Answer questions 4,5,6,7
15 STAT 7031 22FS – Statistical Theory
Midterm Exam I, 24-hr Take-Home exam starting at 10:10AM, September 28, 2022
Please work as independently as possible. Originality and creativity will be rewarded. If I see
similar ”originality and creativity” in the case of two or more submission, the ”reward” will
be appropriately divided. Sorry about it, but that’s the way it is going to be.
1. Let X1 , . . . , Xn be iid N (θ, θ2 ), θ > 0. Show that (a) the statistic T = (X̄, S 2 ) is a sufficient statistic
for θ but (b) the family of the distribution of T is NOT complete.
2. Let X be a random variable with the density of the form pη (x) = exp [ηT (x) − A(η)] h(x), x ∈ R where
η is the parameter, A(·) is a real-valued function of η and the T is a real-valued statistic.
(a) Show that Eη (T ) = ∂η
A(η) and V ar(T ) = ∂η
(b) Show that the moment generating function MT (u) is given by MT (u) = eA(η+u) /eA(η) .
3. Let Y1 , . . . , Yn be independent random variables and Yi be distributed as Poisson(iθ), i = 1, . . . , n.
Here, θ(> 0) is the unknown parameter.
(a) Find the complete sufficient statistics for θ.
(b) Find the Uniformly Minimum Variance Unbiased Estimator(UMVUE) of θ.
4. Let X1 , . . . , Xn be i.i.d. N (µ, σ 2 ) with −∞ < µ < ∞, 0 < σ 2 < ∞. Assume that both µ and σ 2 are unknown, n ≥ 2. Let θ = (µ, σ) and τ (θ) = µσ. 1 (a) Find E(X) and E(S) where S 2 = n−1 Pn 2 i=1 (Xi − X) . (b) Derive the UMVUE for τ (θ). 5. Let X1 , X2 , . . . , Xm and Y1 , . . . , Yn be independently distributed as E(a, b) and E(a0 , b), respectively. i.e., X ∼ E(a, b), f (x; a, b) = 1b exp[−(x − a)/b]I(x > a), a ∈ (−∞, ∞) and b ∈ (0, ∞).
(a) Show that (X(1) , Y(1) ,
i=1 (Xi − X(1) ) +
i=1 (Yi − Y(1) )) is complete S.S. for (a, a , b).
(b) Find the UMVUE for b and a −a
6. Let X1 , X2 , . . . , Xn be iid N (µ, 1) where µ ∈ (−∞, ∞) is assumed unknown.
(a) Derive the UMVUE, T = T (X1 , X2 , . . . , Xn ), for the parametric function τ (µ) = e4µ .
(b) Does V ar(T ) attain the CRLB for unbiased estimators of e4µ ?
7. Suppose that X1 , . . . , Xm are iid Uniform(−θ, θ) and Y1 , . . . , Yn are iid Uniform(−2θ, 2θ) where θ ∈
(−∞, ∞) is assumed unknown. Assume that the X’s and Y ’s are independent.
(a) Show that Y1 /2 ∼ Uniform(−θ, θ).
(b) Derive the UMVUE, T = T (X1 , X2 , . . . , Xm , Y1 , . . . , Yn ), for the parametric function g(θ) = θ2