Question

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 of θ.

Answer #1

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, 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)

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 X1, X2, · · · , Xn be a random sample from the distribution,
f(x; θ) = (θ + 1)x^ −θ−2 , x > 1, θ > 0. Find the maximum
likelihood estimator of θ based on a random sample of size n
above

Let X1, X2 · · · , Xn be a random sample from the distribution
with PDF, f(x) = (θ + 1)x^θ , 0 < x < 1, θ > −1.
Find an estimator for θ using the maximum likelihood

1. Let X1, X2, . . . , Xn be a random sample from a distribution
with pdf f(x, θ) = 1 3θ 4 x 3 e −x/θ , where 0 < x < ∞ and 0
< θ < ∞. Find the maximum likelihood estimator of ˆθ.

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, 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...

Let X2, ... , Xn denote a random sample
from a discrete uniform distribution over the integers - θ, - θ +
1, ... , -1, 0, 1, ... , θ - 1, θ,
where θ is a positive integer. What is the maximum
likelihood estimator of θ?
A) min[X1, .. , Xn]
B) max[X1, .. , Xn]
C) -min[X1, .. , Xn]
D) (max[X1, .. , Xn] -
min[X1, .. , Xn]) / 2
E) max[|X1| , ... , |Xn|]

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.

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