Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

6.3. Maximum Likelihood Tests 383

of efficiency for tests; see Chapter 10 and more advanced books such as Hettman-

sperger and McKean (2011). However, all three tests have the same asymptotic

efficiency. Hence, asymptotic theory offers little help in separating the tests. Finite

sample comparisons have not shown that any of these tests are “best” overall; see

Chapter 7 of Lehmann (1999) for more discussion.

EXERCISES

6.3.1.The following data were generated from an exponential distribution with pdf

f(x;θ)=(1/θ)e−x/θ,forx>0, whereθ= 40.

(a)Histogram the data and locateθ 0 = 50 on the plot.

(b)Use the test described in Example 6.3.1 to testH 0 :θ=50versusH 1 :θ = 50.

Determine the decision at levelα=0.10.

19 15 76 23 24 66 27 12 25 7 6 16 51 26 39

6.3.2.Consider the decision rule (6.3.5) derived in Example 6.3.1. Obtain the

distribution of the test statistic under a general alternative and use it to obtain

the power function of the test. Using R, sketch this power curve for the case when

θ 0 =1,n= 10, andα=0.05.

6.3.3. Show that the test with decision rule (6.3.6) is like that of Example 4.6.1
except that hereσ^2 is known.

6.3.4.Obtain an R function that plots the power function discussed at the end of
Example 6.3.2. Run your function for the case whenθ 0 =0,n= 10,σ^2 =1,and
α=0.05.

6.3.5.Consider Example 6.3.4.

(a)Show that we can writeS∗=2T−n,whereT=#{Xi>θ 0 }.

(b)Show that the scores test for this model is equivalent to rejectingH 0 ifT<c 1

orT>c 2.

(c)Show that underH 0 ,Thas the binomial distributionb(n, 1 /2); hence, deter-

minec 1 andc 2 so that the test has sizeα.

(d)Determine the power function for the test based onTas a function ofθ.

6.3.6.LetX 1 ,X 2 ,...,Xnbe a random sample from aN(μ 0 ,σ^2 =θ) distribution,

where 0<θ<∞andμ 0 is known. Show that the likelihood ratio test ofH 0 :θ=θ 0

versusH 1 : θ =θ 0 can be based upon the statisticW =

∑n

i=1(Xi−μ^0 )

(^2) /θ 0.
Determine the null distribution ofW and give, explicitly, the rejection rule for a
levelαtest.
6.3.7.For the test described in Exercise 6.3.6, obtain the distribution of the test
statistic under general alternatives. If computational facilities are available, sketch
this power curve for the case whenθ 0 =1,n= 10,μ=0,andα=0.05.