On The Estimation Of The Mean Interval Of The Zero-inflated Gamma Distribution

Gamma distribution is one of the most important distribution in statistics,which has been widely employed in many applied fields such as environmental science,hydrology,insurance,etc.It is the most basic statistical techhology to calculate the confidence interval(CI)for the mean of gamma distribution.A large number of researchers have focused on the interval estimation for the mean of gamma distribution.The inference for the mean has been studied in depth.But in real life,right skewed data with a high clump-at-zero occur in many fields.There are certain responses are recorded as zeros and the positive responses are right skewed.Besides the zero part of the data,the positive values in the dataset usually follow gamma distribution in some fields.For the inference for the mean of zero-inflated gamma distribution,if we neglect the zero values,it may return biased results.Therefore,this paper concerns about the construction of confidence interval for the mean of zero-inflated gamma distribution.The problems of constructing the confidence intervals for the mean of a gamma distribution containing zero values are considered based on combining gamma distribution with binomial distribution.Three different methods for constructing such CIs based on Parametric Bootstrap(PB)method,fiducial approach and the method of variance of estimates recovery(MOVER)are developed.First of all,the performances of the proposed CIs are evaluated by Monte Carlo simulation and compared with a published method,asymptotic normal(AN)method proposed by muralidharan and kale.Our simulation studies indicate that the proposed methods perform better than the existing method according to coverage probabilities(CPs),tail error rates(ERs).The CPs for fiducial CIs are very satisfactory even for small samples.Secondly,the robust of proposed methods in this paper are evaluated by Monte Carlo simulation,which indicating fiducial approach and MOVER are more satisfactory than Pb method.Finally,the time consumption for the proposed CIs for different sample are compared.The results show that the fiducial method and MOVER work much more efficiently than PB method.According to the results from coverage studies,robust studies and time comparison in the above subsections,we can easily make a recommendation that when we need to construct CIs for a zero-inflated gamma mean,it is safe and time-saving to use our proposed fiducial method in all different scenarios even for small sample data,compared with the parametric bootstrap(PB)method and method of variance of estimates recovery(MOVER).In order to test the applicability of proposed method,all the methods are illustrated using the rainfall data for the months of June to September(for 122 days)for the years 1961–1970 at the Meteorological stations in Jalgaon in India and a data set for the immunologic response to combination therapy for HIV children.The results showed that the mean values of two cases were capturing in the confidence intervals for three proposed methods.