Please quote each assignment separately.The due date for assignment 6 is Wednesday 17OCT2012, 23:59 EST.The due date for assignment 7 is Wednesday 17OCT2012, 23:59 EST Document Preview: Statistics and Biostatistics Ph.D. Theory Qualifier 2005 Answer any four problems. If you answer more.than four questions, . only the first four will be graded. Total time: 3 hours. This is a closed book. closed nOtesú test. I(i~~knsider the family of distribution on (0, 00) with pdf ~ /(:c,0) fX ex:p(-O:ca ), () > 0, where a > 1 is fixed. . . ~Find the normalizing constant for úthe density. V .. f .:.ns~ Does the pdf belong to the regul8.r .eXponential family? Justify your . ~d the :MLE of o based on an ind~pendent sainple of size Tift-om this . . dist:ibution. . . . -~~~h;;;;;;;;:;~~~tm-mF-~-~-.-~=.. . L(d,tfJ) = (d-tfJ)2. Find the form of the Bayes estimator tfJ . b. Let Xl! X2, .;.X, be i.i~d. exponential 0 distribution (I(x, 0) = 0 exp(()x), x > 0), wher:e Ois an unknown parameter. Let Z denote some hypothetical fu-ú ture observation from the same distribution, andúsuppose we wish.to estimate tfJ(O) = P(Z > z),for some given z. . . .. . Suppose we. assume a Gamma( Of., (3) prior (r( 0) ,.;, r~:) Oo,:le-tl8, 0 > 0) for O. :Find the posterior distribution of O,and show that the Bayes estimatorú of tfJis . .. _ ( P+8 .. )a+n . . . tfJ -tI+s .. +z; .: .. where Sf! =I:~=l Xi. . . . ~ 3. .. . . .. Define the term unbiased test. . . . Describ~ brieflyú the notion of UMP unbiased test. , A town supervisor suspects that the.traflic conditionúhave become more hazardous in Newtown than in neighborfug Oldtown, so she records. the number . úof accidents. occurring in each place in the course of one month. Assume that the random variables X and Y denote the number of acc;idents occurring in Newtown and Oldtown in one month period, and that X and Yare independent Poisson distribl,ltions with parameters). andJL respeCtivelYi .oonstruct an UMP unbiased test of size a for thehypotliesis: . Ho: ). 5, JL vs. Ha: ). > JL (Hint: Compute the conditional distribution of X given X + Y) d. Carry out the test wlif)n X = 5 and Y = 2, and Of. = 0.05. 1 ,. – … Attachments: Assignment6.pdf Assignment7.pdf