Parameter estimation for generalized negative binomial distribution

We investigate the proximity in terms of zeta-structured metrics of generalized negative binomial random sums to generalized gamma distribution with the corresponding parameters, extending thus the zeta-structured estimates of the rate of convergence in the Rényi theorem. In particular, we derive upper bounds for the Kantorovich and the Kolmogorov metrics in the law of large numbers for negative binomial random sums of i.i.d. random variables with nonzero first moments and finite second moments. Our method is based on the representation of the generalized negative binomial distribution with the shape and exponent power parameters no greater than one as a mixed geometric law and the infinite divisibility of the negative binomial distribution.

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On hypergeometric generalized negative binomial distribution

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171202106193 ◽

2002 ◽

Vol 29 (12) ◽

pp. 727-736 ◽

Cited By ~ 11

Author(s):

M. E. Ghitany ◽

S. A. Al-Awadhi ◽

S. L. Kalla

Keyword(s):

Recurrence Relation ◽

Negative Binomial Distribution ◽

Binomial Distribution ◽

Field Data ◽

Goodness Of Fit ◽

Negative Binomial ◽

Generalized Negative Binomial Distribution ◽

The Right ◽

Three Term Recurrence Relation

It is shown that the hypergeometric generalized negative binomial distribution has moments of all positive orders, is overdispersed, skewed to the right, and leptokurtic. Also, a three-term recurrence relation for computing probabilities from the considered distribution is given. Application of the distribution to entomological field data is given and its goodness-of-fit is demonstrated.

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Suitability of Several Statistical Models to Simulate Observed Distribution of Sample Test Results in Inspections of Aflatoxin-Contaminated Peanut Lots

Journal of AOAC International ◽

10.1093/jaoac/79.4.981 ◽

1996 ◽

Vol 79 (4) ◽

pp. 981-988 ◽

Cited By ~ 7

Author(s):

Thomas Whitaker ◽

Francis Giesbrecht ◽

Jeremy Wu

Keyword(s):

Parameter Estimation ◽

Negative Binomial Distribution ◽

Binomial Distribution ◽

Statistical Models ◽

Negative Binomial ◽

Estimation Method ◽

Likelihood Method ◽

Model Parameters ◽

Test Results ◽

Parameter Estimation Method

Abstract The acceptability of 10 theoretical distributions to simulate observed distribution of sample aflatoxin test results was evaluated by using 2 parameter estimation methods and 3 goodness of fit (GOF) tests. All theoretical distributions were compared with 120 observed distributions of aflatoxin test results of farmers' stock peanuts. For a given parameter estimation method and GOF test, the negative binomial distribution had the highest percentage of statistically acceptable fits. The log normal and Poisson-gamma (gamma shape parameter = 0.5) distributions had slightly fewer but an almost equal percentage of acceptable fits. For the 3 most acceptable statistical models, the negative binomial had the greatest percentage of best or closest fits. Both the parameter estimation method and the GOF test had an influence on which theoretical distribution had the largest number of acceptable fits. All theoretical distributions, except the negative binomial distribution, had more acceptable fits when model parameters were determined by the maximum likelihood method. The negative binomial had slightly more acceptable fits when model parameters were estimated by the method of moments. The results also demonstrated the importance of using the same GOF test for comparing the acceptability of several theoretical distributions.

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