Estimation And Application For A Special Kind Of The Exponential Family Distribution

Exponential distribution family includes many common distributions, such as normaldistribution, Weibull distribution, Poisson distribution and so on. And these distributions canbe applied in many fields, so it has great theoretical significance and practical value to studythe parameter estimation and efficient estimation of the exponential distribution family. So far,most of the available research findings have been built on a specific distribution while lessstudy has been done on uniform inference of the different distribution with similar patterns.Firstly, we have defined a special class of exponential distribution family, which could beused both in life-span estimations and reliability analysis of flora, fauna and electronicelements, as well as the study of the wealth distribution and the residential area size. Then,five point estimations of the probability density function(PDF) and cumulative distributionfunction (CDF) for this distribution family have been derived, including ML estimation,UMVU estimation, Bayes estimation under generalized entropy loss function, Bayesestimation under Q-symmetric entropy loss function and Bayes estimation under LINEX lossfunction. And then by using special functions, we obtain the exact expression of the r-thmoment for parts of the estimators. We also consider the asymptotic normality as well as themean square error(MSE) convergence. Moreover, based on the asymptotic expression of MSE,a new method to choose the hyper-parameters in generalized entropy loss function andLINEX loss function is given. We also analyze the efficient estimation. Finally, we show partsof the theoretical results by using application examples. The main results are as follows:1. The five estimations of the PDF have the same asymptotic normal distribution as wellas the estimations of the CDF.2. The five estimations of the PDF and CDF are consistent estimator and have the sameMSE convergence rate.3. In the large sample, Bayes estimations under generalized entropy loss function andLINEX loss function can have an obvious advantage when compared with the estimations ofML and UMVU by selecting reasonable hyper-parameters in prior distribution and the lossfunction. Besides, the comparison of the efficient estimations can be transformed into asimple arithmetic operation. As for small sample, the results are similar as in the large sampleby modifying the selection method of hyper-parameters in the large sample.4. In the large sample, the hyper-parameter in the Q-symmetric entropy loss function has little effect on the MSE. However, when setting the hyper-parameter close to zero, the Q-symmetric entropy loss function shows good results in the small sample.5. The MSE of the estimator of PDF, or CDF, or reliability function has a great differenceunder different independent values. The advantages and disadvantages of different estimatorsdepend on the independent values when applying the MSE evaluation criterion.6. In the small sample, a combined method of variety of estimation in a piecewisecombination way can obtain an overall smaller mean square error estimator when requiringhigh estimation precision.