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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 . 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 {k1,k2,,kn,}.  相似文献   

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Let X1,,Xn be i.i.d. observations, where Xi=Yi+σnZi 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 σn depend on n and let σn0 as n. We consider the deconvolution problem, i.e. the problem of estimation of the density f based on the sample X1,,Xn. 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 σn and the sequence of bandwidths hn. 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 σn0 have to be preferred to the models with fixed σ.  相似文献   

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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 d, in the form of the times-to-failure for n1,,nk items from suppliers 1,,k, respectively is available. There are two suggested selection criteria studied in this paper and when comparing only two suppliers they reduce toP(θ1<bθ2|d)andP(Y1>cY2|d),where b and c are prespecified practical constants; θ1 and θ2 are the respective mean failure rates; Y1 and Y2 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.  相似文献   

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In this paper, we develop uniform bounds for the sequence of distribution functions of g(Vn+μn), where g is some smooth function, {Vn,n1} is a sequence of identically distributed random variables with common distribution having a bounded derivative and {μn} are constants such that μn. These bounds allow us to identify a suitable sequence of random variables which is asymptotically of the same type of g(Vn+μn) 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.  相似文献   

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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 pn approaches a positive constant γ. In addition, we compare its misclassification rate to the standard D-rule under various settings via simulation.  相似文献   

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We study the convergence of weighted sums of associated random variables. The convergence for the typical n1/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 n1/qlog1/γn, 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.  相似文献   

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