Question

3 a. As on the previous page, let X1,…,Xn be i.i.d. with pdf
F_θ(x)= θx^θ-1*1(0<x<1)
Assume we do not actually get to observe X1,…,Xn. Instead let Y1,…,Yn be our observations where Yi=1(Xi≤0.5). Our goal is to estimate θ based on this new data.
What distribution does Yi follow?
First, choose the type of the distribution:
• Bernoulli
• Poisson
• Normal
• Exponential
Second, enter the parameter of this distribution in terms of θ. Denote this parameter by mθ. (If the distribution is normal, enter only 1 parameter, the mean).
mθ=

b. Write down a statistical model associated to this experiment. Is the parameter θ identifiable?
Yes
No

c. Compute the Fisher information I(θ).
(To answer this question correctly, your answer to part (a) needs to be correct.)
I(θ)=

d. Compute the maximum likelihood estimator ˆθ for θ in terms of ¯Yn.
(Enter barY_n for ¯Yn.)
ˆθ=

e. Compute the method of moments estimator ˜θ for θ.
(Enter barY_n for ¯Yn.)
˜θ=

f. What is the asymptotic variance V(˜θ) of the method of moments estimator ˜θ?
V(˜θ)=

g. Give a formula for the p-value for the test of
H0:θ≤1vs.H1:θ>1
based on the asymptotic distribution of ˆθ.
To avoid double jeopardy, you may use V for the asymptotic variance V(θ0), I for the Fisher information I(θ0), hattheta for ˆθ, or enter your answer directy without using V or I or hattheta.
(Enter barY_n for ¯Yn, hattheta for ˆθ. If applicable, enter Phi(z) for the cdf Φ(z) of a normal variable Z, q(alpha) for the quantile qα for any numerical value α. Recall the convention in this course that P(Z≤qα)=1−α for Z∼N(0,1).)
p-value:

Assume n=50, and ¯Yn=0.46. Will you reject the null hypothesis at level α=5%?
Yes, reject the null hypothesis at level α=5%.
No, cannot reject the null hypothesis at level α=5%.