Fiducial Inference On Gamma Distributions:Two-sample Problems With Multiple Detection Limits

Statistics is a science of collecting,sorting and analyzing data.However,due to the lack of accuracy of equipment or human error,it will affect the integrity and accuracy of the collected data,among which the problem of data censoring is the most concerned.In the case of data integrity damage,statistical reasoning and calculation has gradually become one of the hotspots of statistical research.With the advent of the big-data age,data analysis has become the focus of public research.Statistical analysis under the condition of censored cases is the focus of scholars’ research,and has an important application in the field of environmental science,medicine and economics.Among them,the censored problem in the field of environmental science is characterized by the fact that the number of censored data is fixed and the value of deleted data is less than the specific detection limits.This point is more typical.In the past,most of the statisticians used to deal with the censored data mainly by discard method and substitution method.They just drop the censored data or only replaced it with detection limit,and the accuracy of the results was relatively low.These two traditional methods have the defect of low precision for single sample problems,especially in the field of environmental science,the defect of two sample cases is more prominent.Because most of the environmental indicators are right skewed,and the fit gamma distribution well,in order to solve the problem of comparison of environmental indicators in the case of censored,this paper takes gamma distribution as the research object,focusing on the determination of the difference between two independent gamma means and the difference between two upper tolerance limit under the multiple detection limits.Based on the cube root transformation and the fiducial inference method,a simple method is proposed to construct the confidence interval of the difference between two gamma means and the difference between two upper tolerance limits.The performance of the proposed method is evaluated by Monte Carlo simulation,and compared with the parametric bootstrap method.The simulation results show that the fiducial method is better than the parametric bootstrap method according to the coverage probabilities,and the fiducial method can provide more accurate results even in the case of small samples with high non-detect ratio.