2010 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1302 Probability and Statistics II May 8, 2010 Timezs9z30 a.m. - 11:30 a.m. Only approved calculators as announced by the Examinations Secretary can be used in this examination. It is candidates’ responsibility to ensure that their calculator operates satisfactorily, and candidates must record the name and type of the calculator used on the front page of the examination script. Answer ALL FOUR questions. Marks are shown in square brackets. 1. Let f be a density function such that B‘le’”/’9, u > 0, O, u S 0, f (ulfl) = { where B denotes a positive parameter. It is known that /oouf(ulfi)du = ,3 and /00 u2f(u]fl) du == 2fi2. 0 0 Let X, Y be two random variables such that X has the density function f and, conditional on X = x, Y has the density function f (-|0$‘1), for an unknown parameter 6 > 0. Answer the following questions based on the pair of random observations (X, Y). (a) Show that the likelihood function of 0 is given by = g—ZX e—0_IX(1+Y). [3 marks] » (b) Find a scalar sufficient statistic for «9. [3 marks] (0) Show that the maximum likelihood estimator (mle) d of 0 is (5: X(1 + Y). 2 [3 marks] (d) Show that both X and X Y are unbiased estimators of 6. Hence verify that the mle 0 is also an unbiased estimator of 6. [4 marks] S&AS: STAT1302 Probability and Statistics II 2 (e) Show that the Fisher information about 0 is “6) = 2 6‘2. [4 marks] (f) Which of the following three unbiased estimators, X, XY, or é: ———X(12+Y), is/ are efficient? Why? - [8 marks] [Total: 25 marks] 2. Country A owns 8 nuclear plants which she claims are built for peaceful pur- poses. Country B suspects that some of those nuclear plants are used for producing nuclear weapons, and is considering an invasion of A. To help her make the decision, country B sends a secret agent into A to investigate the true functions of two randomly selected nuclear plants there. If any one of the two investigated nuclear plants is found to produce nuclear weapons, country B will invade country A; otherwise, she will abandon the idea of invasion, trust— ing that A’s nuclear plants are all used for peaceful purposes. The secret agent is absolutely professional, and never makes any mistake in his investigations. Let 0 6 {0,1,.. . ,8} be the number of weapon-producing nuclear plants in country A. The problem of country B can be viewed as a decision between the hypotheses H':6=O and H*:6>0. Let X be the number of nuclear plants found to be producing nuclear weapons . by the secret agent. Thus X is a random variable with possible values 0, 1 or 2. (a) Show that, for 0 6 {0,1, . . . ,8}, 6(15 — 0) 1? X = O 6 = 1 — —. V < l > 56 [3 marks] (b) Suppose the decision problem of country B is formulated as a hypothesis test of H0 : H' against H1:H*. (i) What is the type of error made by B if she does not invade A when some of the nuclear plants in A are indeed producing weapons? [3 marks] (ii) Describe the critical region adopted by country B in terms of the observation X. , [4 marks] S&AS: STAT1302 Probability and Statistics II 3 (iii) Calculate the size of the test based on the critical region in (b)(ii). [4 marks] (0) Suppose the decision problem of country B is formulated as a hypothesis test of H0 : H* against H1 : H’. (i) What is the type of error made by B if she does not invade A when some of the nuclear plants in A are indeed producing weapons? [3 marks] (ii) Describe the critical region adopted by country B in terms of the observation X. [4 marks] (iii) Calculate the size of the test based on the critical region in (c) (ii). [4 marks] [Total: 25 marks] 3. Let X be a random variable distributed under the density function N30) f($l9) = F(9)P(29) . 0, 53 ¢ (0:1): cue—1(1 — mfg—1, :c 6 (0,1), for some unknown parameter 0 > 0. The gamma function l"(-), defined by I‘(a)=/ ua‘le’“du, 0 satisfies I‘(a) = (a — 1)! if a is a positive integer. (a) Suppose that we wish to test H020=1 against le6=2 by means of the most powerful test, based on a single observation of X. (i) Show that the critical region of the test has the form {:13 E (0,1) : 93(1 — :13)2 > k}, for some constant k. [3 marks] 1 (ii) Show that if X is observed to be 7, the p—value given by the test is alto 27 [8 k ] e —. Q“ 49 mar s x(1—x)2—%= S&AS: STAT1302 Probability and Statistics II 4 (b) Denote by X1, . . . ,Xn a set of n independent observations of X, for a large sample size n. Describe how you would compute the p-value for the generalised likelihood ratio test of H026=1 against H126741. You may denote the maximum likelihood estimator of 0 by 6,, Without giving its closed—form expression. [4 marks] [Total: 15 marks] 4. The lengths of 600 sentences randomly sampled from the book “The Work, Wealth and Happiness of Mankind”, written by H. G. Wells, are counted and denoted by X1, . . . ,XGOO. Here the length of a sentence is defined to be the number of words in that sentence. A summary of the data is given in the contingency table below: ln(sentencelength)€ (—-oo,1] (1,1.5] (1.5,2] (2,2.5] (253] No. of sentences 1 7 23 81 193 1n(sentencelength)€ (3,3.5] (3.5,4] (4,4.5] (4.5,5] (5,00) No. of sentences 167 104 23 1 0 Under a proposed model (*), ln(X1), . . . ,ln(X600) are independent and identi- cally distributed according to the normal distribution with unknown mean u and unknown variance 02. When answering (a)(iii) and (b) (ii), you may refer to the following statistical r?- | O. ._. 9? 5% upper quantiles of x? 1 2 6 7 8 9 10 upper quantile 3.84 5.99 12.59 14.07 15.51 16.92 18.31 591 592 596 597 598 599 600 648.66 649.71 653.90 654.95 656.00 657.05 658.09 upper quantile (a) Define f1(M,U) = P(ln(X1) S 1 I model correct), f10(p,a) = lP’(ln(X1) > 5 I model (*) correct), and, forj= 2,3,...,9, fj(Ma0) = P <1n(X1) S — model correct) . S&AS: STAT1302 Probability and Statistics II > 5 (i) Write down expressions for f1(,a, a), f2(,u., a), . . . ,f10(p, a) in terms of the standard normal cumulative distribution function (I). [4 marks] (ii) Describe how you would estimate ,u and a under model (*), based on the count data given in the contingency table. You are n_ot required to calculate the estimates numerically. [4 marks] (iii) Suppose that ,a and a are estimated to be 3 and 0.6, respectively, under model The following table gives the values of 600 fj(3, 0.6), forj= 1,2,...,10. 600fj(3,0.6) 0.26 3.50 25.17 93.32 179.13 600fj(3,0.6) 178.32 92.06 24.60 3.39 0.25 l Conduct a Pearson chi-squared test to test if model is acceptable for explaining the observed data at the 5% significance level. [8 marks] (b) Suppose that model is correct, and that we calculate from the data the following statistics: 600 600 656 2mm) = 3.0177 and Z {1n(X,-) — 3.0177}2 = 209.7922. i=1 i=1 (i) Calculate an unbiased estimate of 02. [3 marks] (ii) Calculate a 95% lower confidence bound for 02. [6 marks] (c) An anonymous manuscript was recently discovered in a library. We sus- pect that the manuscript is authored by H. G. Wells, and want to carry out a hypothesis test of our conjecture at the 5% significance level, based on this manuscript and the sentence—length data obtained from the book “The Work, Wealth and Happiness of Mankind”. Suggest a step—by-step procedure for the test. [10 marks] [Total: 35 marks] ...
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2010 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS...

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