RP-HW4-sol - Kyung Hee University Department of Electronics...

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1 Kyung Hee University Department of Electronics and Radio Engineering C1002900 Random Processing Homework 4 Solutions Spring 2010 Professor Hyundong Shin Issued: May 12, 2010 Due: May 26, 2010 (No acceptance of overdue submission) Reading: Course textbook Chapter 8, Lecture Notes VIII–IX HW 4.1 (a) Suppose that  ˆ X was unbiased. Then from the quality  ˆ X  , we have   / / // / / ˆ for all ˆ for all ˆˆ for all , ˆ for all , ˆ for all . xd x xdx xx      12 2 1 1 0 1 0 11 00 1 1 0 10 0 (4.1) The final equality implies that ˆ X cannot be unbiased, so we have reached a contradiction. (b) Yes, we can find a MVU as follows. Let ˆ Xg X  be any unbiased estimator. Then    ; . X gx f x dx gx dx gxdx  0 0 2 0 1 (4.2) This last equality holds for all  0 , so that we can differentiate it with respect to to ob- tain that   gx 2 , which must hold for any unbiased estimator. Therefore, we conclude that ˆ XX 2 is the only unbiased estimator, and so must be the MVU. HW 4.2 Let
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2 ,, . N NN XW r r r      0 00 1 11 1  Xr W (4.3) Then, we wish to estimate A given A  W . By inspection,   , N A 2 I . This prob- lem falls exactly into the canonical scalar/vector. We can find the Cramer-Rao bound (CRB) as follows:  / ;e x p N N fA A     2 22 2 2 X x xr (4.4)   ln ; / ln /l n N A A NA A    2 2 2 1 2 1 2 2 X xx r x r x r (4.5)   ln ; A A  2 1 X x rx rr (4.6)   ln ; A  2 1 X x rr (4.7)   ln ; , i f i f . N n n N IA A r r r r N r       2 2 2 1 2 2 0 2 2 1 1 1 1 1 X X X (4.8) The CRB:  i f ˆ Var i f . N r r r A N r  2 2 1 1 1 1 X X (4.9) Now we can easily check for the existence of an efficient estimator:   eff ln ; ˆ . AA A 2 2 1 X X X X rX rr rX (4.10)
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3 Since the right-hand side of (4.10) does not depend on the parameter A , this estimator is valid and must be the efficient estimator for the problem. Since it is efficient, its MSE must be given by the CRB:   eff , i f ˆ Var i f . N r r r A N r   2 22 2 11 1 1 1 X (4.11) The estimator is consistent if eff ˆ lim Var N A  0 X . We see that   eff i f ˆ lim Var i f . N rr A r 01 X (4.12) Hence, our efficient estimator is consistent for r 1 .
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RP-HW4-sol - Kyung Hee University Department of Electronics...

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