Stat 543 Exam 3 Spring 2016

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Grant Malone
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1 Stat 543 Exam 3 Sprig 06 I have either give or received uauthorized assistace o this exam. Name Siged Date Name Prited This exam cosists of parts. Do at east 8 of them. I wi score aswers at 0 poits apiece ad cout you best 8 scores (makig 80 poits possibe).

2 . A particuar cotiuous mode for radom pairs ( X, Y ) with parameter γ ( 0, ) γ f ( x, y γ) = exp ( γ x + y ) ( x, y) R π x + y (This "radiay symmetric" desity is costat o circes cetered at the origi.) a) For iid data pairs ( X, Y ),,( X, Y ) has joit pdf idetify a statistic i which there is mootoe ikeihood ratio (ad show that your statistic reay does the job). b) For the = case, fid the UMP size α =.05 test of H 0 : γ vs H a : γ > i as expicit a form as is possibe. (You may reca that chage of variabes to poar coordiates i a doube itegra ivoves repacig dxdy with rdrdθ.)

Presume that the θ, tightest cotour is at 0 ad that they fa off at 5 uits per cotour, so that they are at 0, 5, 0,.

3 . Beow is a cartoo of a cotour pot for a ogikeihood fuctio (, ) for a maximum ikeihood estimate of = ( θ, θ ) ( θ ) max ( θ, θ ) θ θ θ. Fid approximate vaues θ ad the vaues of the profie ogikeihood for =. (The tight cotours are at arge vaues of the ogikeihood. Presume that the θ, tightest cotour is at 0 ad that they fa off at 5 uits per cotour, so that they are at 0, 5, 0,.) ( 70) ( 80) ˆ MLE θ Profie ogikeihood vaues are: ( 30) ( 40) ( 50) ( 60) 3. A method of momets estimate of a parameter θ is ˆ θ = 7. The ogikeihood fuctio, ( θ ), is compicated, but it is possibe to fid (umerica) first ad secod derivatives ( 7) = ad ( ) 7 =.. Fid a correctio/improvemet o the method of momets estimate based o these derivatives. 3

4 4. A ogikeihood for a parameter vector = ( θ, θ ) θ is approximatey quadratic ear ˆ MLE θ =,, (, ) θ θ θ + θθ θ + θ Based o a arge sampe approximatio to the distributio of a vector MLE, use the Wad method to give a approximatey 95% two-sided cofidece iterva for θ. 5. Suppose that k ( λ ) is the smaest iteger k so that for X Poisso ( λ ), P[ X k].05 thik of pottig k ( λ ) ad producig a icreasig step fuctio takig iteger vaues. If I observe a. You may Poisso variabe X, exacty how shoud I use that step fuctio to produce a cofidece iterva for λ with associated cofidece at east 95%? 4

5 6. For parameters 0< c < ad 0 p f < <, a pdf o ( ) ( x c, p) 0, is of the form p for 0 < x < c c = p for c< x< c (The desity is costat to the eft ad the to the right of c, assigig probabiities p = P[ X c] ad p = P[ X > c].) Fid the ikeihood ratio statistic for testig H 0 : c =.5 based o iid observatios from this desity, X,, X i as expicit a form as is possibe. 5

6 7. A discrete distributio o the positive itegers has pmf f ( x p) ( p p ) x=, = x = ( ) p p x 3,4, Fid a -dimesioa sufficiet statistic based o a sampe X, X,, X iid from this distributio, ad say carefuy how you kow that it is sufficiet. 8. For a distributio o {,,, M} specified by vaues p, p,, pm, the quatity E M = ( p) p ( p ) m= m is caed the "etropy" of the distributio. Usig a statistica "iformatio" measure, show that this is o arger tha ( M ) (the etropy of the uiform distributio). (I the above expressio the covetio that 00 ( ) = 0is i force.) m 6

7 9. A pmf o the itegers with a iteger parameter, θ, is f x I x x ( θ) = [ = θ or = θ + ] For X ad X iid from this distributio, compare MSE's for the two estimators of θ, ˆ θ X + if X = X ˆ ad θ= θ X if X X = 0. Suppose that X U θ, θ + estimator of θ uder squared error oss. ad that a prior distributio for θ is ( ) N 0,. Fid the Bayes 7

EECS564 Estimation, Filtering, and Detection Hwk 2 Solns. Winter p θ (z) = (2θz + 1 θ), 0 z 1

EECS564 Estimation, Filtering, and Detection Hwk 2 Solns. Winter p θ (z) = (2θz + 1 θ), 0 z 1 EECS564 Estimatio, Filterig, ad Detectio Hwk 2 Sols. Witer 25 4. Let Z be a sigle observatio havig desity fuctio where. p (z) = (2z + ), z (a) Assumig that is a oradom parameter, fid ad plot the maximum