2
Let X1,…,Xn be i.i.d. random variable with pdf fθ defined as follows:
fθ(x)=θxθ−11(0≤x≤1)
where θ is some positive number.
(a)
Is the parameter θ identifiable?
Yes
No
(b)
Compute the maximum likelihood estimator ˆθ of θ.
(Enter Sigma_i(g(X_i)) for the sum ∑ni=1g(Xi), e.g. enter Sigma_i(X_i^2) for ∑ni=1X2i, enter Sigma_i(ln(X_i)) for ∑ni=1ln(Xi). Do not forget any necessary n in your answer, e.g. ˉXn will need to be entered as Sigma_i(X_i)/n . Do not worry about the parser not rendering correctly, as the grader will still work independently. If you would like proper rendering, enclose Σi(g(Xi)) in parentheses i.e. use (Σi(g(Xi))).)
Maximum likelihood estimator ˆθ=
c)
As in previous ecercise let X1…Xn be iid with pdf where θ>0
ocultos)
Compute the Fisher information.
I(θ)=
d)
What kind of distribution does the distribution of √nˆθ approach as n grows large?
Bernoulli
Poisson
Normal
Exponential
(e)
What is the asymptotic variance V(ˆθ) of ˆθ ?
To avoid double jeopardy, you may use I for the Fisher information I(θ) evaluated at θ, or you may enter your answer without using I.
V(ˆθ)
f)
Using the MLE ˆθ, find the shortest confidence interval for θ with asymptotic level 85% using the plug-in method.
To avoid double jeopardy, you may use V for the appropriate estimator of the asymptotic variance V(ˆθ), and/or I for the Fisher information I(ˆθ) evaluated at ˆθ, or you may enter your answer without using V or I.
(Enter 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).)
Iplug-in=[A,B] where
A=
sin responder
B=
[3:58 p. m., 10/7/2021] Ejh: 3. 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=