486 Optimal Tests of Hypotheses8.2.2.LetXhave a pdf of the formf(x;θ)=1/θ, 0 <x<θ, zero elsewhere. Let
Y 1 <Y 2 <Y 3 <Y 4 denote the order statistics of a random sample of size 4 from
this distribution. Let the observed value ofY 4 bey 4. We rejectH 0 :θ=1and
acceptH 1 :θ = 1 if eithery 4 ≤^12 ory 4 >1. Find the power functionγ(θ), 0 <θ,
of the test.
8.2.3.Consider a normal distribution of the formN(θ,4). The simple hypothesis
H 0 :θ= 0 is rejected, and the alternative composite hypothesisH 1 :θ>0is
accepted if and only if the observed meanxof a random sample of size 25 is greater
than or equal to^35. Find the power functionγ(θ), 0 ≤θ,ofthistest.
8.2.4.Consider the distributionsN(μ 1 ,400) andN(μ 2 ,225). Letθ=μ 1 −μ 2 .Let
xandydenote the observed means of two independent random samples, each of
sizen, from these two distributions. We rejectH 0 :θ= 0 and acceptH 1 :θ>0if
and only ifx−y≥c.Ifγ(θ) is the power function of this test, findnandcso that
γ(0) = 0.05 andγ(10) = 0.90, approximately.
8.2.5.Consider Example 8.2.2. Show thatL(θ) has a monotone likelihood ratio in
the statistic
∑n
i=1X2
i. Use this to determine the UMP test forH^0 :θ=θ
′,whereθ′is a fixed positive number, versusH 1 :θ<θ′.
8.2.6.If, in Example 8.2.2 of this section,H 0 :θ=θ′,whereθ′is a fixed positive
number, andH 1 :θ =θ′, show that there is no uniformly most powerful test for
testingH 0 againstH 1.
8.2.7.LetX 1 ,X 2 ,...,X 25 denote a random sample of size 25 from a normal dis-
tributionN(θ,100). Find a uniformly most powerful critical region of sizeα=0. 10
for testingH 0 :θ= 75 againstH 1 :θ>75.
8.2.8.LetX 1 ,X 2 ,...,Xndenote a random sample from a normal distribution
N(θ,16). Find the sample sizenand a uniformly most powerful test ofH 0 :θ=25
againstH 1 :θ<25 with power functionγ(θ) so that approximatelyγ(25) = 0. 10
andγ(23) = 0.90.
8.2.9.Consider a distribution having a pmf of the formf(x;θ)=θx(1−θ)^1 −x,x=
0 ,1, zero elsewhere. LetH 0 :θ= 201 andH 1 :θ> 201. Use the Central Limit
Theorem to determine the sample sizenof a random sample so that a uniformly
most powerful test ofH 0 againstH 1 has a power functionγ(θ), with approximately
γ( 201 )=0.05 andγ( 101 )=0.90.
8.2.10.Illustrative Example 8.2.1 of this section dealt with a random sample of
sizen= 2 from a gamma distribution withα=1,β=θ. Thus the mgf of the
distribution is (1−θt)−^1 ,t< 1 /θ, θ≥2. LetZ=X 1 +X 2. Show thatZhas
a gamma distribution withα =2,β=θ. Express the power functionγ(θ)of
Example 8.2.1 in terms of a single integral. Generalize this for a random sample of
sizen.
8.2.11.LetX 1 ,X 2 ,...,Xn be a random sample from a distribution with pdf
f(x;θ)=θxθ−^1 , 0 <x<1, zero elsewhere, whereθ>0. Show the likelihood has
mlr in the statistic
∏n
i=1Xi. Use this to determine the UMP test forH^0 :θ=θ
′
againstH 1 :θ<θ′,forfixedθ′>0.