Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

7.9. Sufficiency, Completeness, and Independence 467

7.9.12.LetY 1 <Y 2 <···<Ynbe the order statistics of a random sample from a
N(θ, σ^2 ) distribution, whereσ^2 is fixed but arbitrary. ThenY=Xis a complete
sufficient statistic forθ. Consider another estimatorTofθ,suchasT =(Yi+
Yn+1−i)/2, fori=1, 2 ,...,[n/2], orTcould be any weighted average of these latter
statistics.

(a)Argue thatT−XandXare independent random variables.

(b)Show that Var(T)=Var(X)+Var(T−X).

(c)Since we know Var(X)=σ^2 /n, it might be more efficient to estimate Var(T)

by estimating the Var(T−X) by Monte Carlo methods rather than doing that

with Var(T) directly, because Var(T)≥Var(T−X). This is often called the

Monte Carlo Swindle.

7.9.13.SupposeX 1 ,X 2 ,...,Xnis a random sample from a distribution with pdf
f(x;θ)=(1/2)θ^3 x^2 e−θx, 0 <x<∞, zero elsewhere, where 0<θ<∞:

(a)Find the mle,θˆ,ofθ.Isθˆunbiased?

Hint: Find the pdf ofY=

∑n

1 Xiand then computeE(θˆ).

(b)Argue thatYis a complete sufficient statistic forθ.

(c)Find the MVUE ofθ.

(d)Show thatX 1 /YandYare independent.

(e)What is the distribution ofX 1 /Y?

7.9.14.The pdf depicted in Figure 7.9.1 is given by

fm 2 (x)=e−x(1 +m− 21 e−x)−(m^2 +1), −∞<x<∞, (7.9.2)

wherem 2 >0 (the pdf graphed is form 2 =0.1). This is a member of a large family
of pdfs, logF-family, which are useful in survival (lifetime) analysis; see Chapter 3
of Hettmansperger and McKean (2011).

(a)LetW be a random variable with pdf (7.9.2). Show thatW=logY,where

Y has anF-distribution with 2 and 2m 2 degrees of freedom.

(b)Show that the pdf becomes the logistic (6.1.8) ifm 2 =1.

(c)Consider the location model where

Xi=θ+Wi i=1,...,n,

whereW 1 ,...,Wnare iid with pdf (7.9.2). Similar to the logistic location

model, the order statistics are minimal sufficient for this model. Show, similar

to Example 6.1.2, that the mle ofθexists.