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 $\sigma$.     

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.     

Accuracy of regularized D-rule for binary classification

We consider a regularized D-classification rule for high dimensional binary classification, which adapts the linear shrinkage estimator of a covariance matrix as an alternative to the sample covariance matrix in the D-classification rule (D-rule in short). We find an asymptotic expression for misclassification rate of the regularized D-rule, when the sample size $n$ and the dimension $p$ both increase and their ratio $p∕n$ approaches a positive constant $\gamma$. In addition, we compare its misclassification rate to the standard D-rule under various settings via simulation.     

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.     

A bivariate distribution with Lomax and geometric margins

We develop a stochastic model describing the joint distribution of $(X,N)$, where $N$ has a geometric distribution while $X$ is the sum of $N$ dependent, heavy-tail Pareto components. Models of this form arise in many applications, ranging from hydro-climatology to finance and insurance. We present fundamental properties of this vector, including marginal and conditional distributions, moments, representations, and parameter estimation. We also include an example from finance, illustrating modeling potential of this new bivariate distribution.     

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 $1-P^1$.