Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

302 Some Elementary Statistical Inferences

4.8.11.Consider the situation in Example 4.8.6 with the hypotheses (4.8.3). Write

an algorithm that simulates the power of the test (4.8.4) to detect the alternative

μ=0.5 under the same contaminated normal distribution as in the example. Modify

theRfunctionempalphacn(N)to simulate this power and to obtain an estimate of

the error of estimation.

4.8.12.For the last exercise, write an algorithm to simulate the significance level

and power to detect the alternativeμ=0.5 for the test (4.8.4) when the underlying

distribution is the logistic distribution (4.4.11).

4.8.13.For the proof of Theorem 4.8.1, we assumed that the cdf was strictly in-

creasing over its support. Consider a random variableXwith cdfF(x)thatisnot

strictly increasing. Define as the inverse ofF(x) the function

F−^1 (u)=inf{x:F(x)≥u}, 0 <u< 1.

LetUhave a uniform (0,1) distribution. Prove that the random variableF−^1 (U)
has cdfF(x).

4.8.14. Verify the derivative in expression (4.8.12) and show that the function
(4.8.11) attains a maximum at the critical valuex=(α−[α])/(1−b).

4.8.15.Derive expression (4.8.14) and show that the resulting critical valueb=
[α]/α < 1 gives a minimum of the function that is the right side of expression
(4.8.13).

4.8.16. Assume thatY 1 has a Γ(α+1,1)-distribution,Y 2 has a uniform (0,1)
distribution, andY 1 andY 2 are independent. Consider the transformationX 1 =
Y 1 Y 21 /αandX 2 =Y 2.

(a)Show that the inverse transformation is: y 1 =x 1 /x

1 /α

2 andy^2 =x^2 with

support 0<x 1 <∞and 0<x 2 <1.

(b)Show that the Jacobian of the transformation is 1/x^12 /αand the pdf of (X 1 ,X 2 )

is

f(x 1 ,x 2 )=

1

Γ(α+1)

xα 1

x 2

exp

{

−

x 1

x^12 /α

}

1

x^12 /α

, 0 <x 1 <∞and 0<x 2 < 1.

(c)Show that the marginal distribution ofX 1 is Γ(α,1).

4.8.17.Show that the derivative of the ratio in expression (4.8.10) is given by the

function−xexp{−x^2 / 2 }(x^2 −1) with critical values±1. Show that the critical

values provide maxima for expression (4.8.10).

4.8.18.Consider the pdf

f(x)=

{

βxβ−^10 <x< 1

0elsewhere,

forβ>1.