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STAT0100 (2002Dec)

STAT0100 (2002Dec) - THE UNIVERSITY OF HONG KONG DEPARTMENT...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT0100 STATISTICAL METHODS STATO604 STATISTICAL MODELLING (I) STAT0605 MATHEMATICAL STATISTICS December 30, 2002 Time: 6:30 p.m. - 8:30 p.m. Candidates taking examinations that permit the use of calculators may use any cal- culator which fulfils the following criteria: (a) it should be self-contained, silent, battery-operated and pocket-sized and (b) it should have numeral—display facilities only and should be used only for the purposes of calculation. It is the candidate’s responsibility to ensure that the calculator operates satisfactorily and the candidate must record the name and type of the calculator on the front page of the examination scripts. Lists of permitted/prohibited calculators will not be made available to candidates for reference, and the onus will be on the candidate to ensure that the calculator used will not be in violation of the criteria listed above. Answer ALL questions. Marks are shown in square brackets. 1. (a) At a shooting competition, a shooter is allowed to fire n rounds from a naval gun at a fixed target. The first round must be fired from a distance of 100 metres. If the jth round misses the target by 3: metres, then the (j + 1)th round must be fired from a distance of 1003: metres, for j: l,2,...,n~—l. Assume that if the shooter fires from a distance of d metres, he will miss the target by a distance exponentially distributed with the density function (bl/c0591”, a: > 0, 0, 0:30, f($l9:d) ={ for some unknown parameter 9 > 0. Suppose that the shooter’s jth shot misses the target by X, metres, for j=1,2,...,n. (i) Show that the likelihood function of 6 based on X1, . . . ,Xn has the expression 60(6) : 9"exp{—100-19(X1 + Xg/X1+---+ Xn/X -0}. [4 marks] S&AS: STATOlOO Stat. Methods / 0604 Stat. Modelling (I) / 0605 Math. Stat. 2 (ii) Identify, by applying the factorization criterion to 66(6) in (a)(i) or otherwise, a scalar sufficient statistic for 0. [2 marks] (b) Suppose that the rule of the competition is modified such that after fir- ing each round, the shooter has to toss a coin, the outcome of which will determine the distance from which he will fire the next round. If a head comes up, the shooter will fire the next round from a distance of 100 me— tres; otherwise he will fire from a distance of 1003: metres if the previous round misses the target by 3: metres. The coin is specially designed such that its head comes up with a probability of 1 — 6—9. The first round is still fired from a distance of 100 metres. The same shooter as in (a) enters this new competition and misses the target by Yj metres at the jth shot, forj = 1,2, . . . ,n. (i) Show that the likelihood function of 6 based on Y1, . . . , Yn and the outcomes of the tosses of the coin has the expression n_1 6—9 23:31} me) = 910.84) < > x 1—6-9 exp {—100‘16 (Y1 + 3/2/Y111+---+ Yrs/31311)} , Where I]- : 0 if the jth toss of the coin returns a head and = 1 otherwise. [7 marks] (ii) Deduce a bivariate sufficient statistic for 9 based on the results in (NO)- [4 marks] (c) Let n > 1 be fixed. Suppose that at the competition described in (a), the statistic X1 + X2/X1 + - - ' + Xn/Xn-1 is observed to be :13; whereas at the competition described in (b), tails come up at all n — 1 tosses of the coin and the statistic Y1 + Y2/Y1 + - - ‘ + Yn/Yn-1 is observed to be y = x. According to the Likelihood Principle, would you expect to make ex- actly the same inference about 0 based on the data observed from either competition? Explain. [5 marks] 2. Let X = (X 1, . . . , Xn) constitute a random sample drawn from the density function 2 a/7r exp {405 — a(a: + fl/$)2}, as > 0, f (933ml?) = 0, 3? S 0, where a, fi are positive parameters. S&AS: STATOlOO Stat. Methods/0604 Stat. Modelling (I) / 0605 lVIath. Stat. 3 (a) Write down an expression for the loglikelihood function of (0,13) based on X. [2 marks] (b) If both a and 5 are unknown, find the maximum likelihood estimators of a and fl based on X. [6 marks] (0) Suppose that fl = 1 and a is unknown. (i) Find the Fisher information about a contained in X. [4 marks] (ii) Find the maximum likelihood estimator, all say, of (1 based on X. [3 marks] (iii) Describe the large-sample behaviour of 6:1. [4 marks] (iv) Describe how you would calculate the standard error of (341. [3 marks] (d) Suppose that a = 1 and B is unknown. (i) Find the maximum likelihood estimator, 31 say, of 5’ based on X. [3 marks] (ii) Describe how you would calculate the standard error of Bl. [4 marks] 3. Let X1, . . . , Xn be independent random variables distributed with the common density function (0—1):c’9, w> 1, 0, cc 3 1, f($19) = { for some unknown parameter 6 > 1. (a) It is known that the moment generating function of the gamma (5, 1) distribution, for any ,8 > 0, is A/Ig(t) = (1 ~ U”, for |t| <1. Show that (9 — 1) In H; Xi has the gamma (n, 1) distribution. [5 marks] (b) Suppose that we wish to test the hypotheses 1110:6360 vs H1:6>60, for a given constant 60 > 1. (i) Show that the given parametric model has monotone likelihood ratio in some statistic T = T(X1, . . . ,Xn). Specify T. [5 marks] S&AS: STATOlOO Stat. Methods / 0604 Stat. Modelling (I) / 0605 lVIath. Stat. 4 (ii) Using (a) and (b)(i), suggest a test of size 5%, in terms of the 5% quantile of the gamma (n, 1) distribution, for testing H0 against H1. [5 marks] (c) Suppose that we wish to test the hypotheses H010=60 vs H1:9>1, for a given constant (90 > 1. (i) Show that the maximum likelihood estimator of (9 is fin = 1+n/lnHXi. i=1 [3 marks] (ii) Describe the generalized likelihood ratio test of size 5% for testing H0 against H1. [4 marks] (d) Using (b)(ii), construct a 95% lower confidence bound for 6. [4 marks] (e) Using (c) (ii), construct an approximate 95% confidence set for (9. Describe the shape of this confidence set. [6 marks] 4. In a clinical trial, a group of 8 patients were treated with a new medicine and their cholesterol levels (in mg/ ml) were recorded one week before and after the treatment as follows: Patient 1 2 3 4 5 6 7 8 Before treatment 2.8 3.4 2.2 3.7 3.5 3.8 3.0 2.3 2.6 3.3 2.3 3.8 3.0 3.5 2.7 2.2 After treatment Denote by X j and Y} the cholesterol levels of patient j taken one week before and after the treatment respectively, for j = 1, 2, . . . ,8. (a) Is it appropriate to assume that (X1, , . . ,X8) and (Y1, . . . ,Yg) constitute two independent random samples? Explain. [3 marks] (b) Assume that Y]- = Xjezv', j=1,2,...,8, where Z1, . . . , Zg denote 8 independent normal random variables with common mean ,u and common variance 02, for some unknown parameters ,u E (~oo,oo) and a > 0. We wish to investigate the effectiveness of the new medicine in lowering cholesterol level. S&AS: STAT0100 Stat. Methods / 0604 Stat. Modelling (I) / 0605 Math. Stat. 5 (i) Set up an appropriate pair of null and alternative hypotheses, in terms of p, to address the issue. Explain your Choice of hypotheses. [4 marks] (ii) Conduct a t-test of size 5% for testing the hypotheses set up in (b)(i) Do you think that the new medicine succeeds in lowering Cholesterol level? [10 marks] Hint: Upper quantiles a: of Student’s t—distributions degrees of freedom, f 1P(tf > m) 6 7 8 14 15 16 0.025 2.4469 2.3646 2.3060 2.1448 2.1315 2.1199 0.05 1.9432 1.8946 1.8595 1.7613 1.7531 1.7459 0.1 1.4398 1.4149 1.3968 1.3450 1.3406 1.3368 ...
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