Question

Problem 5 (15pts). Suppose that we observe a random sample X. from the density Xn 1 0 2 0, else, where m is a known constant which is greater than zero, and 0>0. (a) Find the most powerful test for testing Ho : θ Bo against b) Indicate how you would find the power of the most powerful test when θ-e-Do not perform (c) Is the resulting test uniformly most powerful for testing Ho :0-00 against Ha :e> et Explain any calculation). why or why not. D at a nuclear power

Answer 1

Note: one required statement is proved at last.

n bf γα nd om sample drocon fom the densit else To test - Now Cike li hood Nouw, ikelihood functon u unde o i under HAs m- EX

KJouw, by Meyman pearson lemmo, he best citral rgian is gien by, or is lve by or al somple points befori ta c egion and 2 k k is non-ngjative mx) Co,) 2 k takin loq. on buth side ide we get 一nlog

Douo since, , Most pourfe esrh hw bu に' o otherwie cwhere, C can be given level sisnificance

tJouw, This statenent is proved. icI 2n pges T hus, 2 2 C 2n 2. n, o 2n 2n 2- Thus, Most uuesful test functron for teti o S iven b

(B) power ff thR Most pour ful -est cu hen Co hen 2 2 に! 2 22 20 "This is 3equired廂rmulation to(find power of thole most roco

iuen by /2 2n o 2 o otheroise And, since, this est funetondoej 4 not defend S on parameter under HA. (i-e. ong "Thus, this tert i,seUni ml most pou) end (UMP) test for 'tert , - -

Notc r, we need to knou the distibudten X20 otheri let Y m Th i-IT 15) Jaco bian/ Now, pd X" 7 IJ ) m-t

-is ie. exponentral dish budrn with mean expte) 2. 2η .