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- Title
Asymptotic Normality in Mixtures of Power Series Distributions.
- Authors
Böhning, Dankmar; Patilea, Valentin
- Abstract
The problem of estimating the individual probabilities of a discrete distribution is considered. The true distribution of the independent observations is a mixture of a family of power series distributions. First, we ensure identifiability of the mixing distribution assuming mild conditions. Next, the mixing distribution is estimated by non-parametric maximum likelihood and an estimator for individual probabilities is obtained from the corresponding marginal mixture density. We establish asymptotic normality for the estimator of individual probabilities by showing that, under certain conditions, the difference between this estimator and the empirical proportions is asymptotically negligible. Our framework includes Poisson, negative binomial and logarithmic series as well as binomial mixture models. Simulations highlight the benefit in achieving normality when using the proposed marginal mixture density approach instead of the empirical one, especially for small sample sizes and/or when interest is in the tail areas. A real data example is given to illustrate the use of the methodology.
- Subjects
POWER series; DISTRIBUTION (Probability theory); MATHEMATICS; STATISTICS; ESTIMATION theory; PROBABILITY theory; EXPONENTIAL functions; POISSON distribution
- Publication
Scandinavian Journal of Statistics, 2005, Vol 32, Issue 1, p115
- ISSN
0303-6898
- Publication type
Article
- DOI
10.1111/j.1467-9469.2005.00418.x