Let X1,…,Xn be i.i.d. random variable with pdf fθ defined as follows:

Question

fθ(x)=θxθ−11(0≤x≤1)
where θ is some positive number.
(a)
1 punto posible (calificable, resultado oculto)
Is the parameter θ identifiable?
Yes
No
(b)
2.0 puntos posibles (calificables, resultados ocultos)
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) Ha realizado 1 de 3 intentosAlgunos problemas tienen opciones como guardar, restablecer, sugerencias o mostrar respuesta. Estas opciones aparecen después de oprimir el botón Enviar.
(e)
1 punto posible (calificable, resultado oculto)
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=