Question

Problem 1 Let X1, · · · , Xn IID∼ p(x; θ) = 1/2 (1 +θx), −1 < x < 1, −1 < θ < 1. 1. Estimate θ using the method of moments. 2. Show that the above MoM is consistent by showing it’s mean square error converges to 0 as n goes to infinity. 3. Find its asymptotic distribution.

Answer #1

Problem 2 Let X1, · · · , Xn IID∼ N(θ, θ) with θ > 0. Find a
pivotal quantity and use it to construct a confidence interval for
θ.

Let X1,...,Xn be a random sample from the pdf f(x;θ) = θx^(θ−1)
, 0 ≤ x ≤ 1 , 0 < θ < ∞ Find the method of moments estimator
of θ.

Let X1, X2, . . . , Xn be iid random variables with pdf
f(x|θ) = θx^(θ−1) , 0 < x < 1, θ > 0.
Is there an unbiased estimator of some function γ(θ), whose
variance attains the Cramer-Rao lower bound?

Let X1,...,Xn∼iid Gamma(3,1/θ) and we assume the prior for θ is
InvGamma(10,2). (a) Find the posterior distribution for θ. (b) If
n= 10 and ̄x= 18.2, find the Bayes estimate under
squared error loss. (c) The variance of the data distribution is φ=
3θ2. Find the Bayes estimator (under squared error loss)
for φ.Let X1,...,Xn∼iid Gamma(3,1/θ) and we assume the prior for θ
is InvGamma(10,2). (a) Find the posterior distribution for θ. (b)
If n= 10 and ̄x= 18.2, find...

Let {X1, ..., Xn} be i.i.d. from a distribution with pdf f(x; θ)
= θ/xθ+1 for θ > 2 and x > 1.
(a) (10 points) Calculate EX1 and V ar(X1).
(b) (5 points) Find the method of moments estimator of θ.
(c) (5 points) If we denote the method of moments estimator as
ˆθ1. What does √ n( ˆθ1 − θ) converge in distribution to? (d) (5
points) Is the method of moment estimator efficient? Verify your
answer.

Let X1, . . . , Xn ∼ iid N(θ, σ^2 ) for σ ^2 known. Find the UMP
size-α test for H0 : θ ≥ θ0 vs H1 : θ < θ0.

2 Let X1,…, Xn be a sample of iid NegBin(4, ?) random variables
with Θ=[0, 1]. Determine the MLE ? ̂ of ?.

Let X1, . . . , Xn be i.i.d from pmf f(x|λ) where f(x) =
(e^(−λ)*(λ^x))/x!, λ > 0, x = 0, 1, 2
a) Find MoM (Method of Moments) estimator for λ
b) Show that MoM estimator you found in (a) is minimal
sufficient for λ
c) Now we split the sample into two parts, X1, . . . , Xm and
Xm+1, . . . , Xn. Show that ( Sum of Xi from 1 to m, Sum...

Let X1,…, Xn be a sample of iid random variables with
pdf f (x ∶ ?) = 1/? for x ∈ {1, 2,…, ?} and Θ = ℕ. Determine the
MLE of ?.

Let X1, X2, . . . Xn be iid
random variables from a gamma distribution with unknown α and
unknown β. Find the method of moments estimators for α and β

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