EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY

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EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA, 016 MODULE : Statistical Iferece Time allowed: Three hours Cadidates should aswer FIVE questios. All questios carry equal marks.

1 EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA, 016 MODULE : Statistical Iferece Time allowed: Three hours Cadidates should aswer FIVE questios. All questios carry equal marks. The umber of marks allotted for each part-questio is show i brackets. Graph paper ad Official tables are provided. Cadidates may use calculators i accordace with the regulatios published i the Society's "Guide to Examiatios" (documet Ex1). The otatio log deotes logarithm to base e. Logarithms to ay other base are explicitly idetified, e.g. log 10. Note also that r is the same as C. r 1 GD Module 016 This examiatio paper cosists of 1 prited pages. This frot cover is page 1. Questio 1 starts o page. RSS 016 There are 8 questios altogether i the paper.

2 1. 1 X, X,..., X is a radom sample from the uiform distributio betwee ad 1 (i.e. 1 f ( x) (1 ) for x 1), where (< 1) is a ukow parameter. Deote the sample mea by X. (i) Show that the method of momets estimator, ˆ, of is X 1. Show that ˆ is ubiased ad fid its variace. (iii) State the Cramér-Rao lower boud for the variace of ubiased estimators, ad explai why it is ot applicable whe estimatig defied above. (iv) Let Y mi{ X1, X,..., X }. By fidig P( Y y), show that the probability desity fuctio of Y is (1 y) f ( y) (1 ) 1 for y 1. (v) A estimator 1 c(1 Y) is to be used to estimate, where c is a costat to be chose. Show that the mea square error of, E[( ) ], is miimised whe c. 1 (6) [You are give that (1 ) E(1 Y) 1 ad (1 ) E[(1 Y) ].]

3 . (i) Explai what is meat by the ivariace property of maximum likelihood estimators (MLEs). () Cosider a biomial experimet with trials ad probability of success i which y successes are observed. Fid the MLE of. (iii) A radom sample X1, X,..., X of observatios is take from a expoetial 1 x distributio with probability desity fuctio f ( x) e, x 0. The actual values of the observatios are ot available; all that is kow is that y of the values are less tha a threshold T ad the remaiig ( y) values are greater tha T. Use part ad the ivariace property of MLEs to show that the MLE ˆ y of, give this restricted iformatio, is T 1 log 1. (8) (iv) State the asymptotic distributio of ˆ ad hece write dow a expressio for a approximate 95% cofidece iterval for, defiig ay terms that appear i your expressio. 3

4 3. (i) A radom sample X1, X,..., X is available from a distributio with probability desity fuctio (pdf) f ( x; ), where is a sigle ukow parameter. Defie what is meat by a statistic ad by a sufficiet statistic ad explai what sufficiecy meas. A radom variable X is said to belog to the oe-parameter expoetial family of distributios if its pdf ca be writte i the form f ( x; ) exp{ A( ) B( x) C( x) D( )} where A( ), D( ) are fuctios of the sigle parameter (but ot x) ad Bx ( ), C( x ) are fuctios of x (but ot ). Write dow the likelihood fuctio, give a radom sample X1, X,..., X from a distributio with such a pdf. () (iii) (iv) (v) If the likelihood ca be expressed as the product of a fuctio which depeds o ad which depeds o the data oly through a statistic T ( X1, X,..., X ) ad a fuctio that does ot deped o, the it ca be show that T is sufficiet for. Use this result to show that B( xi ) is a sufficiet statistic i1 for i the oe-parameter expoetial family of part. The Rayleigh distributio has pdf x x f ( x; ) exp, x 0, 0. Show that this distributio is a member of the oe-parameter expoetial family ad hece fid a sufficiet statistic for. I Bayesia iferece defie what is meat by a prior distributio ad by a cojugate prior distributio. Show that the family of distributios with pdfs proportioal to 1 exp, 1 0, 0 is cojugate for the family of Rayleigh distributios defied i part (iv). (8) 4

5 4. (i) Defie what is meat by a geeralised likelihood ratio test. (iii) (iv) Show how the test statistic from a geeralised likelihood ratio test, with a simple ull hypothesis for a sigle parameter, ca be used to costruct a approximate cofidece iterval for the parameter of the form ˆ 1 l( ; x) l( ; x ) χ, where l( ; x ) is the log-likelihood fuctio ad χ 1; 1; is a appropriate critical value from the χ distributio with 1 degree of freedom. (6) The umber of mior accidets per moth i a large factory ca be assumed to have a Poisso distributio with mea. Obtai the maximum likelihood estimator of give a radom sample of observatios from this distributio. Over a log period, the mea umber of mior accidets per moth has bee aroud 5. Some of the safety procedures i the factory have chaged recetly. I the first 9 moths after the chage, the umbers of mior accidets were 8, 3,,, 3, 1,, 5, 1. Use a geeralised likelihood ratio test to test the ull hypothesis that 5 agaist a two-sided alterative at the 5% sigificace level. (8) Show that the edpoits of a approximate 95% iterval for are the solutios of the equatio 3log A radom sample of observatios X1, X,..., X is available from a probability distributio with probability desity fuctio f ( x; ), where is a sigle ukow parameter. (a) Defie a pivotal quatity ad explai how to use such a quatity to costruct a (frequetist) cofidece iterval for. (6) (b) (c) Explai how to costruct a credible iterval (Bayesia cofidece iterval). Explai how to costruct a bootstrap percetile cofidece iterval. Give iterpretatios for the itervals i (a) ad (b), cotrastig those iterpretatios. 5

6 6. Give a radom sample X1, X,..., X from a distributio with probability desity fuctio f ( x ), it is required to test the hypothesis H 0 : f ( x) f0( x), where f 0 ( x ) is a kow fuctio, agaist the geeral alterative H 1: f ( x) f0( x). (i) Defie the test statistic used i the oe-sample Kolmogorov-Smirov test of these hypotheses. The data below are times to breakdow, x, of five mechaical compoets. It is required to test whether they could have bee geerated by a expoetial distributio with mea. Use the oe-sample Kolmogorov-Smirov test to examie this hypothesis. [You are give that the 5% critical value for this test is ad the 1% 1x e critical value is You are also give 1 for each of the five values of x, as follows. x x e ] (iii) It is ow required to test H 0 :m m0 agaist H 1:m m0 usig the data ad assumptios i part, for some measure of locatio m. It is suggested that either the t test or the sig test might be used. Discuss which measure of locatio would be used for each of these tests ad which test it would be most appropriate to carry out. (8) (iv) Use the sig test to ivestigate whether the data i part could have come from a distributio with mea, assumig that the distributio is expoetial. [You eed ot calculate the p-value associated with this test.] 6

7 7. (a) Give two simple hypotheses, H 0, H 1, defie what is meat by the odds of H 0 ad by the Bayes factor of H 0 compared to H. 1 Show that i Bayesia iferece whe comparig two simple hypotheses, the posterior odds equals the product of the prior odds ad the Bayes factor. (b) A radom sample X1, X,..., X is available from a Poisso distributio with mea (> 0). It is required to test the ull hypothesis H 0 : 5 agaist the alterative hypothesis H 1 : 10. (i) Show that the Bayes factor is 5 e (0.5). x i Show that the posterior odds of the ull hypothesis will be greater tha 5 1 the prior odds if x, where x xi. log i 1 Suppose ow that the ull hypothesis is as above, but the alterative hypothesis is H 1 : 5, ad the prior distributio of uder H 1 is expoetial with mea 5. (iii) Show that the Bayes factor is 5 (5 1) 1 e (1 x ) i x i. [Hit: recall that the gamma fuctio ca be writte as 1 x ( ) x e dx. 0 ] (7) 7

8 8. A compay that maufactures a arrow rage of products is operatig i a ecoomy i ecoomic recessio ad is faced with a persistet drop i sales. The maagers are cosiderig five possible strategies to cope with the crisis, as follows. d 1 : cotiue as at preset d : cotiue with preset products aloe, make 10% of the workforce redudat d 3 : cotiue with preset products aloe, make 0% of the workforce redudat d 4 : start small-scale productio of ew products, make 10% of the workforce redudat d 5 : start large-scale productio of ew products, keep preset workforce Four states of ature are evisaged, as follows. 1: ecoomic recessio cotiues, small market for ew products : ecoomic recessio cotiues, large market for ew products 3: ecoomic recovery, small market for ew products 4: ecoomic recovery, large market for ew products By takig ito accout, for each state of ature, expected profits o the old ad ew products, together with redudacy costs, the maagemet costructs a table of utilities as follows, where utility = expected profit i millio d d d d d (i) Defie what is meat by a iadmissible strategy ad by a maximi strategy. Which of the five strategies are (a) (b) iadmissible, maximi? Suppose that the prior probabilities for cotiued recessio ad for recovery are 1 ad 1 1 respectively, that the prior probabilities for large ad small demad are ad 1 respectively, ad that the evets 'cotiued recessio' ad 'large demad' are idepedet. Defie the Bayes strategy ad fid the Bayes strategy if ad 0.5. Questio 8 cotiues o the ext page 8

9 (iii) If is fixed at 0.5, for what values of 1, if ay, are d 3, d 4, d 5, respectively, Bayes strategies? (iv) A ecoomic cosultacy firm offers to provide advice o the likely demad for the ew product, for a fee of If the firm advises that demad will be large, there is a probability of 0.95 that the advice is correct; if the advice is that demad will be small, there is a probability of 0.90 that it is correct. Assumig that ad 0.5, as i part, the posterior probabilities of 1,, 3, 4 whe the advice is that there will be large demad are ,,, respectively, where is the probability that the advice is that demad will be large. Whe the advice is that there will be large demad, the Bayes strategy is d 5 with utility For the case whe the advice is that demad will be small, fid the posterior probabilities of the 4 states of ature, ad determie which strategy is Bayes i this case. [You eed ot evaluate.] Calculate whether the expected extra utility gaied by usig the cosultacy firm exceeds their fee, ad by how much. (7) 9

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EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY

EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY HIGHER CERTIFICATE IN STATISTICS, 017 MODULE 4 : Liear models Time allowed: Oe ad a half hours Cadidates should aswer THREE questios. Each questio carries