Question

Suppose that X1,..., Xn form a random sample from the
uniform distribution on the interval [0,θ], where the value of the
parameter θ is unknown (θ>0).

(1)What is the maximum likelihood estimator of θ?

(2)Is this estimator unbiased? (Indeed, show that it underestimates
the parameter.)

Answer #1

Suppose that X1,...,Xn ∼ U(0,θ); that is, a sample of N
observations from a random variable with a uniform distribution
where the lower bound is 0 and the upper bound θ is unknown. Find
the maximum likelihood estimate of θ, also demonstrating this in R.
Draw the pdf and the likelihood, and explain what they represent,
in R.

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|]

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...

Suppose X1, . . . , Xn are a random sample from a N(0, σ^2)
distribution. Find the MLE of σ^2
and show that it is an unbiased efficient estimator.

Suppose that (X1, · · · , Xn) is a random sample from uniform
distribution U(0, θ).
(a) Prove that T(X1, · · · , Xn) = X(n) is minimal sufficient
for θ. (X(n) is the largest order statistic, i.e., X(n) = max{X1, ·
· · , Xn}.)
(b) In addition, we assume θ ≥ 1. Find a minimal sufficient
statistic for θ and justify your answer.

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.

Suppose X1 ...... Xn is a random sample from the uniform
distribution on [a; b].
(a) Find the method of moments estimators of a and b.
(b) Find the maximum likelihood estimators of a and b.
please step by step

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 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?

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...

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