Question

1. Find the maximum likelihood estimator (MLE) of θ based on a random sample X1, . . . , Xn from each of the following distributions

(a) f(x; θ) = θ(1 − θ) ^(x−1) , x = 1, 2, . . . ; 0 ≤ θ ≤ 1

(b) f(x; θ) = (θ + 1)x ^(−θ−2) , x > 1, θ > 0

(c) f(x; θ) = θ^2xe^(−θx) , x > 0, θ > 0

Answer #1

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 X1,X2,...,Xn be i.i.d. Geometric(θ), θ = 1,2,3,... random
variables.
a) Find the maximum likelihood estimator of θ.
b) In a certain hard video game, a player is confronted with a
series of AI opponents and has an θ probability of defeating each
one. Success with any opponent is independent of previous
encounters. Until ﬁrst win, the player continues to AI contest
opponents. Let X denote the number of opponents contested until the
player’s ﬁrst win. Suppose that data of...

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...

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.

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.)

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 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 iid random variables with pdf
f(x|θ) = θx^(θ−1) , 0 < x < 1, θ > 0.
Is there an unbiased estimator of some function γ(θ), whose
variance attains the Cramer-Rao lower bound?

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.

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.

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