Question

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)

Answer #1

Let ?? ~ ??? (?,θ ) independent random variable. for ? = 1,2,…
?. Find an estimator for ? by the maximum likelihood
method.(MLE)

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

Suppose Y1, . . . , Yn are independent random variables with
common density fY(y) = eμ−y , y > μ
1. Find the Method of Moments Estimator for μ.
2. Find the MLE for μ. Then find the bias of the estimator

Let Y1, Y2, . . ., Yn be a
random sample from a Laplace distribution with density function
f(y|θ) = (1/2θ)e-|y|/θ for -∞ < y < ∞
where θ > 0. The first two moments of the distribution are
E(Y) = 0 and E(Y2) = 2θ2.
a) Find the likelihood function of the sample.
b) What is a sufficient statistic for θ?
c) Find the maximum likelihood estimator of θ.
d) Find the maximum likelihood estimator of the standard
deviation...

6. Let X1, X2, ..., Xn be a random sample of a random variable X
from a distribution with density
f (x) ( 1)x 0 ≤ x ≤ 1
where θ > -1. Obtain,
a) Method of Moments Estimator (MME) of parameter θ.
b) Maximum Likelihood Estimator (MLE) of parameter θ.
c) A random sample of size 5 yields data x1 = 0.92, x2 = 0.7, x3 =
0.65, x4 = 0.4 and x5 = 0.75. Compute ML Estimate...

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 B > 0 and let X1 , X2 , … , Xn be a random sample from
the distribution with probability density function.
f( x ; B ) = β/ (1 +x)^ (B+1), x > 0, zero otherwise.
(i) Obtain the maximum likelihood estimator for B, β ˆ .
(ii) Suppose n = 5, and x 1 = 0.3, x 2 = 0.4, x 3 = 1.0, x 4 =
2.0, x 5 = 4.0. Obtain the maximum likelihood...

Let X have a gamma distribution
with and which is unknown. Let be
a random sample from this distribution.
(1.1) Find a consistent estimator for using the
method-of-moments.
(1.2) Find the MLE of denoted by .
(1.3) Find the asymptotic variance of the MLE, i.e.
(1.4) Find a sufficient statistic for .
(1.5) Find MVUE for .

Let X1, X2, . . . Xn be iid
exponential random variables with unknown mean β. Find the method
of moments estimator of β

Let X1, X2,..., Xn be a random sample from a population with
probability density function f(x) = theta(1-x)^(theta-1), where
0<x<1, where theta is a positive unknown parameter
a) Find the method of moments estimator of theta
b) Find the maximum likelihood estimator of theta
c) Show that the log likelihood function is maximized at
theta(hat)

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