In the name of GOD.
Sharif University of Technology
Stochastic Processes CE 695 Fall 2015 Dr. H.R. Rabiee
Homework 4 (Estimation Theory)
(135 points)
1. (20 pts) Let X1 , . . . , Xn are iid samples from following distributions. Find
a Sufficient Statistic for each part.
a
(a) f (x|θ) = θaxa−1 e−θx , x > 0, θ > 0, a > 0 is fixed.
(b) f (x|θ1 , θ2 ) =
(c) f (x|θ) =
(d) f (x|θ) =
1 −
θ2 e
(x−θ1 )
θ2
2θ 2
x3 , 0 < x
1
1
π 1+(x−θ)2
, x > θ1 , θ1 ∈ (−∞, ∞), θ2 > 0
<θ
2. (20 pts) Let X1 , . . . , Xn are iid samples from following distributions. Find
a Minimal Sufficient Statistic for each part.
2
(a) f (x|θ) = π −1/2 e−(x−θ)
1
(b) f (x|θ) = π1 1+(x−θ)
2
(c) f (x|θ) =
e−(x−θ)
(1+e−(x−θ) )2
θ
(d) f (x|θ1 , θ2 ) =
θ1 2
θ2 −1 −θ1 x
e
,0
Γ(θ2 ) x
< x < ∞, θ1 > 0, θ2 > 0
3. (20 pts) Let X1 , . . . , Xn are iid samples from following distributions. For
each part, find a Complete Sufficient Statistic, or show that one does not
exist.
(a) f (x|θ) =
(b) f (x|θ) =
2x
θ 2 , 0 < x < θ, θ > 0
(log θ)θ x
θ−1 , 0 < x < 1, θ >
−(x−θ)
−(x−θ)
1
(c) f (x|θ) = e
exp (−e
)
θ |x|
1−|x|
(d) f (x|θ) = ( 2 ) (1 − θ)
, x ∈ {−1, 0, 1} , θ ∈ (0, 1)
4. (10 pts) Let X1 , · · · , Xn be iid observations from f (x|θ) that belongs to
exponential family, i.e.
!
k
X
f (x|θ) = h(x)c(θ) exp
wi (θ)ti (x)
i=1
where, θ = (θ1 , . . . , θd ), d ≤ k. show that,


n
n
X
X
T (X) = 
t1 (Xj ), · · · ,
tk (Xj )
j=1
j=1
is a sufficient statistic for θ.
1
5. (10 pts) Let X1 , ..., Xn be samples from a distribution with pdf
f (x|µ, λ) =
λ
2πx3
1/2
e
−λ(x−µ)2
2µ2 x
,0 < x < ∞
(a) Find sufficient statistics for µ, λ.
(b) Is the statistics complete?
6. (10 pts) Let X1 , ..., Xn be samples from the following distribution:
f (x|σ) =
1 − x−σ
e σ ,σ < x < ∞
σ
(a) Find a MSS for σ.
(b) Find a CSS for σ.
7. (15 pts) Let X1 , . . . , Xm ; Y1 , . . . , Yn be iid samples from N (µ, σ 2 ) and
N (η, τ 2 ), respectively. Find the MSS for the followings:
(a) µ, η, σ, τ are arbitrary: −∞ < µ, η < ∞, 0 < σ, τ
(b) σ = τ and µ, η, τ are arbitrary.
(c) µ = η and µ, σ, τ are arbitrary.
8. (10 pts) Let X1 , . . . , Xn be a random sample from a uniform distribution
on the interval (θ, 2θ), θ > 0. Find a minimal sufficient statistic for θ. Is
the statistic complete?
9. (10 pts) Let X1 , . . . , Xn be K dimensional samples from a Dirichlet distribution with parameter α = (α1 , . . . , αk ).
K
1 Y αk −1
f (x|α) =
xk
B(α)
k=1
find a sufficient statistic for α. Is the statistic complete?
10. (10 pts) Let X1 , . . . , Xn be iid N (θ, aθ2 ), where a is a known positive
constant and θ > 0.
(a) Show that the parameter space does not contain a two-dimensional
open set.
(b) Show that the statistic T = (X̄, S 2 ) is a sufficient statistic for θ, but
the family of distributions is not complete.
2