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 $\{k_1,k_2,\dots ,k_n,\infty \}$.     

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 $\sigma$.     

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.     

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.     

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.     

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 $1-P^1$.     

Inferences for hierarchical models with partial prior information

Suppose items can be purchased from one of k-suppliers and it is required to purchase from the one with the smaller failure rate or equivalently from the one with the larger mean-time-to-failure. It is assumed that data $\mathit{d}$, in the form of the times-to-failure for $n_1,\dots ,n_k$ items from suppliers $1,\dots ,k$, respectively is available. There are two suggested selection criteria studied in this paper and when comparing only two suppliers they reduce to$P({\theta }_1<b{\theta }_2|d)\text{and}P(Y_1>{\mathit{cY}}_2|d)\text{,}$where b and c are prespecified practical constants; ${\theta }_1$ and ${\theta }_2$ are the respective mean failure rates; $Y_1$ and $Y_2$ are the predicted times to failure for individual items purchased from each supplier.In addition partial prior information about the $k$-suppliers collectively is assumed to have been elicited. This situation is modelled using the hierarchical Bayesian approach, which easily facilitates interpreting the elicited partial prior information as constraints on the hyperpriors, i.e. hyperpriors that are known only to be contained in families with specified properties. In this paper these properties are assumed to be in the form of specifying certain quantiles arising from the elicited information. Minimum and maximum values of the above selection criteria are obtained and are used to indicate whether or not the elicited prior information is useful. Specific examples are given for comparing two suppliers but generalisation to k-suppliers follows easily.