Research On The Parameters Estimates Of NOMMLE Distribution And Their MCMC Simulation

With time advancement and society development, mathematic statistics has absorbed and inosculated knowledge of the connected sciences with it. There emerge many new statistical distributions which many researchers would research profoundly in theory. It proved that they have broad development prospect in many fields.The paper studies a novel lifetime distribution denoted NOMMLE distribution which was introduced by my tutor. The distribution was proposed by making use of the noise property of gear wears. It is characterized by its moment estimator with negative order being equal to its maximum likelihood estimator. He hadn't done further research on the novel distribution so far. The paper will attempt to develop the problem.The author discusses it from the followings:(1) Consider the existence of its origin moment of order r, percentile and mode. The author proves NOMMLE distribution belongs to the exponential distribution family. The n-dimensional random variable T = (T₁.…,T_n)is also distributed in the exponential distribution family, where T₁,…,T_n are independent identically distributed random variables, in the meantime we find its complete and sufficient statistic.(2) In view of the complicated form of NOMMLE distribution with three parameters, the paper chooses Bayes estimation using MCMC sampling. Then we make use of four methods which are variance ratio method, Geweke spectral density convergence diagnostics, Heidelberger and Welch convergence diagnostics as well as Raftery and Lewis convergence diagnostics respectively. From the four diagnostics results, we can draw a conclusion that Markov Chains approach to their target distributions, which represents the Bayes estimation is credible and effective.(3) By applying Cramer-von Mises,Anderson-Darling and Watson statistics for the goodness-of-fit test, we can then test whether NOMMLE distribution fits what are recorded in the experimental data. In MC simulation trial, we can't calculate steady solutions of the equation group from the MLE or F estimator, so the paper proposes Bayes estimation using MCMC sampling. WinBUGS is run in R2WinBUGS package of R, and then we simulate successfully the critical values of the three statistics. Finally we try to select the exponential distribution, Weibull distribution and lognormal distribution as the alternative hypothesis, and compute the test power of the three tests.The paper considers the properties of the parameter estimator in NOMMLE distribution further theoretically and researches the distribution using MCMC simulation in numerical test systematically and in detail. So the study on the subject is fulfilled successfully.