The virtual waiting time of the M/G/1 queue with customers of n types of impatience

We consider the $M/G/1$ queue in which the customers are classified into $n+1$ classes by their impatience times. First, we analyze the model with two types of customers; one is the customer with constant impatience time $k$ and the other is the patient customer whose impatience time is $\infty$. The expected busy period of the server and the limiting distribution of the virtual waiting time process are obtained. Then, the model is generalized to the one in which the impatience time of each customer is anyone in $\{k_1,k_2,\dots ,k_n,\infty \}$.     

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.     

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.     

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 $1-P^1$.     

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 $\sigma$.