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Exam05solution - Midterm Exam 1 Math 756/856 March 9 2005...

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Unformatted text preview: Midterm Exam 1 Math 756/856, March 9, 2005 Nameﬂﬁmﬂ ID Score Since I haven’t provided you the Z— table t- table and x2 -tabte, you can. use 20,, to m and xi‘n in your solution. 2a means that P(Z > 20,) — a, the 1000: upper percentage point of standard normal distribution ta,“ means that P(t (n) > to, W) — cc, and xan means that P(X2(n) > xi“) = a, where n is the degrees of freedom 1. (40pts) Let X1, X2, - - 3X“ be an i.i.d. sample from a Normal distribution with mean it and known variance 0 = 1, i.e., X1, X2, ---,Xn ~ N01,, 1), with the density function 1 _(L—_Eﬁ :1; = —e 2 , n ) T,” (a) (8pts) Find the method of moments estimate of is, denoted with Li. (b) (8pts) Find the maximum likelihood estimate of 4“: denoted with ﬁt. (c) (8pts) What is the distribution of {1? Based on this distribution, construct a 95% conﬁdence interval for n. (d) (8pts) Is it an unbiased estimator? Why? What is the variance of ﬁt? (e) (8pts) What is the Fisher information I (p)? Is there any other unbiased estimate of n have Smaller variance than ,1}? Why? (a) XwA/fﬂ,!) Mug gxs/gg ﬁMere/a =X ._. (.5) Mad-- —- EM; ,u)__ ﬁ/ﬁ) 6 mzjj M“) WM“) ‘ MM?” " ego ya)“ .-.= —§,{§m) ~é§m w) / a a 51/66 0'1 3%: gugij/ 2% M6):- [UV m! 1415421):er Maw - ' ”0% .5wa gang Ma WWWM MW 2. (40pts) Let X1, X2, - - - ,Xﬂ be an i.i.d. sample from a distribution with probability density distribution ﬁx] 6) = 6:1:‘9’1, a; 21 6 > 1. (a) (Spts) Find the method of moments estimate of 6, denoted with 6. (b) (8pts) Find the maximum likelihood estimate for 6, denoted with 6. (c) (Spts) Based on the factorization theorem, ﬁnd a sufﬁcient statistic for 6. (d) (8pts) What is the Fisher information 1(6), what is the asymptotic variance of the m.].e, 6. A (e) (Spts) Based on the asymptotic normality of rule, 6, construct a 95% conﬁdence interval for 6. (a) Xxx/ﬂac/MwEQ" 79!. W m 96 5? [500 146 49,3205 ”19% 4% (9%.. 4% 3&1? t? f" X ﬁat! Xﬁﬁé 655:?- (5) We?) WM): £97237)“ [6) 6M) 72 6!? w Wm. M) -~ -——- ”56“” r 11...,— J/ =3 WWI-=07? a 1%“; 7a ﬁre—ﬁr ﬁf— C a , ( ) ﬂxbg-‘Mw’ 9) :‘QZZWJ/ “ﬂ mm gym X”) y: -ﬂ/ W mevﬁ W): 4% m. M / MM, 77%):Z7ﬂé 425 a (mg/424M Wﬁw (d) 750%) =-- M“. WW) == [579-— —- (w) W 89 W 119% *5 i35331;gififW =- -EH-» '7‘; A . f 2 3. (20pts) Let X1, X2, ' ' - ,Xn be an i.i.d. sample from following distribution with the density function f(:z:|,\) : AEJIB—M, 0 < a; < oo, A > U. 5pts) Argue that Y : 2232131} is a suﬂicient statistic for A, why? (a K t (c t l C 1)) [5p 5) Find the maximum likelihood estimate for A, denoted with i. ) (5p 8) Compute E(1i,) based on the distribution of Y. (d) ( Spts) From part (c), ﬁnd the constant C such that C;\ is an unbiased estimator of A. ‘- 4t 9, 1/0“ a”! 4: 0&5)“ - .- - ’ rH (6L) {haXzJ'ﬁXn'ﬂQf- ll fl, Xth if], Egg-X46 xiv! vital-W). wigs) Mg) £15”. ...
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