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Monrad.410.35 - STAT 410/MATH 464 Ditlev Monrad Final...

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Unformatted text preview: STAT 410/MATH 464 Ditlev Monrad Final Examination May 8, 2006 OPEN BOOK SHOW ALL WORK Problem 1. (50 points) Let 6 > 0 be an unknown parameter and let X1, X2, ..., Xn be independent random variables, each with the p.d.f. f(x;6) = 6x“, o< x <1. (a) Compute u = E(Xk). (b) Compute 02 = Var(Xk). (c) Find the method-of—moments estimator 6:, of 6. N (d) Show that the sequence {6" } is asymptomatically normally distributed and determine the parameters. N (e) Explain why the sequence {6" } is consistent. Problem 2. (60 points) Let (9 > 0 be an unknown parameter and let X1, X2, ..., X,n be independent random variables, each with the p.d.f. f(x;6) = 6W”), x 2 1. (a) Determine the joint p.d.f. ofX1,X2, ...,Xn, L(6) = L(6; )6}, 362, ...,xn). (b) Show that 2;, ln(Xk)= ln(X1X2 X“) is sufficient for 6. (c) Show that for any constant c > 0, the set C = {(x1, x2, ..., xn) : ln(x1) + ln(x2) + + ln(xn) S c} is a uniformly most powerful critical region for testing against H1 26> l 1 H:6=— ° 2 2 (d) What is the distribution of the test statistic 2L1 ln(Xk) if H0 is true? (e) For a sample of size n=5 and at = 0.025, determine the threshold c so that the critical region C is of size a. (i) For a sample of size n=100, find a threshold c such that the test has approximate significance level at = 0.025. Hint: Apply the Central Limit Theorem to approximate the distribution of 2 2:1 ln(Xk) under HO. Problem 3. (30 points) Let 6 > 0 be an unknown parameter and let X1, X2, ..., Xn be independent random variables, each with the p.d.f. f(x;6) = é—e‘xm, x> 0. (a) Given the observations x1, x2, ...,xn, determine the likelihood function L(6) = L(6; x1, xz, ...,xn). (b) Find the maximum likelihood estimator 6. (c) Show that the likelihood ratio test for testing H0: 6 = 1 against HA : 6 96 1 has the critical region {( x1, ...,xn) : (2130" 6‘23; 5 ca} where the constant ca depends on the significance level 0L (and on n). Problem 4. (60 points) If gene frequencies are in equilibrium in a population, the genotypes AA, Aa, and aa occur with frequencies 92, 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. (a) What is the joint p.m.f. of X and Y? (b) Assume that 6 is unknown. Given the observations x, y, and 2, find the likelihood function L(6) = L09; x, y, z). (c) Find the maximum likelihood estimator é ((1) Show that E(é) = 6. (e) Show that Var(é ) = 602— 6) . n (f) 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 1 87 500 342 1029 Find an approximate 95% confidence interval for 0, using these data. Math 364/Stat 311 Ditlev Monrad Final Examination August 7. 1987 OPEN BOOK SHOW ALL WORK Problem 1 (45 points). Let 6 > 0 be an unknown parameter and let X1.X2.....Xn be .n independent random variables. each with the p.d.f. f(x;6) = %_e-x/6. x > 0. (a) Show that for any constant c ) 0 the set C={urunfi):fi+x2hu+figc} is a uniformly most powerful critical region for testing HO 2 G = 2 against H1 : 6 < 2. (b) What is the distribution of the test statistic X1 + X2 +...+ Xn if H0 is true? (c) If we take a = 0.05 and n = 10. determine the constant c. Problem 2 (60 points). Let 0 < 9 S 1 be an unknown parameter and let X1.X2.....Xn be n independent random variables. each with the p.d.f. f(x:9) = 9(1 - e)". x = 0.1.2.... (a) Given n observations x1.x2.....xn. determine the likelihood function L(9) = L(9;x1.....xn). (b) Find the maximum likelihood estimator an of 6. (c) Show that an is sufficient for 6. (d) Show that the sequence {an} is asymptotically normally distributed if 0 < 6 < 1. Find the parameters. (e) Show that an is a consistent estimator of G (as n a co). Problem 3 (60 points). In a survey about a certain behavior pattern people may be too embarrassed to reply truthfully. It is possible to reduce the potential embarrassment while retaining half the information by the following kind of questions: "I will ask you to toss this coin. but do not show or tell me how it lands. If it lands heads. answer (yes or no): Is you first child male? If the coin lands tails. answer (yes or no): Have you had an extramarital affair?" Let 9 denote the proportion of all parents that have had affairs. (a) Find the probability p (as a function of 6) that a randomly chosen parent will answer yes. (b) Suppose you got 237 yes’s from a random sample of 400 respondents. Give an estimate 3 of p and give the standard error of S. (c) Give an estimate 9 of 9 and give the standard error of 9. (d) Find a 95% confidence interval for 6. Problem 4 (120 points); Let -w < 9 < N be an unknown parameter and let X1. 2.....Xn be n independent random variables. each uniformly distributed on the interval [9 — é , 9 + ii. Let Y1 < Y2 (...< Yn be the corresponding order statistics. (a) Given n ordered observations y1 < y2 (...< yn. find the likelihood function L(9) = L(9;y1.y2.....yn). (What inequalities must 6 satisfy?) (b) Show that the random vector (Yl'Yn) is sufficient for B. (c) Show that the midrggge An = %(Y1 + Yfi) is a maximum likelihood estimator of B. (d) Find the Joint p.d.f. f(y1.yn) of (Y1.Yn). (e) Find the Joint p.d.f. g(z.r) of the random variables 2n = An — B and Rh = Yn - Y1. . (Pay close attention to the range of (zn’Rn)') (f) Find the p.d.f. gn(z) of Zn' (3) _Show that l(1+2t)n. for -lgtgo 2 2 P{Zn S t} = 1 n 1 ' 1 - -{1 - 2t) . for O S t S — 2 2 (h) Is An an unbiased estimator of 6? Explain. (1) Compute Var(An). (.1) Put Wn = 2n(An — 9). Find Fn(t) = P{wn g 1:}, ~00 < t ( m. (k) Find F(t) = lim rum. -oo < t < «a. W (1) Determine a constant c > 0 such that for large n. the interval c c [An - 5" An + ha is an approximate 95% confidence interval for 6. Problem 5 (40 points) A grocery chain claims that at most 10 percent of their 8 oz. Jars of instant coffee contain less than 8 oz. coffee: To test this claim, 20 jars are randomly selected and the contents weighed. The claim is accepted if 4 or fewer then 4 Jars contain too little coffee. Use the table for the binomial distribution function to find the probability that the claim will be rejected if the actual percentage of jars containing too little coffee is (a) 5x I (b) 102 (c) 15% (d) 20% (e) 252 (f) 30% (E) 40% (h) 50% (i) 60%- Skefch the graph of the power function. Tabl- I BINOMIAL DISTRIBUTION FUNCTION I n B(x;n.p)- Z: (Jim—pr” k-O 0.30 0.35 0.40 0.45 0.50 0.[5 0.20 0.25 0.3585 0.lZI6 0.0388 0.0] [5 0.0032 0.0008 0.0002 0.0000 0.0000 0.0000 0.7358 0.39” 0.[756 0.0692 0.0243 0.0076 0.002[ 0.0005 0.000l 0.0000 0.9245 0.6769 0.4049 0.206I 0.09 I 3 0.0355 0.0I ZI 0.0036 0.0009 0.0002 0.984[ 0.8670 0.6477 0.4[ [4 0.2252 0.[07[ 0.0444 0.0[60 0.0049 0.00I3 0.9974 0.9568 0.8298 0.6296 0.4[48 0.2375 0.|[82 0.05I0 0.0[89 0.0059 0.9997 0.9887 0.9327 0.8042 0.6[72 0.4[64 0.2454 01256 0.0553 0.0207 [.0000 0.9976 0.9781 0.9[33 0.7858 0.6080 (”[66 0.2500 0.I299 0.0577 [.0000 0.9996 0.994[ 0.9679 0.8982 0.7723 0.60“) 0.4[59 0.2520 0.I3l6 [.0000 0.9999 0.9987 0.9900 0.9591 0.0867 0.7624 0.5956 0.4[43 0.2517 [.0000 [.0000 0.9998 0.9974 0.90“ 0.9520 08782 0.7553 0.59“ 0.4] [9 [0 [.0000 [.0000 [.0000 0.9994 0.996I 0.9829 0.9468 0.II725 0.7507 0.587“ [I [.0000 [.0000 [.0000 0.9999 0.999I 0.9949 0.9804 0.9435 0.8692 0.7483 [2 [.0000 [.0000 [.0000 [.0000 0.9998 0.9987 0.9940 0.9790 0.94 20 0.8684 [3 [.0000 [.0000 [.0000 [.0000 [.0000 0.9997 0.9985 0.9935 0.9786 0.9423 [4 [.0000 [.0000 [.0000 [.0000 [.0000 [.0000 0.9997 0.9984 0.9936 0.9793 [5 [.0000 [.0000 [.0000 [.0000 I .0000 [.0000 [.0000 0.9997 0.9985 0.994 I I6 [.0000 I .0000 [.0000 I .0000 [.0000 [.0000 [.0000 [.0000 0.9997 0.9987 [.0000 [.0000 [.0000 [.0000 [.0000 [.0000 [.0000 [.0000 [.0000 0.9998 [.0000 [.0000 [.0000 [.0000 [.0000 [.0000 [.0000 [0000 [.0000 [.0000 ...
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