides with the asymptotic variance of maximum likelihood. We provide examples of triangular arrays of observation points achieving the lower and upper bounds. We illustrate our asymptotic results with simulations, and provide extensions to the case of an unknown mean function. To our knowledge, this work constitutes the first fixed-domain asymptotic analysis of cross validation.

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arXiv2012

On the asymptotic behavior of the Nadaraya-Watson estimator associated with the recursive SIR method

We investigate the asymptotic behavior of the Nadaraya-Watson estimator for the estimation of the regression function in a semiparametric regression model. On the one hand, we make use of the recursive version of the sliced inverse regression method for the estimation of the unknown parameter of the model. On the other hand, we implement a recursive Nadaraya-Watson procedure for the estimation of the regression function which takes into account the previous estimation of the parameter of the semiparametric regression model. We establish the almost sure convergence as well as the asymptotic normality for our Nadaraya-Watson estimator. We also illustrate our semiparametric estimation procedure on simulated data.

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