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Unformatted text preview: Ditlev Monrad STAT 4lO/MATH 464 November 27, 2007 OPEN BOOK SHOW ALL WORK Problem 1. (40 points) (a) Compute the mean ,u and the variance 0' 2 of the continuous distribution with p.d.f. f(x;6) = 23261—9)”, for 0 <x <1, where the constant 9 is greater than 0. (b) Given n independent observations from this distribution, X1, X2, WK”, ﬁnd the method—of—moments estimator 19" of the unknown parameter 9. (c) Explain Why 5” is a consistent estimator of 6. (d) Show that the sequence {6” } is asymptotically normally distributed. Find the parameters. Problem 2. (30 points) Let X1, X2,...Xr1 be independent random variables, each With the p.d.f. f(x) = 2(l-x), for O <x < l. (a) Compute the distribution function F(t)=P(X15t) , -00<t<oo. (Consider the cases: t S O, O < t < l, and t Z l separately.) (b) Let Y1 < Y2 < <YI1 denote the corresponding order statistics. Put .2n = J; (Yn — 1). Compute Fn(t) = P(Zn 5 t), —00 < t < 00 (c) Determine the distribution function Fco (t)=1im Fn (t), —oo<t<oo 11—)00 (You probably have not seen this distribution function before.) Problem 3. (60 points) Let G be an unknown parameter and let X1, X2, ...,Xn be n independent random variables, each with the p.d.f. ﬁx; 6) = ex'e , forxfe, -oo<9 <00 (a) Given n observations x1, x2, ...,xn, determine the likelihood function L(0) = L(9; x1, xz, ...,x,,), -oo < 6 < 00. (b) Explain why Yn (the largest order statistic) is the maximum likelihood estimator of 6. (c) Show that Yn is a sufﬁcient statistic for 9. Identify functions k1(y; G) and k2(x1, x2, ...,xn), both input restrictions and output. ((1) Determine gn(y; 6), the p.d.f. of Y“. (6) Compute the mean of Yn. (f) Obtain an unbiased estimator of 6 that is a function of Y“. (The UMVUE.) Problem 4. (50 points) Let G > 0 be an unknown parameter and let X1, X2, ...,Xn , be independent random variables, each with the p.d.f. ﬁx; 9) = éx—(em/a’ for X > 1. (a) Given the observations x1, x2, ...,xn , determine the likelihood function Me) = L(9;X1,X2, --.,xn) (b) Find the maximum likelihood estimator é of 6. (c) Show that é is unbiased. ((1) Compute Var( é ). (6) Show that é is a consistent estimator of 6. Ditlev Monrad STAT 4lO/MATH 464 November 28, 2006 OPEN BOOK SHOW ALL WORK Problem 1. (20 points) (a) Compute the mean ,u of the continuous distribution with p.d.f. f(x;6) = 262/38, for e < x < 00, where the constant (9 is positive. (b) Given :1 independent observations from this distribution, X1, X2, ...,Xn, ﬁnd the method-of-moments estimator 6: of the unknown parameter 6. Problem 2. (30 points) Let X1, X2, . ., Xn be independent random variables, each with the p.d.f. f(x) =x'2, for x21. (a) Compute the distribution function F(t)=P(X1§ 1‘), -oo <t< oo (Consider the case: IS 1 and the case t> l separately.) (b) Let Y1 < Y2 < <Yn denote the corresponding order statistics. Put Zn = Yn/n. Compute Fn(t) = P(Zn S t), — 00 < r< oo_ (0) Determine the distribution function Fooa): "A“, Fm), —oo<t<oo. (You probably have not seen this function before). Problem 3. (70 points) Let 0 < 6 < 1 be an unknown parameter and let X1, X2, . ., Xn be n independent random variables, each with the pmf f(x;6) = 60—6)“, x=l,2,3,... (a)Showthat E(X1) =—;— and Var (x1) = 1: (b) Show that the family { f (x; 6) : 0 < 6 < 1} is a regular exponential family. Identify functions p(6), K(x), S(x), and q(6). (0) Given n observations x1, x2, ..., xn, determine the likelihood function 14(6) 2 L(0 ; xla-XZ: "'3xn)' (d) Find the maximum likelihood estimator 6" . (e) Explain why 6" is a complete sufﬁcient statistic for 6. (f) Show that the sequence {6” } is asymptotically normally distributed. Find the parameters. (g) Explain why 6” is a consistent estimator of 6 (as n ~—> oo ). Problem 4. (70 points) If gene frequencies are in equilibrium in a population, the genotypes AA, Aa, and aa occur with the relative frequencies 62 ,26(1- 6), and (1— 6?. Imagine that we take a random sample of n persons. Let us pretend that we sample with replacement so that the n draws are independent. Let X, Y, and Z denote the number of persons in the sample with genotypes AA, Aa, and aa, respectively. (Notice that X, Y, and Z are not independent.) (a) What is the joint pmf of X and Y? (See Chapter 3.) (b) Assume that 6 is unknown. Given the observations x, y, and 2, ﬁnd the likelihood function L(6)=L(6;x,y,z), 059:1 Notice that we only observe one value of each of the random variables X, Y, and Z. (0) Explain why the statistic 2X + Y is sufﬁcient for 6’. Identify functions k1(u ; 6) and k2 (x, y, 2), both input restrictions and output. (d) Find the maximum likelihood estimator <9 . (e) Show that E ((9 ) = 6. (D Show that Var( c9) = 602— 0) . n (g) In a sample from the Chinese population of Hong Kong in 1937, certain erythrocyte antigens (blood types) occurred with the following frequencies Blood Type AA Aa aa Total Frequency 187 500 342 1029 Find an approximate 95% conﬁdence interval for 6, using these data. Name: Ditlev Monrad STAT 31 l/lVlATH 364 November 20, 2003 SHOW ALL WORK Problem 1. (40 points) Let X1, X2, ..., Xr1 be independent random variables, each with the p.d.f. f(x)=e'x, for x _>_0. (a) Compute the distribution function F(t)=P(X1£t) , —oo<t< 00. (b) Let Y1 < Y2 < < Yn denote the corresponding order statistics. Put Zn = Yn — In (n). Compute Fn(t) = P(Zn S t) , -oo< t < 00 (b) Determine the distribution function IFOO (t)= lim Fn(t), —oo<t<oo n —-) 00 (You probably have not seen this distribution function before.) Problem 2. (50 points) Let X1, ..., Xn, W1, ..., Wm be independent normal random variables. Assume that X1, ..., Xn have mean u 1 and variance 9 > 0 and W1, ..., Wm mean u 2 and the same variance 9. Assume that the parameters u 1, u 2 and 6 are unknown. (a) Given observations x1, . ., xn, W1, ..., Wm , determine the likelihood function L(u1,1u2,9)= L011, “2,9;X1,---,Xn,W1,-~,Wm). (b)Find the maximum likelihood estimators (£31, ‘ﬁ‘ 2, ' ’6‘) of (u 1, u 2, 6) . (c) ComputeE(’§). Is ’9‘ unbiased? (d) Compute Var (6‘). (e) Show that 9 is a consistent estimator of 9 (as m + n —> oo ). Problem 3 (60 points) Let 7» > 0 be an unknown parameter and let X1, X2, ..., Xn be n independent random variables, each with the p.d.f. f(x; t) = Re "v" , x> o. (a) Show that E (X1)=% and Var(X1)= %2, (b) Given n observations x1, xz, ...xn , determine the likelihood function L (9») :L()\4; X1, X2, ...xn). (c) Find the maximum likelihood estimator/K?» n . ((1) Explain why/7:n is sufﬁcient for )L . (6) Show that as n —> oo , the sequence {63,1} is asymptotically normally distributed. Find the parameters. A k (0 Explain why n is a consistent estimator of 9» (as n —> oo) . ...
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