8.5.∗Minimax and Classification Procedures 513EXERCISES8.5.1.LetX 1 ,X 2 ,...,X 20 be a random sample of size 20 from a distribution that
isN(θ,5). LetL(θ) represent the joint pdf ofX 1 ,X 2 ,...,X 20. The problem is to
testH 0 :θ= 1 againstH 1 :θ=0. ThusΩ={θ:θ=0, 1 }.
(a)Show thatL(1)/L(0)≤kis equivalent tox≤c.(b)Findcso that the significance level isα=0.05. Compute the power of this
test ifH 1 is true.(c)If the loss function is such thatL(1,1) =L(0,0) = 0 andL(1,0) =L(0,1)>0,
find the minimax test. Evaluate the power function of this test at the points
θ=1andθ=0.8.5.2.LetX 1 ,X 2 ,...,X 10 be a random sample of size 10 from a Poisson distribu-
tion with parameterθ.LetL(θ)bethejointpdfofX 1 ,X 2 ,...,X 10. The problem
is to testH 0 :θ=^12 againstH 1 :θ=1.
(a)Show thatL(^12 )/L(1)≤kis equivalent toy=∑n
1 xi≥c.(b)In order to makeα=0.05, show thatH 0 is rejected ify>9 and, ify=9,
rejectH 0 with probability^12 (using some auxiliary random experiment).(c)If the loss function is such thatL(^12 ,^12 )=L(1,1) = 0 andL(^12 ,1) = 1 and
L(1,^12 ) = 2, show that the minimax procedure is to rejectH 0 ify>6 and, if
y= 6, rejectH 0 with probability 0.08 (using some auxiliary random experi-
ment).8.5.3.In Example 8.5.2 letμ′ 1 =μ′ 2 =0,μ′′ 1 =μ′′ 2 =1,σ 12 =1,σ^22 =1,andρ=^12.(a)Find the distribution of the linear functionaX+bY.(b)Withk= 1, computeP(aX+bY≤c;μ′ 1 =μ′ 2 =0)andP(aX+bY > c;μ′′ 1 =
μ′′ 2 =1).8.5.4. Determine Newton’s algorithm to find the solution of Equation (8.5.2). If
software is available, write a program that performs your algorithm and then show
that the solution isc=76.8. If software is not available, solve (8.5.2) by “trial and
error.”
8.5.5.LetXandYhave the joint pdff(x, y;θ 1 ,θ 2 )=
1
θ 1 θ 2exp(
−
x
θ 1−
y
θ 2)
, 0 <x<∞, 0 <y<∞,zero elsewhere, where 0<θ 1 , 0 <θ 2 .Anobservation(x, y) arises from the joint
distribution with parameters equal to either (θ′ 1 =1,θ′ 2 =5)or(θ′′ 1 =3,θ′′ 2 =2).
Determine the form of the classification rule.