HW11_2007 - PROF HOHG FALL 2007 ECONOMICS 619 PROBLEM SET #...

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Unformatted text preview: PROF HOHG FALL 2007 ECONOMICS 619 PROBLEM SET # 11 1. [# 7.1, p.355] One observation in taken on a discrete random variable X with pmf f (x; ), where 2 f1; 2; 3g. Find the MLE of x 0 1 2 3 f (xj 1) 1 3 1 3 1 6 1 6 f (xj 2) 1 4 1 4 1 4 f (xj 3) 0 0 1 2 1 4 0 2. [# 7.7, p.355] Let X1( Xn be i.i.d. with one of two pdfs. If ; :::; 1 0<x<1 f (x; ) = 0 otherwise while if = 1, then ( p 1=(2 x) 0 < x < 1 f (x; ) = 0 otherwise Find the MLE of . = 0, then 3. [# 7.8, p.356] One observation, X, is taken from a N (0; 2 ) population. (a) Find an unbiased estimation of 2 : (b) Find the MLE of : (c) Discuss how the method of moments estimator of might be founded. 4. [# 7.19, p.358] Suppose that the random variables Y1 ; :::; Yn satisfy Yi = xi + "i ; i = 1; :::; n; where x1 ; :::; xn are ...xed constants, and are i.i.d. N (0; 2 ), 2 are unknown. (a) Find a two-dimensional su cient statistic for ( ; 2 ): (b) Find the MLE of , and show that it is an unbiased estimator of : (c) Find the distribution of the MLE of : 5. [# 7.40, p.362] Let X1 ; :::; Xn be i.i.d. Bernoulli(p). Show that the variance of Xn attains the Cramer-Rao Lower Bound, and hence Xn is the best unbiased estimator of p: 6. [# 7.42, p.363] Let W1 ; :::; Wk be unbiased estimators of a paramater with Var Wi = 2 and Cov(Wi ; Wj ) = 0 if i 6= j: P (a) Show that, of all estimators of the form k ai Wi ; where the ai s are constants and i=1 Pk 2 Pk i=1 Wi = i Pk E ( i=1 ai Wi ) = , the estimator W = 2 has minimum variance. i=1 1= i 1 (b) Show that Var(W ) = 7. [# 7.44, p.363] Let X1 ; :::; Xn be i.i.d. N ( ; 1). Show that the best unbiased estimator of 2 2 is Xn (1=n): Calculate its variance and show that it is greater than the Cramer-Rao Lower 1 2 Bound. [Hint: You can calculate the variance of Xn n using the property of 2 .] 8. [# 7.50(a,b), p.364] Let X1 ; :::; Xn be p i.i.d. N ( ; 2 ); > 0: For this model both Xn and cS 1 are unbiased estimators of , where c = n p2 ((n 1)=2) : (n=2) (a) Prove that for any number a the estimator aXn + (1 a)cSn is an unbiased estimator of . (b) Find the value of a that produces the estimator with minimum variance. 9. Suppose ^1 ; ^2 and ^3 are estimators of ; and we know that E(^1 ) = E(^2 ) = ; E(^3 ) 6= ; var(^1 ) = 12; var(^2 ) = 10; and E(^3 )2 = 6: Which estimator is the best in terms of MSE criterion? 10. Suppose fX1 ; X2 ; :::; Xn g is an i.i.d. random sample from some population with unknown mean and variance 2 : De...ne parameter = ( 2)2 : (a) Suppose ^ = (Xn 2)2 is an estimator for ; where Xn is the sample mean: Show that ^ is not unbiased for : [Hint: Xn 2 = Xn + 2:] (b) Find an unbiased estimator for : 11. A random sample, X1 ; :::; Xn ; is taken from an i.i.d. population with ( ; the following estimator of : X 2 2 ^= (X1 + 2X2 + 3X3 + i Xi = n(n + 1) n(n + 1) i=1 n 2 ): Pk 1 i=1 1= 2 i : Consider + nXn ) : (a) Show ^ is unbiased for : Pn (b) Which estimator, ^ or Xn ; is more e cient? Explain. [Hint: i=1 i = Pn 2 n(n+1)(2n+1) :] i=1 i = 6 12. Suppose (X1 ; X2 ; :::; Xn ) is an i.i.d. N (0; 2 Sn 2) n(n+1) 2 and random sample. De...ne (Xi Xn )2 ; = (n 1) 1 where X1 = n 1 Pn n X i=1 i=1 Xi ; and ^2 = n 1 n X i=1 Xi2 : Show which estimator is more e cient? Give your reasoning. 2 ...
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This homework help was uploaded on 12/08/2007 for the course ECON 6190 taught by Professor Hong during the Fall '07 term at Cornell University (Engineering School).

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