安徽师范大学概率论与数理统计课件Chapter6 Parameters Estimation.ppt

Chapter 6 parameter estimation § 6.1 Point estimation Set the distribution function of population X is F(x), which contains the unknown parameters ,we call any function of a sample point estimator. 1, Moment estimate If population distribution contains k unknown parameters,method of moments estimator are found by equating the first k sample moments to the corresponding k population moments,and solving the resulting system of simultaneous equations. Example 6.1 suppose are iid ,We have according to the method of moments,we yield the moments estimator of is . Example 6.2 Suppose are iid ,We have ,hence we must solve the equation Solving for yields the moments estimator of is Example 6.3 suppose are iid,and the common pdf is ， is a known positive constant，we have Let ,we yield the moment estimator of is . 2, Maximum likelihood estimate Consider a random sample from a distribution having pdf or probability distribution , are unknown parameters,The joint pdf of is ,this may be regard as a function of ,when so regarded,it is called the likelihood function L of the random sample,and we write Suppose that we can find a nontrivial function of ,say ,when is replaced by ,the likelihood function L is a maximum,then the statistic will be called a maximum likelihood estimator of ,and will be denoted by the symbol . The function can be maximized by setting the first partial derivation of ,with respect to ,equal to zero,that is to say and solving the resulting equation for ,which is