460 Sufficiency(a)b(1,θ), where 0≤θ≤1.(b)Poisson with meanθ>0.(c)Gamma withα=3andβ=θ>0.(d)N(θ,1), where−∞<θ<∞.(e)N(0,θ), where 0<θ<∞.7.8.2. LetY 1 <Y 2 <···<Ynbe the order statistics of a random sample of
sizenfrom the uniform distribution over the closed interval [−θ, θ]havingpdf
f(x;θ)=(1/ 2 θ)I−θ,θ.
(a)Show thatY 1 andYnare joint sufficient statistics forθ.(b)Argue that the mle ofθisθˆ=max(−Y 1 ,Yn).(c)Demonstrate that the mleθˆis a sufficient statistic forθandthusisaminimal
sufficient statistic forθ.7.8.3.LetY 1 <Y 2 <···<Ynbe the order statistics of a random sample of sizen
from a distribution with pdf
f(x;θ 1 ,θ 2 )=(
1
θ 2)
e−(x−θ^1 )/θ^2 I(θ 1 ,∞)(x),where−∞<θ 1 <∞and 0<θ 2 <∞. Find the joint minimal sufficient statistics
forθ 1 andθ 2.
7.8.4.Continuing with Example 7.8.4, show that the following statistics are location-
invariant:(a)Thesamplevariance=S^2.(b)The mean deviation from the sample median = (1/n)∑
|Xi−median(Xi)|.(c)(1/n)∑
[Xi−min(Xi)].7.8.5. In Example 7.8.5, a scale model was presented and scale invariance was
defined. Using the notation of this example, show that the following statistics are
scale-invariant:
(a)X 12 /∑n1X^2 i.(b)min{Xi}/max{Xi}.7.8.6.Obtain the location and scale invariance of the other statistics listed in
Example 7.8.6, i.e., the statistics(a)T 1 =[max(Xi)−min(Xi)]/S.