Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

386 Maximum Likelihood Methods

(a)Find Λ and−2logΛ.

(b)Determine the Wald-type test.

(c)What is Rao’s score statistic?

6.3.17.LetX 1 ,X 2 ,...,Xnbe a random sample from a Poisson distribution with
meanθ>0. Consider testingH 0 : θ=θ 0 againstH 1 :θ =θ 0.

(a)Obtain the Wald type test of expression (6.3.13).

(b)Write an R function to compute this test statistic.

(c)For θ 0 = 23, compute the test statistic and determine thep-value for the

following data.

27 13 21 24 22 14 17 26 14 22

21 24 19 25 15 25 23 16 20 19

6.3.18.LetX 1 ,X 2 ,...,Xnbe a random sample from a Γ(α, β) distribution where
αis known andβ>0. Determine the likelihood ratio test forH 0 : β=β 0 against
H 1 :β =β 0.

6.3.19.LetY 1 <Y 2 <···<Ynbe the order statistics of a random sample from a
uniform distribution on (0,θ), whereθ>0.

(a)Show that Λ for testingH 0 : θ=θ 0 againstH 1 : θ =θ 0 is Λ = (Yn/θ 0 )n,

Yn≤θ 0 ,andΛ=0ifYn>θ 0.

(b)WhenH 0 is true, show that−2 log Λ has an exactχ^2 (2) distribution, not

χ^2 (1). Note that the regularity conditions are not satisfied.

6.4 MultiparameterCase:Estimation....................

In this section, we discuss the case whereθis a vector ofpparameters. There
are analogs to the theorems in the previous sections in whichθis a scalar, and we
present their results but, for the most part, without proofs. The interested reader
can find additional information in more advanced books; see, for instance, Lehmann
and Casella (1998) and Rao (1973).
LetX 1 ,...,Xnbe iid with common pdff(x;θ), whereθ∈Ω⊂Rp. As before,
the likelihood function and its log are given by

L(θ)=

∏n

i=1

f(xi;θ)

l(θ)=logL(θ)=

∑n

i=1

logf(xi;θ), (6.4.1)

forθ∈Ω. The theory requires additional regularity conditions, which are listed in
Appendix A, (A.1.1). In keeping with our number scheme in the last three sections,