Laguerre and Hermite bases for inverse problems

We present inverse problems of nonparametric statistics which have a smart solution using projection estimators on bases of functions with non compact support, namely, a Laguerre basis or a Hermite basis. The models are $Y_i=X_i{U}_i,Z_i=X_i+{\Sigma }_i,$ where the $X_i$’s are i.i.d. with unknown density $f$, the ${\Sigma }_i$’s are i.i.d. with known density $f_{\Sigma }$, the $U_i$’s are i.i.d. with uniform density on $[0,1]$. The sequences $\left(X_i\right),\left(U_i\right),\left({\Sigma }_i\right)$ are independent. We define projection estimators of $f$ in the two cases of indirect observations of $(X_1,\dots ,X_n)$, and we give upper bounds for their ${\mathbb{L}}^2$-risks on specific Sobolev–Laguerre or Sobolev–Hermite spaces. Data-driven procedures are described and proved to perform automatically the bias–variance compromise.     

Hypothesis testing via a penalized-likelihood approach

It is illustrated in this paper that hypothesis testing procedures can be derived based on the penalized likelihood approach. Based on this point of view, many traditional hypothesis tests, including the two-sample mean test, score test, and Hotelling’s $T^2$ test are revisited under the penalized likelihood framework. Similar framework is also applicable to the empirical likelihood.     

A method for high-dimensional smoothing

We consider the problem of the computation of smoothed additive functionals, which are some integrals with respect to the joint smoothing distribution. It is a key issue in inference for general state-space models as these quantities appear naturally for maximum likelihood parameter inference. The computation of smoothed additive functionals is very challenging as exact computations are not possible for non-linear non-Gaussian state-space models. It becomes even more difficult when the hidden state lies in a high dimensional space because traditional numerical methods suffer from the curse of dimensionality. We propose a new algorithm to efficiently calculate the smoothed additive functionals in an online manner for a specific family of high-dimensional state-space models in discrete time, which is named the Space–Time Forward Smoothing (STFS) algorithm. The cost of this algorithm is at least $O\left(N^2{d}^{2}T\right)$, which is polynomial in $d$. $T$ and $N$ denote the number of time steps and the number of particles respectively, while $d$ is the dimension of the hidden state space. Its superior performance over other existing methods is illustrated by various simulation studies. Moreover, STFS algorithm is successfully applied to perform Maximum Likelihood estimation for static model parameters both in an online and an offline manner.     

Optimal Berry–Esseen bound for parameter estimation of SPDE with small noise

We investigate a rate of convergence on asymptotic normality of the maximum likelihood estimator (MLE) for parameter $\theta$ appearing in parabolic SPDEs of the form

$d{u}^{?}(t,x)=(A_0+\theta A_1)u^{?}(t,x)dt+?dW(t,x),$

where $A_0$ and$A_1$ are partial differential operators, $W$ is a cylindrical Brownian motion (CBM) and $?\downarrow 0$. We find an optimal Berry–Esseen bound for central limit theorem (CLT) of the MLE. It is proved by developing techniques based on combining Malliavin calculus and Stein’s method.     

Asymptotic normality of the deconvolution kernel density estimator under the vanishing error variance

Let $X_1,\dots ,X_n$ be i.i.d. observations, where $X_i=Y_i+{\sigma }_n{Z}_i$ and the $Y$’s and $Z$’s are independent. Assume that the $Y$’s are unobservable and that they have the density $f$ and also that the $Z$’s have a known density $k$. Furthermore, let ${\sigma }_n$ depend on $n$ and let ${\sigma }_n\to 0$ as $n\to \infty$. We consider the deconvolution problem, i.e. the problem of estimation of the density $f$ based on the sample $X_1,\dots ,X_n$. A popular estimator of $f$ in this setting is the deconvolution kernel density estimator. We derive its asymptotic normality under two different assumptions on the relation between the sequence ${\sigma }_n$ and the sequence of bandwidths $h_n$. We also consider several simulation examples which illustrate different types of asymptotics corresponding to the derived theoretical results and which show that there exist situations where models with ${\sigma }_n\to 0$ have to be preferred to the models with fixed $\sigma$.     

Weighted sums of associated variables

We study the convergence of weighted sums of associated random variables. The convergence for the typical $n^{1/p}$ normalization is proved assuming finiteness of moments somewhat larger than $p$, but still smaller than 2, together with suitable control on the covariance structure described by a truncation that generates covariances that do not grow too quickly. We also consider normalizations of the form $n^{1/q}{log}^{1/\gamma }n$, where $q$ is now linked with the properties of the weighting sequence. We prove the convergence under a moment assumption than is weaker that the usual existence of the moment-generating function. Our results extend analogous characterizations known for sums of independent or negatively dependent random variables.     

Consistency for the LS estimator in the linear EV regression model with replicate observations

In this paper, we consider the following linear errors-in-variables regression model: ${\xi }_{ij}=x_i+{\delta }_{ij},{\eta }_{ij}=y_i+{\epsilon }_{ij}=\theta +\beta x_i+{\epsilon }_{ij}$, with independent identically distributed errors $({\epsilon }_{ij},{\delta }_{ij}),(j=1,2,\dots ,n_i;i=1,2,\dots )$. The strong and weak consistency for the LS estimators $\stackrel{?}{\beta }$ and $\stackrel{?}{\theta }$ of the unknown parameters $\beta ,\theta$ in this model are obtained, which weaken some known conditions and improve some known results.