Methods of Estimation of Generalized Negative Binomial Distribution

Abstract

The negative binomial distribution was perhaps the first probability distribution, considered in statistics, whose variance is larger than its mean. On account of wide variety of available discrete distributions, the research workers in applied fields have begun to wonder which distribution would be most suitable one in a particular case and how to choose it. Generalized Negative Binomial Distribution (GNBD) reduces the binomial or the negative binomial distribution as particular cases and converges to a Poisson-type distribution in which the variance may be more than, equal to or less than the mean, depending upon the value of the parameter. A number of methods for estimation of parameters of GNBD, like weighted discrepancies method, minimum chi-square method etc. are available but these methods produce such equations which are not simple to be solved directly and hence some iterations has to be applied to find the solution. An alternative estimator has been suggested here, which is capable of giving more or less as good results as given by the moment estimators. Although, the values of P (χ2), the probability of the observed value of χ2 to be exceeded, are slightly higher in case of the suggested method that in case of method of moments, these differences do not seem to be much significant and can be considered due to sample fluctuation. Moreover, it is relatively very quick to be obtained and so it may be preferred to others where very quick results are required.