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

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Stat 312: Lecture 04 Moment Matching Moo K. Chung [email protected] September 14, 2004 1. Given a random sample X 1 , · · · , X n , a linear estimator of parameter θ is an estimator of form ˆ θ = n X i =1 c i X i . Then it can be shown that ¯ X is the MVUE for population mean among all linear unbi- ased estimators. Proof. Case n = 2 will be proved. The gen- eral statement follows inductively. Consider linear estimators ˆ μ = c 1 X 1 + c 2 X 2 . To be unbiased, c 1 + c 2 = 1 . To be most ef- ficient among all unbiased linear estimators, the variance has to be minimized. The vari- ance is V ˆ μ = c 2 1 V X 1 + c 2 2 V X 2 = £ c 2 1 + (1 - c 1 ) 2 / σ 2 The quadratic term in the bracket 2 c 2 1 - 2 c 1 + 1 is minimized when c 1 = 1 / 2 . 2. Given a random sample X 1 , · · · , X n , the k - th sample moment is M k = n j =1 X k j /n . The moment estimators of population param- eters are obtained by matching the sample moments to correspond population moments and solving the resulting equations simulta-
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Unformatted text preview: neously. 3. Exponential distribution. X is an exponen-tial distribution with parameter λ if the den-sity function is f ( x ) = λe-λx for x ≥ . It can be shown that X = 1 λ , V X = 1 λ 2 . 4. Given random sample X 1 , ··· ,X n , the like-lihood function is given as the product of probability or density functions, i.e. L ( θ ) = f ( x 1 ; θ ) f ( x 2 ; θ ) ··· f ( x n ; θ ) . The maximum likelihood estimatate of θ is an estimate that maximizes L ( θ ) . If we denote ˆ θ = θ ( x 1 , ··· ,x n ) to be the maximum like-lihood estimate, The maximum likelihood estimator (MLE) of θ is denoted by ˆ θ = ˆ θ ( X 1 , ··· ,X n ) . Review Problems. Example 6.12. Example 6.16. Example 6.17....
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