We propose a structure of a semiparametric two-component mixture model when one component is parametric and the other is defined through linear constraints on its distribution function. Estimation of a two-component mixture model with an unknown component is very difficult when no particular assumption is made on the structure of the unknown component. A symmetry assumption was used in the literature to simplify the estimation. Such method has the advantage of producing consistent and asymptotically normal estimators, and identifiability of the semiparametric mixture model becomes tractable. Still, existing methods which estimate a semiparametric mixture model have their limits when the parametric component has unknown parameters or the proportion of the parametric part is either very high or very low. We propose in this paper a method to incorporate a prior linear information about the distribution of the unknown component in order to better estimate the model when existing estimation methods fail. The new method is based on ϕ−divergences and has an original form since the minimization is carried over both arguments of the divergence. The resulting estimators are proved to be consistent and asymptotically normal under standard assumptions. We show that using the Pearson's χ 2 divergence our algorithm has a linear complexity when the constraints are moment-type. Simulations on univariate and multivariate mixtures demonstrate the viability and the interest of our novel approach.
We consider an M/M/1 queueing system with impatient customers with multiple and single vacations. It is assumed that customers are impatient whenever the state of the server. We derive the probability generating functions of the number of customers in the system and we obtain some performance measures.
In this paper we model and study a general vacation queueing model with impatient customers. We first propose a sufficient condition for the existence of the stationary workload process. We then give an integral equation for the independent and identically distributed case. This integral equation is solved when customers arrive according to a Poisson point process. A relationship between the tail of the waiting time distribution and the tail of service distribution is also given.
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