Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

402 Maximum Likelihood Methods

(a)Show that the likelihood ratio for testingH 0 :θ 1 =θ 2 ,θ 3 =θ 4 against all

alternatives is given by

[n

∑

1

(xi−x)^2 /n

]n/ 2 [m

∑

1

(yi−y)^2 /m

]m/ 2

{[n

∑

1

(xi−u)^2 +

∑m

1

(yi−u)^2

]/

(m+n)

}(n+m)/ 2 ,

whereu=(nx+my)/(n+m).

(b)Show that the likelihood ratio test for testingH 0 :θ 3 =θ 4 ,θ 1 andθ 2 unspec-

ified, againstH 1 :θ 3

 =θ 4 ,θ 1 andθ 2 unspecified, can be based on the random

variable

F=

∑n

1

(Xi−X)^2 /(n−1)

∑m

1

(Yi−Y)^2 /(m−1)

.

6.5.6.LetX 1 ,X 2 ,...,XnandY 1 ,Y 2 ,...,Ymbe independent random samples from
the two normal distributionsN(0,θ 1 )andN(0,θ 2 ).

(a)Find the likelihood ratio Λ for testing the composite hypothesisH 0 :θ 1 =θ 2

against the composite alternativeH 1 :θ 1

 =θ 2.

(b)This Λ is a function of whatF-statistic that would actually be used in this

test?

6.5.7.LetXandYbe two independent random variables with respective pdfs

f(x;θi)=

{(

1

θi

)

e−x/θi 0 <x<∞, 0 <θi<∞

0elsewhere,

fori=1,2. To testH 0 :θ 1 =θ 2 againstH 1 :θ 1
=θ 2 , two independent samples
of sizesn 1 andn 2 , respectively, were taken from these distributions. Find the
likelihood ratio Λ and show that Λ can be written as a function of a statistic having
anF-distribution, underH 0.

6.5.8. For a numerical example of theF-test derived in Exercise 6.5.7, here are
two generated data sets. The first was generated by the R callrexp(10,1/20),
i.e., 10 observations from a Γ(1,20)-distribution. The second was generated by
rexp(12,1/40). The data are rounded and can also be found in the filegenexpd.rda.
(a)Obtain comparison boxplots of the data sets. Comment.

(b)Carry out theF-test of Exercise 6.5.7. Conclude in terms of the problem at

level 0.05.

x: 11.1 11.7 12.7 9.6 14.7 1.6 1.7 56.1 3.3 2.6

y: 55.6 40.5 32.7 25.6 70.6 1.4 51.5 12.6 16.9 63.3 5.6 66.7