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.     

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.     

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.     

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 $\sigma$.     

Confidence intervals for P(Y

《Statistical Methodology》2012,9(3):445-455

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.