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

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Stat 312: Lecture 4 Maximum Likelihood Estimation Moo K. Chung [email protected] 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.
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 5. (Invariance Principle) If ˆ θ 1 , ˆ θ 2 are the MLE’s of θ 1 , θ 2 , the MLE of h ( θ 1 , θ 2 ) is h ( ˆ θ 1 , ˆ θ 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....
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern