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Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat 312: Lecture 4 Maximum Likelihood Estimation Moo K. Chung mchung@stat.wisc.edu January 30, 2003 Concepts 1. For a random sample X 1 , · · · , X n the likeli- hood function is given as the product of prob- ability or density functions, i.e. L ( θ ) = f ( x 1 ; θ ) f ( x 2 ; θ ) · · · f ( x n ; θ ) . 2. The maximum likelihood estimatate of θ maxi- mizes L ( θ ) . If we denote ˆ θ = θ ( x 1 , · · · , x n ) to be the maximum likelihood estimate, The max- imum likelihood estimator (MLE) of θ is ˆ θ = ˆ θ ( X 1 , · · · , X n ) . Note: x i are numbers while X i are random variables. 3. When the sample size is large, the maximum like- lihood estimator of θ is approximately unbiased. The MLE of θ is approximately the MVUE of θ . This is why it is the most widely used parameter estimation technique. 4. If explicit density function is not available, you can not apply MLE. In this case apply the method of moment matching.
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Unformatted text preview: 5. (Invariance Principle) If 1 , 2 are the MLEs of 1 , 2 , the MLE of h ( 1 , 2 ) is h ( 1 , 2 ) . In-class problems Example 6.15. Example 6.16. Example 6.21. Exercise 6.25. Assuming normal distribution, find c such that P ( strength c ) = 0 . 95 . > x<-strength > length(x) [1] 10 > sd(x) [1] 19.87852-3-2-1 1 2 3 0.0 0.1 0.2 0.3 0.4 y dnorm(y) Figure 1: Density of N (0 , 1) > sigma<-sqrt((length(x)-1)/length(x))*sd(x) > sigma [1] 18.85842 > qnorm(0.95,mu,sigma) [1] 415.4193 > qnorm(0.95) [1] 1.644854 > pnorm(415,mu,sigma) [1] 0.9476644 > y<- -30:30/10 > plot(y,dnorm(y),l) Self-study problems Example 6.17., Example 6.18., Exercise 6.23., Exercise 6.29....
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