Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

384 Maximum Likelihood Methods

6.3.8.Using the results of Example 6.2.4, find an exact sizeαtest for the hypotheses

(6.3.21).

6.3.9.LetX 1 ,X 2 ,...,Xnbe a random sample from a Poisson distribution with

meanθ>0.

(a)Show that the likelihood ratio test ofH 0 :θ=θ 0 versusH 1 :θ =θ 0 is based

upon the statisticY=

∑n

i=1Xi. Obtain the null distribution ofY.

(b)Forθ 0 =2andn= 5, find the significance level of the test that rejectsH 0 if

Y≤4orY≥17.

6.3.10.LetX 1 ,X 2 ,...,Xnbe a random sample from a Bernoullib(1,θ) distribu-
tion, where 0<θ<1.

(a)Show that the likelihood ratio test ofH 0 :θ=θ 0 versusH 1 :θ =θ 0 is based

upon the statisticY=

∑n

i=1Xi. Obtain the null distribution ofY.

(b)Forn= 100 andθ 0 =1/2, findc 1 so that the test rejectsH 0 whenY≤c 1 or

Y ≥c 2 = 100−c 1 has the approximate significance level ofα=0.05.Hint:

Use the Central Limit Theorem.

6.3.11.LetX 1 ,X 2 ,...,Xnbe a random sample from a Γ(α=4,β=θ) distribu-

tion, where 0<θ<∞.

(a)Show that the likelihood ratio test ofH 0 :θ=θ 0 versusH 1 :θ =θ 0 is based

upon the statisticW=

∑n

i=1Xi. Obtain the null distribution of 2W/θ^0.

(b)Forθ 0 =3andn= 5, findc 1 andc 2 so that the test that rejectsH 0 when

W≤c 1 orW≥c 2 has significance level 0.05.

6.3.12. LetX 1 ,X 2 ,...,Xn be a random sample from a distribution with pdf
f(x;θ)=θexp

{

−|x|θ

}
/2Γ(1/θ),−∞<x<∞,whereθ>0. Suppose Ω =
{θ:θ=1, 2 }. Consider the hypothesesH 0 :θ= 2 (a normal distribution) versus
H 1 :θ= 1 (a double exponential distribution). Show that the likelihood ratio test
can be based on the statisticW=

∑n

i=1(X

2

i−|Xi|).

6.3.13.LetX 1 ,X 2 ,...,Xnbe a random sample from the beta distribution with
α=β=θand Ω ={θ:θ=1, 2 }. Show that the likelihood ratio test statistic
Λfortesting∑ H 0 :θ=1versusH 1 : θ= 2 is a function of the statisticW =
n
i=1logXi+

∑n

i=1log (1−Xi).

6.3.14.Consider a location model

Xi=θ+ei,i=1,...,n, (6.3.25)

wheree 1 ,e 2 ,...,enare iid with pdff(z). There is a nice geometric interpretation
for estimatingθ.LetX=(X 1 ,...,Xn)′ande=(e 1 ,...,en)′be the vectors of
observations and random error, respectively, and letμ=θ 1 ,where 1 is a vector
with all components equal to 1. LetV be the subspace of vectors of the formμ;