Question

Let ?1, ?2,…. . , ?? (n random variables iid) as a

variable X whose pdf is given by ??^{-a-1}

for ? ≥1.

(a) For ? ≥ 1 calculate ? (??? ≤ ?) = ? (?). Deduce the
function

density of probabilities of Y = lnX.

(b) Determine the maximum likelihood estimator (MLE) of ?

and show that he is without biais

Answer #1

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 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 be a sample of iid
Exp(?1, ?2) random variables with common pdf
f (x; ?1, ?2) =
(1/?1)e−(x−?2)/?1 for x
> ?2 and Θ = ℝ × ℝ+.
a) Show that S = (X(1), ∑ni=1
Xi ) is jointly sufficient for (?1, ?2).
b) Determine the pdf of X(1).
c) Determine E[X(1)].
d) Determine E[X2(1) ].
e ) Is X(1) an MSE-consistent estimator of
?2?
f) Given S = (X(1), ∑ni=1
Xi )is a complete sufficient statistic...

Let X1,…, Xn be a sample of iid random
variables with pdf f (x; ?) = 3x2 /(?3) on S
= (0, ?) with Θ = ℝ+. Determine
i) a sufficient statistic for ?.
ii) F(x).
iii) f(n)(x)

Let X1, X2, . . . , Xn be iid exponential random variables with
unknown mean β.
(1) Find the maximum likelihood estimator of β.
(2) Determine whether the maximum likelihood estimator is
unbiased for β.
(3) Find the mean squared error of the maximum likelihood
estimator of β.
(4) Find the Cramer-Rao lower bound for the variances of
unbiased estimators of β.
(5) What is the UMVUE (uniformly minimum variance unbiased
estimator) of β? What is your reason?
(6)...

Let X ∼ Geo(?) with Θ = [0,1].
a) Show that pdf of the random variable X is in the
one-parameter
regular exponential family of distributions.
b) If X1, ... , Xn is a sample of iid Geo(?) random variables
with
Θ = (0, 1), determine a complete minimal sufficient statistic
for ?.

Suppose the random variable X follows the Poisson P(m) PDF, and
that you have a random sample X1, X2,...,Xn from it. (a)What is the
Cramer-Rao Lower Bound on the variance of any unbiased estimator of
the parameter m? (b) What is the maximum likelihood estimator
ofm?(c) Does the variance of the MLE achieve the CRLB for all
n?

4. Let X and Y be random variables having joint probability
density function (pdf) f(x, y) = 4/7 (xy − y), 4 < x < 5 and
0 < y < 1
(a) Find the marginal density fY (y).
(b) Show that the marginal density, fY (y), integrates to 1
(i.e., it is a density.)
(c) Find fX|Y (x|y), the conditional density of X given Y =
y.
(d) Show that fX|Y (x|y) is actually a pdf (i.e., it integrates...

Let ?? ~ ???? (2?, 4?) independet random variables. for ? =
1,2,… ?.
a)Find an estimator for ? by the method of moments.
b) Find an estimator for ? by the maximum likelihood estimator
(MLE)

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

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