Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

7.7. The Case of Several Parameters 453

7.7.2.LetX 1 ,X 2 ,...,Xnbe a random sample from a distribution that has a

pdf of the form (7.7.2) of this section. Show thatY 1 =

∑n
∑m i=1K^1 (Xi),...,Ym=
i=1Km(Xi) have a joint pdf of the form (7.7.4) of this section.
7.7.3.Let (X 1 ,Y 1 ),(X 2 ,Y 2 ),...,(Xn,Yn) denote a random sample of sizenfrom
a bivariate normal distribution with meansμ 1 andμ 2 , positive variancesσ 12 and
σ 22 , and correlation coefficientρ. Show that

∑n

1 Xi,

∑n

1 Yi,

∑n

1 X

2

i,

∑n

1 Y

2

∑n i,and

∑^1 nXiYiare joint complete sufficient statistics for the five parameters. AreX=

1 Xi/n,Y=

∑n

1 Yi/n,S

2

1 =

∑n

1 (Xi−X)

(^2) /(n−1),S 2
2 =
∑n
1 (Yi−Y)
(^2) /(n−1),
and
∑n
1 (Xi−X)(Yi−Y)/(n−1)S^1 S^2 also joint complete sufficient statistics for
these parameters?
7.7.4.Let the pdff(x;θ 1 ,θ 2 )beoftheform
exp[p 1 (θ 1 ,θ 2 )K 1 (x)+p 2 (θ 1 ,θ 2 )K 2 (x)+H(x)+q 1 (θ 1 ,θ 2 )], a<x<b,
zero elsewhere. Suppose thatK′ 1 (x)=cK′ 2 (x). Show thatf(x;θ 1 ,θ 2 ) can be written
in the form
exp[p(θ 1 ,θ 2 )K 2 (x)+H(x)+q(θ 1 ,θ 2 )], a<x<b,
zero elsewhere. This is the reason why it is required that no oneK′j(x) be a lin-
ear homogeneous function of the others, that is, so that the number of sufficient
statistics equals the number of parameters.
7.7.5.In Example 7.7.2:
(a)Find the MVUE of the standard deviation
√
θ 2.
(b)Modify the R functionbootse1.Rso that it returns the estimate in (a) and
its bootstrap standard error. Run it on the Bavarian forest data discussed
in Example 4.1.3, where the response is the concentration of sulfur dioxide.
Using 3,000 bootstraps, report the estimate and its bootstrap standard error.
7.7.6.LetX 1 ,X 2 ,...,Xnbe a random sample from the uniform distribution with
pdff(x;θ 1 ,θ 2 )=1/(2θ 2 ),θ 1 −θ 2 <x<θ 1 +θ 2 ,where−∞<θ 1 <∞andθ 2 >0,
and the pdf is equal to zero elsewhere.
(a)Show thatY 1 =min(Xi)andYn=max(Xi), the joint sufficient statistics for
θ 1 andθ 2 ,arecomplete.
(b)Find the MVUEs ofθ 1 andθ 2.
7.7.7.LetX 1 ,X 2 ,...,Xnbe a random sample fromN(θ 1 ,θ 2 ).
(a)If the constantbis defined by the equationP(X≤b)=pwherepis specified,
find the mle and the MVUE ofb.
(b)Modify the R functionbootse1.Rso that it returns the MVUE of Part (a)
and its bootstrap standard error.
(c)Run your function in Part (b) on the data set discussed in Example 7.6.4 for
p=0.75 and 3,000 bootstraps.