20137 Advanced Statistics for Economic and Social Sciences (ESS-MS) General Exam 2021
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20137 Advanced Statistics for Economic and Social Sciences (ESS-MS)
General Exam
September 7th, 2021
Notation: Recall that 1(a < x < b) = 1(a,b)(x) and that fX (x; θ) = fθ(X)(x).
Question 1 (11 points)
Consider a size n i.i. d. sample from a population X with density
f (x; θ) = ^x eoθ 2 x I(0,+o)(x), θ > 0,
note that X ~ Gamma(, θ 2 ).
Solution: As E[X] = 2 2θ(3) , we obtain M = 入 2X(3) .
Solution: The likelihood function is
the log-likelihood function is then
y(θ; x) = ln L(θ; x) = 3n ln θ _ n ln Γ() + ln xi _ θ 2 xi ,
its derivative w.r. to θ is
∂ 3n
∂θ θ
which vanishes if θ = ┌ , this is indeed a maximum since
∂2 3n
∂θ 2 θ 2
(c) Is L asymptotically unbiased? Is it unbiased?
= 入 ) 33n2在2) y 32(在) o1 eoθ 2 y dy = 入 2(3n) = 入 θ . 1) y 3在1 o1 eoθ 2 y
è μ
density of a Gamma r.v.
0 Γ( ) y 2 eoθ 2 y dy
dy = 入 θ
è μ
★
and the ratio ★ is different from 1, this means that L is not unbiased.
Solution: As
_E – (ln f (X; θ))! = + 2E[X] = ,
the Fisher information of the sample is wx = .
(e) Find a level 1 _ α confidence interval for θ
Solution: As
-n(L _ θ) s Norm ╱0, 、 ,
θ ∈ L 土 zα/2┌ 2Ln .
Question 2 (5 points)
Consider a single observation distributed according to the probability density
exoθ
f (x; θ) = (1 + exoθ )2 , θ ∈ e,
(a) find the MP (Most Powerful) level α test for the hypotheses
H0 : θ0 = 0 vs. H1 : θ 1 = 1;
Solution: Let us use Neyman-Pearson theorem: bearing in mind that we have a single observation we reject H0 if f (x; θ1 ) > kf (x; θ0 ) in our specific case this boils down to f (x; 1) > kf (x; 0) that is
exo1 ex
(1 + exo1)2 (1 + ex )2
÷
1 + ex
> k1
the function h(x) = e弘1 is increasing (the derivative is always positive) therefore I = Rx ∈ R : x > k}
with k such that Pθ=0(X > k) = α, as Pθ=0(X > k) = 1 _ FX(k) = we get that = α hence ek = and
I = {x ∈ R : x > ln } .
(b) find the UMP (Uniformly Most Powerful) level α test for the hypotheses
H0 : θ = 0 vs. H1 : θ > 0;
Solution: Repeating the above calculations with θ 1 = θ (and not θ 1 = 1, we see that (provided that θ > 0, hence h is increasing) the results are the same, therefore I does not depend on θ 1 and we can assume that I is the critical region also for the UMP test.
(c) Compute the power function of the test found in part (b).
Solution: Recalling that the distribution function of X is
FX(x) = 1 _ 1
the power function is, by definition,
Q(θ) = Pθ(X ∈ I) = Pθ ╱X > ln 、 = = .
Q(θ)
1
α
θ
Question 3 (5 points)
Let θ ∈ e1 . Define the one parameter exponential family of distributions, in symbols Exp(1), and the Monotone Likelihood Ratio (MLR) property. Does the MLR property hold for a density function fX (x; θ) ∈ Exp(1)? Show in detail why (or why not), and provide an example of a member of the Exp(1) family that satisfies (or does not satisfy) the MLR property.
Carefully state the theorem that allows one to obtain the Uniformly Most Powerful test for the testing problem H0 : θ > θ0 vs. H1 : θ < θ0 when X1 , . . . , Xn is an i.i. d. sample from fX (x; θ) ∈ Exp(1).
Solution: Please refer to your textbook and/or notes.
Question 4 (5 points)
State and prove (with all details) the Rao-Blackwell theorem.
Solution: Please refer to your textbook and/or notes.
Question 5 (5 points)
State and prove the Cramer-Rao Lower Bound (CRLB) theorem, with its corollary. For which functions of the parameter θ can the CRLB be achieved, and why?
Solution: Please refer to your textbook and/or notes.
2022-08-20