- (10 points) Let x = (1.27, 1.39, 1.01, 5.05, 5.82) be an observed sample of size n = 5
from distribution with pdf fθ(x) = θ
xθ+1 , x > 1 and zero otherwise, where θ > 1 is
unknown. Let Ψ(θ) = 1
θ
. Find the profile likelihood function LΨ(ψ|x) and use R
to plot LΨ(ψ|x) against ψ from ψ = 0 to ψ = 1 . Also, calculate the 0.95-profile
likelihood interval for Ψ. - Suppose that a measure X on a population takes values x ∈ X = {1, 2, 3, 4} and we
have the following statistical model for X:
x = 1 x = 2 x = 3 x = 4
fθ1
(x) 0.04 0.29 0.55 0.12
fθ2
(x) 0.66 0.14 0.14 0.06
fθ3
(x) 0.03 0.83 0.05 0.09
fθ4
(x) 0.57 0.05 0.22 0.16
fθ5
(x) 0.13 0.08 0.03 0.76
Question 2 continues on the next page. . .
Page 2 of 3
Suppose further that we are interested in the characteristic Ψ(θ), where Ψ(θ1) =
Ψ(θ2) = 1 and Ψ(θ3) = Ψ(θ4) = Ψ(θ5) = 2.
Note: When answering the questions below, you should explain your work in great
details assuming that you are teaching your answers to a friend who has no idea of
what a profile likelihood function is.
(a) (8 points) If x = 4 was the observed value of X, determine the profile likelihood
function of Ψ based on this observation and find the profile likelihood estimate
of Ψ.
(b) (4 points) Calculate a 0.5-profile likelihood region for Ψ based on this observation (i.e. x = 4). - Let x = (x1, x2, . . . , xn) be an observed sample from the location-scale Normal model
N(µ, σ2
), θ = (µ, σ) ∈ R×R+. Let Ψ(θ) be the interquartile range of the distribution.
(a) (3 points) Show that Ψ(θ) = σ(z0.75 − z0.25).
(b) (6 points) Find the profile likelihood function LΨ(ψ) of Ψ.
Note: You MUST express LΨ(ψ) as a function of ψ instead of other parameters (i.e. µ or σ).
(c) (6 points) Find the profile MLE of Ψ directly by finding arg maxψ LΨ(ψ).
Note: This means, this question wants you to find the profile MLE of Ψ by
finding the value of ψ that maximizes the profile likelihood function LΨ(ψ) without using any results that we have discussed in class, even though this indeed
can be done much quicker using the some of the results that we have discussed
in class. - Let X = (X1, X2, . . . , Xn) be a random sample from the Negative-binomial(r, θ) distribution, where r is known and θ ∈ (0, 1) is unknown.
Note: negative-binomial(r, θ) distribution has probability mass function:
f(x) =
r−1+x
x
θ
r
(1 − θ)
x
, x = 0, 1, 2, 3, . . .
(a) (5 points) Find the MLE of θ.
(b) (8 points) Find a sufficient statistic for θ. Use Factorization theorem. Is this
sufficient statistic minimal sufficient? Justify your answer. - Let X = (X1, X2, . . . , X10) be a random sample of n = 10 observations from distribution with p.d.f
fθ(x) = x
4
24θ
5
e
− x
θ , x > 0
and zero otherwise. θ > 0.
(a) (6 points) Statistic T1 = X¯ is used as an estimator of ψ1(θ) = θ. Find MSEθ(T1).
Hint: This is a function of of θ of the form MSEθ(T1) = k1θ
2
. Calculate the
value of k1.
Question 5 continues on the next page. .
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