Question

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)

Answer #1

a)

b)

c)

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 a distribution with
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< ∞ 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
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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 B > 0 and let X1 , X2 , … , Xn be a random sample from
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f( x ; B ) = β/ (1 +x)^ (B+1), x > 0, zero otherwise.
(i) Obtain the maximum likelihood estimator for B, β ˆ .
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Problem 1. The Cauchy distribution with scale 1 has following
density function
f(x) = 1 / π [1 + (x − η)^2 ] , −∞ < x < ∞.
Here η is the location and rate parameter. The goal is to find the
maximum likelihood estimator of η.
(a) Find the log-likelihood function of f(x)
l(η; x1, x2, ..., xn) = log L(η; x1, x2, ..., xn) =
(b) Find the first derivative of the log-likelihood function
l'(η; x1, x2,...

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

Let X1, ..., Xn be a sample from an exponential population with
parameter λ.
(a) Find the maximum likelihood estimator for λ. (NOT PI
FUNCTION)
(b) Is the estimator unbiased?
(c) Is the estimator consistent?

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.

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.

Suppose X_1, X_2, … X_n is a random sample from a population
with density f(x) = (2/theta)*x*e^(-x^2/theta) for x greater or
equal to zero.
Find the Maximum Likelihood Estimator Theta-Hat_1 for
theta.
Find the Method of Moment Estimator Theta-Hat_2 for theta.

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