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HOMEWORK 7, STAT 6332
1. Consider a normal random variable X ∼ Norm(0, σ 2 ). Calculate the Fisher information I(σ).
2. Consider a random variable X = θ + ǫ, where ǫ is a random variable. Find the
information inequality bound for unbiased estimators of θ. Note that this is the example of
the location parameter.
3. Show that for the scale family of distributions with density θ−1 f (x/θ), θ > 0, the
amount of information a single observation X has about θ is
θ
−2
Z
[
yf ′ (y)
+ 1]2 f (y)dy.
f (y)
4. For the setting of Problem 3, show that the information X contains about ξ = log(θ)
is independent of θ.
5. Show that if Eθ (δ) = g(θ), and Var(δ) attains the information inequality lower bound,
then
g ′ (θ)
δ(X) = g(θ) +
∂ ln(pθ (x))/∂θ.
I(θ)
6. Let X be distributed as Norm(θ, 1). Show that conditionally given a < X < b, the
variable X tends in probability to b as θ → ∞.
7. Consider a sample X1 , . . . , Xn from a Poisson distribution conditioned to be positive,
so that P (X = x) = θx e−θ /[x!(1 − e−θ )] for x = 1, 2, . . .. Show that the likelihood equation
has a unique root for all values of x.
8. Show that under appropriate assumptions
[L(θ0 + n−1/2 ) − L(θ0 ) + (1/2)I(θ0 )]/I 1/2 (θ0 )
tends in law to N(0, 1).
9. A density function is strongly unimodal , or equivalently log concave, if ln(f (x)) is a
concave function. Shoe that such a density has a unique mode.
10. Let X1 , . . . , Xn be iid with density f (x − θ). Show that the likelihood function has a
unique root if f ′ (x)/f (x) is monotone, and the root is a maximum if f ′ (x)/f (x) is decreasing.
Hence, densities that are log concave yield unique MLEs.
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