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.     

Asymptotic normality and mean consistency for the weighted estimator in nonparametric regression models

In this paper, we mainly study the asymptotic properties of weighted estimator for the nonparametric regression model based on linearly negative quadrant dependent (LNQD, for short) errors. We obtain the rate of uniformly asymptotic normality of the weighted estimator which is nearly $O\left(n^{?1∕4}\right)$ when the moment condition is appropriate. The results generalize the corresponding ones of Yang (2003) from NA samples to LNQD samples and improve or extend the corresponding one of Li et al. (2012) for LNQD samples. Moreover, we obtain some results on mean consistency, uniformly mean consistency, and the rate of mean consistency for the weighted estimator. Finally we carry out some simulations to verify the validity of our results.     

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.     

On the rate of convergence of uniform approximations for sequences of distribution functions

In this paper, we develop uniform bounds for the sequence of distribution functions of $g(V_n+{\mu }_n)$, where $g$ is some smooth function, $\{V_n,n\ge 1\}$ is a sequence of identically distributed random variables with common distribution having a bounded derivative and $\left\{{\mu }_n\right\}$ are constants such that ${\mu }_n\to \infty$. These bounds allow us to identify a suitable sequence of random variables which is asymptotically of the same type of $g(V_n+{\mu }_n)$ showing that the rate of convergence for these uniform approximations depends on the ratio of the second derivative to the first derivative of $g$. The corresponding generalization to the multivariate case is also analyzed. An application of our results to the STATIS-ACT method is provided in the final section.     

A new goodness-of-fit test for certain bivariate distributions applicable to traffic accidents

T. Cacoullos and H. Papageorgiou [On some bivariate probability models applicable to traffic accidents and fatalities, Int. Stat. Rev. 48 (1980) 345–356] studied a special class of bivariate discrete distributions appropriate for modeling traffic accidents, and fatalities resulting therefrom. The corresponding random variable may be written as $Z={(N,Y)}^{\prime }$, with $Y={\sum }_{j=1}^N{X}_j$, where ${\left\{X_j\right\}}_{j=1}^N$, are independent copies of a (discrete) random variable $X$, and $N$ is independent of ${\left\{X_j\right\}}_{j=1}^N$, and follows a Poisson law. If $X$ follows a Poisson law (resp. Binomial law), the resulting distribution is termed Poisson–Poisson (resp. Poisson–Binomial). L2-type goodness-of-fit statistics are constructed for the ‘general distribution’ of this kind, where $X$ may be an arbitrary discrete nonnegative random variable. The test statistics utilize a simple characterization involving the corresponding probability generating function, and are shown to be consistent. The proposed procedures are shown to perform satisfactorily in simulated data, while their application to accident data leads to positive conclusions regarding the modeling ability of this class of bivariate distributions.     

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.     

Simultaneous selection of treatments better and worse than the best and worst controls

In this paper, we consider $p(p\ge 2)$ and $q(q\ge 2)$ independent treatment and control populations respectively, such that an appropriate probability model for the data from $i\mathrm{th}\left(j\mathrm{th}\right)$ treatment (control) population is a member of absolutely continuous location and scale family of distributions which have common scale parameter and possibly differ in location parameters. For example, there may be $p$ newly invented drugs/varieties of seeds/components which have to compete with their existing $q$ standard competitors in terms of their average responses. A newly invented drug/variety of seed/component is said to be good (bad) if the distance of its average response from the largest (smallest) average response of $q$ control populations is more (less) than ${\delta }_1\left({\delta }_2\right)$ units, where ${\delta }_1$ and ${\delta }_2$ are positive constants to be specified by the experimenter. In this setting a selection procedure is proposed to select simultaneously two subsets $S_U$ and $S_L$ of the $p$ treatment populations such that the subset $S_U$ contains all the good treatments and the subset $S_L$ contains all the bad treatments with probability at least $P^1$, where $P^1$ is a pre-assigned value. The proposed procedure was applied to normal and two parameters exponential probability models separately and the relevant selection constants have been tabulated. The implementation of the proposed methodology is demonstrated through a numerical example based on real life data. The authenticity of numerically computed critical constants have been verified through simulation. Further, if we define the $i\mathrm{th}$ treatment population as bad (good) if the distance of its average response from the largest (smallest) average response of $q$ control populations is less (more) than ${\delta }_3\left({\delta }_4\right)$ units, where ${\delta }_3$ and ${\delta }_4$ are to be specified by the experimenter such that ${\delta }_4>{\delta }_3>0$, then we have proposed a simultaneous selection procedure to select $S_U$ and $S_L$ and a sample size is determined so that the probability of omitting a good (bad) treatment population from $S_U\left(S_L\right)$ or selecting a bad (good) treatment population in $S_U\left(S_L\right)$ is at most $1-P^1$.     

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.