Statistical Inference For Negative Binomial Distribution With Applications

Many products need for reliability testing in life, to obtain indicators of the reliability of products. If a product's test means is Bernoulli type of test, then its life expectancy is measured through the "success and failure" number. If we take into consideration the economic benefits, we often need to identify test "failure" number.The negative binomial distribution can excatly portray these tests life . Therefore, the negative binomial distribution has been subject to more and more people's concern today, and its research ascendant. The theoretical risk of fitting in the number of claims often use the second common types of negative binomial distribution, this paper shall be referred to the negative binomial distribution of this type. Namely: Let p is the probability of success of each test in Bernoulli test,before there is just r successful in Bernoulli test series, the number of failures X obey parameters of the negative binomial distribution ( r , p ). Written X～NB ( r , p)Easy to see that the negative binomial distribution has a very simple nature, its variance greater than average. This is precisely because this laid the foundation of the role of risk management. When the same collective risk, and its number of claims obey Poisson distribution, the mean constant equal variance.And the actual situation of collective risk extents a certain non-homogeneity a greater or lesser, which has created conditions for the application of the negative binomial distribution. The greater the variance of the negative binomial distribution than its mean , the more serious the non-homogeneity of the collective risk.This paper discusses the issue of the negative binomial distribution is the problem of estimating parameters p . Papers mainly has three parts.Part I discussed the negative binomial distribution's general parameters estimates. Mainly it presents the negative binomial distribution parameters p maximum likelihood estimation, moment estimated , interval estimates and the smallest variance consistent and unbiased estimation and its nature. The main result of the following conclusions.1 The negative binomial distribution moment generating function.Know from the theory of probability, the negative binomial distribution random variables can be seen as r independent random variables each other and r random variables are subject to the geometric distribution, i.e. X = X₁ + X₂ +…+X_r, where X_i obey geometric distribution, i.e. as qe,then the negative binomial distribution for the moment generating functionThe negative binomial parameters p maximum likelihood estimate 3 The negative binomial distribution parameters p moments estimates4 The negative binomial distribution parameters p interval estimationIf r known, the unknown parameters p following inequality can be identified5 The negative binomial distribution parameters p UMVUE estimatedThe UMVUE of the negative binomial distribution meanμ and varianceσ² is respectively,as r = 1,the UMVUE of , the UMVUE of p is , it shows no bias.Part II discusses the negative binomial distribution parameters p Bayes estimation. it presents the negative binomial distribution of parameters p Bayes estimation under the square loss of function , as well as balanced loss function , and entropy loss function. The main result of the following conclusions.6 The negative binomial distribution parameters p Bayes estimation under the square loss function 7 The negative binomial distribution parameters p Bayes estimation under balanced loss functionTheorem 1 Given any prior distribution and entropy loss of function , the negative binomial distribution reliablityθthe only Bayes estimated isPart III discusses the application of the negative binomial distribution, obtain the negative binomial distribution and Possion distribution relations and the applications of the negative binomial distribution in motor vehicle insurance, as well as in biology, and other fields. The main result of the following conclusions.Lemma 1 Any a compound negative binomial distribution can be considered a compound Poisson distribution In automobile insurance the negative binomial models fitted good is simply incredible. Negative binomial distribution as a collective claim frequency distribution of risk is reasonable. Negative binomial distribution fitting the distribution of the number of inpatient hospitalization sloves the problem of inpatient hospitalization frequency distribution described, accurately described the crowd directly to the distribution of medical expenses may be provided.