Advances and Applications in Statistics
Volume 17, Issue 1, Pages 61 - 77
(July 2010)
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ASYMPTOTIC NORMAL STATISTICS FOR POISSON AND COMPOUND POISSON DISTRIBUTIONS
John J. Hsieh
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Abstract: The stationary independent increments and infinite divisibility property of compound Poisson processes make them useful in modeling real life problems. This article establishes the conditions for the weak convergence to normality of compound Poisson distributions and derives for use as approximate test statistics three types of simple asymptotic unit normal statistics (SAUNS), using linear, square root and log transformations. These SAUNS are obtained for the general compound Poisson distribution and five specific infinitely divisible discrete distributions (see Table 1). Assessments of accuracy based on asymptotic expansion and detailed numerical comparisons indicate that the statistics for compound Poisson distributions have better normal approximations than those of the Poisson distribution and that the SAUNS derived from square root transformation perform better than those derived from linear and log transformations. The SAUNS derived from square root transformation for compound Poisson distributions, especially for negative binomial distribution, represent the best compromise between simplicity and accuracy. |
Keywords and phrases: infinite divisibility, compound Poisson processes, transformation, asymptotic tests, Cornish-Fisher expansion, stochastic expansion. |
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