Showing papers by "Palaniappan Vellaisamy published in 2013"
TL;DR: In this paper, negative binomial approximation to sums of independent Z +$-valued random variables using Stein's method is employed to obtain the error bounds Convolution of negative Binomial and Poisson distribution is used as a three-parametric approximation.
Abstract: This paper deals with negative binomial approximation to sums of independent ${\bf Z}_+$-valued random variables Stein's method is employed to obtain the error bounds Convolution of negative binomial and Poisson distribution is used as a three-parametric approximation
22 citations
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TL;DR: In this paper, the authors defined a fractional negative binomial process (FNBP) by replacing the Poisson process by a FPP in the gamma subordinated form of the negative Binomial process.
Abstract: In this paper, we define a fractional negative binomial process (FNBP) by replacing the Poisson process by a fractional Poisson process (FPP) in the gamma subordinated form of the negative binomial process. First, it is shown that the one-dimensional distributions of the FPP are not infinitely divisible. The long-range dependence of the FNBP, the short-range dependence of its increments and the infinite divisibility of the FPP and the FNBP are investigated. Also, the space fractional Polya process (SFPP) is defined by replacing the rate parameter $\lambda$ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to $pde$'$s$ governing the density of the FNBP and the SFPP are also investigated.
16 citations
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TL;DR: In this paper, the sum of m-dependent integer-valued random variables are approximated by compound Poisson, negative binomial and binomial distributions and signed compound poisson measures.
Abstract: Sums of m-dependent integer-valued random variables are approxi- mated by compound Poisson, negative binomial and binomial distributions and signed compound Poisson measures. Estimates are obtained for the total variation metric. The results are then applied to statistics of m-dependent (k1,k2) events and 2-runs. Heinrich's method and smoothing properties of convolutions are used for the proofs. 1. The setup
10 citations
TL;DR: This work considers compound negative binomial and compound Poisson approximations to the generalized Poisson–binomial distribution and derives some total variation upper bounds which improve on the existing results in terms of the order of approximation.
9 citations
TL;DR: In this paper, sufficient conditions for obtaining improvements over scale equivariant estimators for the one-parametric model are derived, and estimators that improve upon the maximum likelihood estimator (MLE), an analog of the uniformly minimum variance unbiased estimators (UMVUE), and the best-scale estimators are also obtained and are shown to be admissible.
Abstract: The estimation of the reliability of a series system of k(≥ 2) independent components, where the life time of each component is exponentially distributed, has been considered. First, sufficient conditions for obtaining improvements over scale equivariant estimators for the one-parametric model are derived. As a consequence, we derive estimators that improve upon the maximum likelihood estimator (MLE), an analog of the uniformly minimum variance unbiased estimator (UMVUE) and the best-scale equivariant estimator (BSEE). Bayes and generalized Bayes estimators are also obtained and are shown to be admissible. We consider also the case where the lifetimes follow two-parametric exponential distributions and derive the UMVUE of the system reliability. Further, the MLE and the modified MLE (MMLE) are discussed for this case. Finally, the risks of these estimators are compared numerically for the case k = 2.
2 citations