8.3. Likelihood Ratio Tests 499(b)Show that the likelihood ratio for testingH 0 :θ 3 =θ 4 withθ 1 andθ 2 unspec-
ified can be based on the test statisticFgiven in expression (8.3.9).8.3.12.LetY 1 <Y 2 <···<Y 5 be the order statistics of a random sample of size
n= 5 from a distribution with pdff(x;θ)=^12 e−|x−θ|,−∞<x<∞, for all realθ.
Find the likelihood ratio test Λ for testingH 0 :θ=θ 0 againstH 1 :θ =θ 0.
8.3.13.A random sampleX 1 ,X 2 ,...,Xnarises from a distribution given by
H 0 :f(x;θ)=
1
θ, 0 <x<θ, zero elsewhere,or
H 1 :f(x;θ)=
1
θe−x/θ, 0 <x<∞, zero elsewhere.Determine the likelihood ratio (Λ) test associated with the test ofH 0 againstH 1.8.3.14. Consider a random sampleX 1 ,X 2 ,...,Xnfrom a distribution with pdf
f(x;θ)=θ(1−x)θ−^1 , 0 <x<1, zero elsewhere, whereθ>0.
(a)Find the form of the uniformly most powerful test ofH 0 :θ= 1 against
H 1 :θ>1.
(b)What is the likelihood ratio Λ for testingH 0 :θ= 1 againstH 1 :θ =1?
8.3.15.LetX 1 ,X 2 ,...,XnandY 1 ,Y 2 ,...,Ynbe independent random samples from
two normal distributionsN(μ 1 ,σ^2 )andN(μ 2 ,σ^2 ), respectively, whereσ^2 is the
common but unknown variance.
(a)Find the likelihood ratio Λ for testingH 0 :μ 1 =μ 2 = 0 against all alterna-
tives.
(b)Rewrite Λ so that it is a function of a statisticZwhich has a well-known
distribution.
(c)Give the distribution ofZunder both null and alternative hypotheses.8.3.16.Let (X 1 ,Y 1 ),(X 2 ,Y 2 ),...,(Xn,Yn) be a random sample from a bivariate
normal distribution withμ 1 ,μ 2 ,σ^21 =σ^22 =σ^2 ,ρ=^12 ,whereμ 1 ,μ 2 ,andσ^2 >0are
unknown real numbers. Find the likelihood ratio Λ for testingH 0 :μ 1 =μ 2 =0,σ^2
unknown against all alternatives. The likelihood ratio Λ is a function of what
statistic that has a well-known distribution?
8.3.17.LetXbe a random variable with pdffX(x)=(2bX)−^1 exp{−|x|/bX},for
−∞<x<∞andbX>0. First, show that the variance ofXisσX^2 =2b^2 X.Next,
letY, independent ofX,havepdffY(y)=(2bY)−^1 exp{−|y|/bY},for−∞<x<∞
andbY>0. Consider the hypotheses
H 0 :σ^2 X=σ^2 YversusH 1 : σX^2 >σ^2 Y.
To illustrate Remark 8.3.2 for testing these hypotheses, consider the following data
set (data are also in the fileexercise8316.rda). Sample 1 represents the values
of a sample drawn onXwithbX= 1, while Sample 2 represents the values of a
sample drawn onYwithbY= 1. Hence, in this caseH 0 is true.