Let X1. . . . Xn be i.i.d f(x; θ) = θ(1 − θ)^x x = 0.. Is there
a function of θ for which there exists an unbiased estimator of θ
whose variance achieves the CRLB? If so, find it
For a fixed θ>0, let X1,X2,…,Xn be i.i.d., each with the beta
(1,θ) density.
i) Find θ^ that is the maximum likelihood
estimate of θ.
ii) Let X have the beta (1,θ) density. Find the
density of −log(1−X). Recognize this as one of the famous ones and
provide its name and parameters.
iii) Find f that is the density of the MLE θ^
in part (i).
Let X1, . . . , Xn be i.i.d. samples from Uniform(0, θ). Show
that for any α ∈ (0, 1), there is a cn,α, such that
[max(X1,...,Xn),cn,α max(X1,...,Xn)] is a 1−α confidence interval
of θ.
Let X1,..., Xn be an i.i.d. sample from a geometric distribution
with parameter p.
U = ( 1, if X1 = 1, 0, if X1 > 1)
find a sufficient statistic T for p.
find E(U|T)
Let X1, ..., Xn be i.i.d random variables with the density
function f(x|θ) = e^(θ−x) , θ ≤ x. a. Find the Method of Moment
estimate of θ b. The MLE of θ (Hint: Think carefully before taking
derivative, do we have to take derivative?)
Let X1, X2, · · · , Xn (n ≥ 30)
be i.i.d observations from N(µ1,
σ12 ) and Y1, Y2, · · ·
, Yn be i.i.d observations from N(µ2,
σ22 ). Also assume that X's and Y's are
independent. Suppose that µ1, µ2,
σ12 ,
σ22 are unknown. Find an
approximate 95% confidence interval for
µ1µ2.
R simulation:
Let X1, . . . , Xn be i.i.d. random variables from a uniform
distribution on [0, 2]. Generate
and plot 10 paths of sample means from n = 1 to n = 40 in one
figure for each case. Give
some comments to empirically check the Law of Large Numbers.
(a) When n is large,
X1 + · · · + Xn/n converges to E[Xi].
(b) When n is large,
X1^2+ · · · + Xn^2/n converges to...
Let X1, . . . , Xn ∼ iid Beta(θ, 1). (a) Find the UMP test for
H0 : θ ≥ θ0 vs H1 : θ < θ0. (b) Find the corresponding Wald
test. (c) How do these tests compare and which would you
prefer?
Suppose X1; : : : ; Xn is i.i.d Exponential distribution with
density
f(xjθ) = (1/θ) * e(-x/θ); 0 ≤ x < 1; θ > 0:
(a) Find the UMVUE (the best unbiased estimator) of θ.
(b) What is the Cramer-Rao lower bound of all unbiased estimator of
all unbiased estimator
of θ. Does the estimator from (a) attain the lower bound? Justify
your answer.
(c) What is the Cramer-Rao lower bound of all unbiased estimator of
θ^2?
3
(d)...