The Maximum Likelihood Estimations Of The Parameters For Gamma Distribution Based On Ranked Set Sampling

The Gamma distribution plays a significant role in the Science of Probability Theory and Mathematical Statistics which was used in various areas of research,especially in the fields of Hydrology,Reliability Theory and Life Insurance Actuarial and other fields,therefore,it is accepted widely by scholars around the world.It is an important content that investigate the parameters estimation of Gamma distribution.The moment estimation,interval estimation method,maximum likelihood estimation(MLE)and other methods was used to study the parameters estimation of Gamma distribution by a lot of experts based on simple random sampling(SRS).Ranked set sampling(RSS)rise in response to the proper time and conditions for the reason of the limitations of SRS which samples are obtained cannot represent the population well.RSS was first suggested by McIntyre(1952),an agronomist from Australian,used to estimate the output in a farm.It requires small sample size needed when estimate the same population compare with SRS.Moreover,the unit carries more the fisher information with respect to SRS when the same sample size is used.Undoubtedly,RSS get the favour of many researchers all over the world which is developing vigorous.In this article,the MLEs of Gamma distribution are studied under RSS.The background,theory and development of MLE on RSS are introduced in section one.Then the main contents and researches are given in other sections: Firstly,the MLE of the scale parameter of the gamma distribution by RSS is researching when the shape parameter is known,and the existence and uniqueness of MLE is proved at the same time in section two.What's more,the MLE of shape parameter of Gamma distribution is studied on RSS when the scale parameter is given,as well as the existence and uniqueness of MLE is proof in section three.A new sampling method is proposed: Maximum likelihood estimation based on the Fisher information maximization(FMLE)which is also used to investigate the MLE of the shape parameter in Gamma distribution in section three.Moreover,the MLEs of Gamma distribution are studied when the scale parameters and shape parameters are unknown in section four.Simultaneously,the theoretical proof of the existence of their MLEs is given.Besides,the numerical simulation of the MLEs in each scenario is compared with SRS in each section.The results show that the parameter estimation of the RSS is efficient and the MSE is smaller.Finally,summarize the full text and prospects of future research in section five.