STAT 400 Lecture AL1 p m f or p d f 1 Spring 2015 Dalpiaz Examples for 6 4 Part 1 f x Suppose 1 2 3 and the p d f parameter space f x is 1 f 1 1 0 6 f 2 1 0 1 f 3 1 0 1 f 4 1 0 2 2 f 1 2 0 2 f 2 2 0 3 f 3 2 0 3 f 4 2 0 2 3 f 1 3 0 3 f 2 3 0 4 f 3 3 0 2 f 4 3 0 1 What is the maximum likelihood estimate of based on only one observation of X if a X 1 b X 2 c X 3 d X 4 Likelihood function L L x1 x2 x n It is often easier to consider n f x i i 1 ln L f x 1 f x n n ln f x i i 1 Maximum Likelihood Estimator arg max L arg max ln L Method of Moments E X g Set X g Solve for 2 Let X 1 X 2 X n be a random sample of size distribution with mean 0 That is P X k k e k k 0 1 2 3 a Obtain the method of moments estimator of b Obtain the maximum likelihood estimator of 3 n from a Poisson Let X 1 X 2 X n be a random sample of size n from a Geometric distribution with probability of success p 0 p 1 That is P X k 1 p k 1 p k 1 2 3 a p Obtain the method of moments estimator of p b Obtain the maximum likelihood estimator of p p 4 Let X 1 X 2 X n be a random sample of size n from the distribution with probability density function 1 1 x f x 0 0 x 1 otherwise 0 a Obtain the method of moments estimator of Method of Moments E X g b Set X g Solve for Obtain the maximum likelihood estimator of Likelihood function L L x1 x2 x n Maximum Likelihood Estimator n f x i i 1 f x 1 f x n arg max L arg max ln L 4 continued c Suppose n 3 and x 1 0 2 x 2 0 3 x 3 0 5 Compute the values of the method of moments estimate and the maximum likelihood estimate for 5 Let X 1 X 2 X n be a random sample of size n from N 1 2 where 1 2 1 0 2 That is here we let 1 and 2 2 a Obtain the maximum likelihood estimator of 1 1 and of 2 2 b Obtain the method of moments estimator of 1 1 and of 2 2
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