tailieunhanh - Second order approximations in sequential point estimation of the probability of zero in Poisson distribution

In this paper, we consider sequential point estimation of the probability of zero in Poisson distribution. Second order approximations to the expected sample size and the risk of the sequential procedure are derived as the cost per observations tends to zero. Finally, a simulation study is given. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 719 – 731 ¨ ITAK ˙ c TUB doi: Second order approximations in sequential point estimation of the probability of zero in Poisson distribution Eisa MAHMOUDI,∗ Mohammad HATAMI KAMIN Department of Statistics, Yazd University, . Box 89195-741, Yazd, Iran Received: • Accepted: • Published Online: • Printed: Abstract: In the analysis of the count data, the Poisson model becomes overtly restrictive in the case of over-dispersed or under-dispersed data. When count data are under-dispersed, specific models such as generalized linear models (GLM) are proposed. Other examples are the zero-inflated Poisson model (ZIP) and zero-truncated Poisson model (ZTP), which have been used in literature to deal with an excess or absence of zeros in count data. Thus having a knowledge of the probability of zeros and its estimation in Poisson distribution can be significant and useful. Some estimation problems with unknown parameter cannot attain minimum risk where the sample size is fixed. To resolve this captivity, working with a sequential sampling procedure can be useful. In this paper, we consider sequential point estimation of the probability of zero in Poisson distribution. Second order approximations to the expected sample size and the risk of the sequential procedure are derived as the cost per observations tends to zero. Finally, a simulation study is given. Key words: Poisson distribution, regret, second-order approximations, sequential estimation 1. Introduction The Poisson distribution with the probability density function f(x; θ) = e−θ θx , x! x = 0, 1, 2, . . ., θ > 0, (1) was first studied by Poisson [13] as a limiting case of the binomial distribution. This distribution is very important among the discrete distributions. Johnson et al. [10] have discussed the genesis of Poisson distribution in