566Chap7S_2016.pdf - Chapter 7 Point Estimation Important Questions • How to construct an estimator θˆ using the random sample X1 � � � Xn • How

566Chap7S_2016.pdf - Chapter 7 Point Estimation Important...

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Chapter 7: Point EstimationImportant Questions:How to construct an estimatorˆθusing the random sampleX1,· · ·, Xn?How to measure/evaluate the closeness ofˆθtoθ?How to find the “best” possibleˆθ? What is the “best”?Def. 31
1 Method of Moments Estimator Let X 1 , . . . , X n be a random sample from a population with pdf or pmf f ( x | θ ), where θ = ( θ 1 , . . . , θ k ) and k 1. Sample moments m 0 and population moments μ 0 are defined: Sample moment vs Population moment m 0 1 = 1 n n i =1 X i μ 0 1 = EX m 0 2 = 1 n n i =1 X 2 i μ 0 2 = EX 2 · · · · · · m 0 k = 1 n n i =1 X k i μ 0 k = EX k . · · · · · · Each μ 0 j is a function θ , i.e. μ 0 j = μ 0 j ( θ 1 , . . . , θ k ) for j = 1 , . . . , k . Method of Moments Estimators (MME): Assume θ R k . Step 1 : Equate the first k sample moments to the corresponding k population moments. m 0 1 = μ 0 1 ( θ ) , m 0 2 = μ 0 2 ( θ ) , . . . m 0 k = μ 0 k ( θ ) , Step 2 : Solve the above equation system for θ . The solutions are de- noted as ˆ θ MME . Applications : German Tank ProblemSuppose it is the end of June 1940, and the Allies have captured 13 tankswith the following serial numbers:497513892369417639521218116309395493How to estimate the total number of mark V tanks produced by the Germansup to this time? 32
Example: X 1 , . . . , X n iid N ( μ, σ 2 ), both μ and σ 2 unknown. Example: X 1 , . . . , X n iid Bin( k, p ), both k and p unknown. 33

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