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### lecture05

Course: STAT 312, Fall 2008
School: Wisconsin
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312: Stat Lecture 05 Maximum Likelihood Estimation Moo K. Chung mchung@stat.wisc.edu September 14, 2004 ^ 1. (Invariance Principle) If is the MLE's of ^ parameter then the MLE of h() is h() for some function h. Proof (partial). Consider likelihood function ^ L(). satisfies dL() = 0. Let = h(). d Then the likelihood function for = h() is given by L(h-1 ()). Differentiating the likelihood with respect to , we...

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312: Stat Lecture 05 Maximum Likelihood Estimation Moo K. Chung mchung@stat.wisc.edu September 14, 2004 ^ 1. (Invariance Principle) If is the MLE's of ^ parameter then the MLE of h() is h() for some function h. Proof (partial). Consider likelihood function ^ L(). satisfies dL() = 0. Let = h(). d Then the likelihood function for = h() is given by L(h-1 ()). Differentiating the likelihood with respect to , we have L(h-1 ()) dL() d dL() 1 = = = 0. d d d d h () 2. Loglikelihood. Maximizing L() is equivalent to maximizing ln L() since ln is an increasing function. Example. This technique is best illustrated by finding the MLE of parameters in N (, 2 ). = X, ^ 2 = ^ 1 n n 5. Additional problem (previous midterm). Let X1 , X2 be a random sample from N (0, 1/). Note that the sample size is 2 and the density function for Xi is f (xi ) = exp(-x2 /2). i 2 Find the likelihood function and use it to obtain the maximum likelihood estimator of . Solution. The likelihood is function L() = /(2) exp - (x2 + x2 )/2 . 1 2 Now get log-likelihood function ln L() = const + ln - (x2 + x2 )/2. 1 2 Differentiate with respect to we get d ln L() 1 1 = - (x2 + x2 ) = 0. 2 d 2 1 Solving the equation, we get ^ = 2/(x2 + x2 ). 1 2 Review problems. Example 6.17., Example 6.18., Exercise 6.23., Exercise 6.29. Read Chapter 7. Homework II. Exercise 6.30., 6.38., 7.4. 7.10. 7.12. 7.14. Due Sept 30. 9:30am. (Xi - X)2 i=1 are the MLE of and 2 respectively. Note that 2 is not un unbiased estimator of 2 . ^ 3. Asymptotic unbiasness. When the sample ...

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Wisconsin - STAT - 312
Stat 312: Lecture 06 Confidence IntervalsMoo K. Chung mchung@stat.wisc.edu September 21, 20041. A confidence interval (CI) is an interval used to estimate the likely size of a population parameter. A confidence level is a measure of the degree of r
Wisconsin - STAT - 312
Stat 312: Lecture 06 Quantile-quantile plotsMoo K. Chung mchung@stat.wisc.edu September 23, 2004F (q) = P (X q) = p. The textbook represent it in terms of percentile. Note that p-th quantile = 100 p-th percentile. So given p, q = F -1 (p). For X
Wisconsin - STAT - 312
Stat 312: Lecture 08 Large sample confidence intervalsMoo K. Chung mchung@stat.wisc.edu September 27, 20041. The sample size is inversely related to the width of confidence interval. Example 7.4. 2. Central Limit Theorem. Let X1 , , Xn be a ran
Wisconsin - STAT - 312
Stat 312: Lecture 09 Confidence intervals from normal populationMoo K. Chung mchung@stat.wisc.edu September 30, 2004Normal Q-Q Plot16 18 20 22 24 26 28 300.30 dt(x, 100) -4 -2 0 x 2 4 dt(x, 1) 0.4 0.0 -4 -2 0.2Sample Quantiles0.000.150 x
Wisconsin - STAT - 312
Stat 312: Lecture 2Moo K. Chung mchung@stat.wisc.edu January 23, 2003Definitions. 1. A statistic is a random variable from a random sample. If X 1 , , Xn is a random sample, a statistic is a function of them. 2. A point estimate of population p
Wisconsin - STAT - 312
Stat 312: Lecture 10 Confidence intervals for varianceMoo K. Chung mchung@stat.wisc.edu August 13, 20041. Suppose the fat content of a hotdog follows normal distribution. 10 measurements are given.dchisq(y, 1) 0.25 dchisq(y, 5) 0 20 y 400.000
Wisconsin - STAT - 312
Stat 312: Lecture 11 Hypothesis testingMoo K. Chung mchung@stat.wisc.edu Oct 20, 2004Concepts1. The null hypothesis H0 is a claim about the value of a population parameter. The alternate hypothesis H1 is a claim opposite to H0 . 2. A test of hypo
Wisconsin - STAT - 312
Stat 312: Lecture 15 P -valuesMoo K. Chung mchung@stat.wisc.edu November 2, 20041. The P -value is the smallest level of significance at which H0 would be rejected. Example 8.5, 8.15. Nicotine contents X1 , X32 of cigarettes follow N (, 0.22 ).
Wisconsin - STAT - 312
Stat 312: Lecture 17 Two sample t testMoo K. Chung mchung@stat.wisc.edu November 9, 20041. Let X1 , , Xn and Y1 , , Ym be two independent samples from normal distributions with the same population variance, i.e. Xi N (X , 2 ) and Yj N (Y
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Stat 992: Lecture 41 Image Analysis via Random Fields and Kernel Smoothing: OverviewMoo K. Chung mchung@stat.wisc.edu December 7, 2003Although it may be difcult to formulate all the problems in image analysis in terms of random elds and kernel smo
Wisconsin - STAT - 992
Topics mentioned briefly Topics covered Topics covered In detailSTAT 992: Streamlined Image Analysis FrameworkMoo K. Chung mchung@stat.wisc.edu 1. Image collection MRI, fMRI, PET, CT, EEG, MEG 2. Reading Image into computer Image formats ascii, bi
Wisconsin - STAT - 471
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Wisconsin - STAT - 471
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Wisconsin - STAT - 471
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Wisconsin - STAT - 471
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Wisconsin - STAT - 471
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Wisconsin - STAT - 471
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Wisconsin - STAT - 471
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Wisconsin - STAT - 471
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Wisconsin - STAT - 351
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Wisconsin - STAT - 351
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Wisconsin - STAT - 351
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Wisconsin - STAT - 351
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