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ONLY A) B) D) 4 Let X be a single observation from the density f(x; 0)=...

Question:

4 Let X be a single observation from the density f(x; 0)= Ox® -110, 1)(x), where 0 >0. (a) In testing Ho: 0 <1 versus H 1:8 >ONLY A) B) D)

4 Let X be a single observation from the density f(x; 0)= Ox® -110, 1)(x), where 0 >0. (a) In testing Ho: 0 <1 versus H 1:8 > 1, find the power function and size of the test given by the following: Reject H , if and only if X > . (6) Find a most powerful size-a test of Ho:8=2 versus H 1:0= 1. (c) For the loss function given by [(do; 2) = f(d1; 1)=0, ((do; 1)=((dı; 2)=1, find the minimax test of H.:0=2 versus H 1:0= 1. (d) Is there a uniformly most powerful size-a test of Ho:8 > 2 versus H 1:8< 2? If so, what is it? (e) Among all possible simple likelihood-ratio tests of H.: 0= 2 versus H 1: 0= 1, find that test that minimizes a + B, where a and B are the respective sizes of the Type I and Type II errors. () Find the generalized likelihood-ratio test of size a of Ho: Q=1 versus H 1:0 +1.

Answers

Let x be a Saluhien: 4 single Olsesdalian from the densidy function an f(0) = {cox 0-1 ~ Os xal í Olherwise 0 where oro The lC as We wants to lista Hoi cosi vs Hiori Lot la critical region (way) C Xa xz 133 Let 7(e) dee the powerfuuchos of the lisp aLet & be the size of the list, & P(x? / (Ho) - ll dx 42 (2) / = -1 = o.so tencer size of the list an X = 0.50We want to lest Hoid=2 VB Hino=1 The likelihood funchie heo (0-11loga Theems likelihood funcliba inder Hon Lo=al loga ། The l& C > - woguz log(ek) - Logx where ce logek) Ver to be lo most powerful testy W = { cogxi (loga) = Let Y = -logxn than criticLet & be the s size of the rest X = P(Y=C(Ho) Then? x=p(44746) = 7, : 465 M.2 » C= 4 7 (-0,2) en as a (pencer Ma most powerfu@ The likelihood function Oc1 h = 0x (6-1) Loga L = o e The voglikelihood femelisin cogha logco + (0-1) Loga é + loga For MLEWe wanig to list Horong VS Hoca Then, likelilood function unde Ho as loqu Los 4 e The lirelitwood furelia under Hia (cê-1) loLoga Qe - cô logx - Loga e a logy = 2 e Choge) el-lope 172609C = (2 logx) e (+2(loge) - 2 (- loga) a Let T = (cogx) 0 Then delhen. Theur ceuforudy most pouresful Ori ücal neğion (Bay) Cal Ca{ti Tak} Let a be the size of the list x= P(TZK(Ho) > x=P(4T

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