Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

6.5. Multiparameter Case: Testing 403

6.5.9.Consider the two uniform distributions with respective pdfs

f(x;θi)=

{ 1

2 θi −θi<x<θi,−∞<θi<∞

0elsewhere,

fori=1,2. The null hypothesis isH 0 :θ 1 =θ 2 , while the alternative isH 1 :θ 1
=θ 2.
LetX 1 <X 2 <···<Xn 1 andY 1 <Y 2 <···<Yn 2 be the order statistics of two
independent random samples from the respective distributions. Using the likelihood
ratio Λ, find the statistic used to testH 0 againstH 1. Find the distribution of
−2logΛ whenH 0 is true. Note that in this nonregular case, the number of degrees
of freedom is two times the difference of the dimensions of Ω andω.

6.5.10.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?

6.5.11.Letnindependent trials of an experiment be such thatx 1 ,x 2 ,...,xkare
the respective numbers of times that the experiment ends in the mutually exclusive
and exhaustive eventsC 1 ,C 2 ,...,Ck.Ifpi=P(Ci) is constant throughout then
trials, then the probability of that particular sequence of trials isL=px 11 px 22 ···pxkk.

(a)Recalling thatp 1 +p 2 +···+pk= 1, show that the likelihood ratio for testing

H 0 :pi=pi 0 > 0 ,i=1, 2 ,...,k, against all alternatives is given by

Λ=

∏k

i=1

(

(pi 0 )xi

(xi/n)xi

)

.

(b)Show that

−2logΛ =

∑k

i=1

xi(xi−npi 0 )^2

(np′i)^2

,

wherep′iis betweenpi 0 andxi/n.

Hint: Expand logpi 0 in a Taylor’s series with the remainder in the term

involving (pi 0 −xi/n)^2.

(c)For largen,arguethatxi/(np′i)^2 is approximated by 1/(npi 0 ) and hence

−2logΛ≈

∑k

i=1

(xi−npi 0 )^2

npi 0

whenH 0 is true.

Theorem 6.5.1 says that the right-hand member of this last equation defines

a statistic that has an approximate chi-square distribution withk−1 degrees

of freedom. Note that

dimension of Ω – dimension ofω=(k−1)−0=k− 1.