Parametric estimation of extinction probability of branching process by using joint estimation of Mean and Variance

Parametric estimation of extinction probability of branching process

In this paper, we introduce a new generalization of univariate and bivariate modified discrete Weibull distribution. Various properties of the univariate generalized modified discrete Weibull distribution, such as survival function, probability mass function, hazard rate function, probability generating function, and moment generating function, are derived. The joint distribution function, joint probability mass function, marginal distributions, moment generating function, and conditional distribution of the proposed bivariate distribution are derived. Parameters of the distributions are estimated using maximum likelihood estimation. The use of these distributions is illustrated using real-life datasets.

The paper deals with the joint asymptotic normality of estimators of offspring mean and the offspring variance of the Bienayme-Galton-Watson Branching process, when n generation sizes are observable. Using these results, the probability of ultimate extinction is also estimated in the parametric set up. As an illustration, an application of the method to the case of fractional linear generating function is demonstrated. The method is also applied for H1N1 positive cases data generated during 2009.