Question

Let X_{1},..., X_{n} be a random sample from a
distribution with pdf as follows:

fX(x) = e^-(x-θ) , x > θ

0 otherwise.

Find the sufficient statistic for θ.

Find the maximum likelihood estimator of θ.

Find the MVUE of θ,θˆ

Is θˆ a consistent estimator of θ?

Answer #1

Let X1, . . . , Xn be a random sample from the following
pdf:
f(x|θ)= (x/θ)*e^(-x^2/2θ). x>0
(a) Find a sufficient statistic for θ.

Let X1, X2, ..., Xn be a random sample from a distribution with
probability density function f(x; θ) = (θ 4/6)x 3 e −θx if 0 < x
< ∞ and 0 otherwise where θ > 0
. a. Justify the claim that Y = X1 + X2 + ... + Xn is a complete
sufficient statistic for θ. b. Compute E(1/Y ) and find the
function of Y which is the unique minimum variance unbiased
estimator of θ.
b. Compute...

6. Let θ > 1 and let X1, X2, ..., Xn be a random sample from
the distribution with probability density function f(x; θ) =
1/(xlnθ) , 1 < x < θ.
a) Obtain the maximum likelihood estimator of θ, ˆθ.
b) Is ˆθ a consistent estimator of θ? Justify your answer.

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 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...

Let X1, X2, ·······, Xn be a random sample from the Bernoulli
distribution. Under the condition 1/2≤Θ≤1, find a
maximum-likelihood estimator of Θ.

Consider a random sample X1,
X2, ⋯ Xn from the
pdf
fx;θ=.51+θx, -1≤x≤1;0,
o.w., where (this distribution arises in particle
physics).
Find the method of moment estimator of θ.
Compute the variance of your estimator. Hint: Compute the
variance of X and then apply the formula for X, etc.

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 a random sample from an exponential
distribution f(x) = (1/θ)e^(−x/θ) for x ≥ 0. Show that likelihood
ratio test of H0 : θ = θ0 against H1 : θ ≠ θ0 is based on the
statistic n∑i=1 Xi.

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)

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