Question

Let X1, X2, ..., Xn be a random sample (of size n) from U(0,θ). Let Yn be the maximum of X1, X2, ..., Xn.

(a) Give the pdf of Yn.

(b) Find the mean of Yn.

(c) One estimator of θ that has been proposed is Yn. You may note from your answer to part (b) that Yn is a biased estimator of θ. However, cYn is unbiased for some constant c. Determine c.

(d) Find the variance of cYn, wherec is the constant you determined in part (c).

Answer #1

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?

let X1,X2,..............,Xn be a r.s from
N(θ,1). Find the best unbiased estimator for
(θ)^2

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 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.
c) Let Zn = nlnY1. Find the limiting distribution of Zn.
d) Let Wn = nln( θ/Yn ). Find the limiting distribution of
Wn.

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 and Y1, . . . , Yn be two random samples with
the same mean µ and variance σ^2 . (The pdf of Xi and Yj are not
specified.)
Show that T = (1/2)Xbar + (1/2)Ybar is an unbiased estimator of
µ.
Evaluate MSE(T; µ)

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 a random sample from the Bernoulli
distribution. Under the condition 1/2≤Θ≤1, find a
maximum-likelihood estimator of Θ.

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...

5. Let a random sample, X1, X2, ..., Xn of size n = 10 from a
distribution that is N(μ1, σ2 ) give ̄x = 4.8 and s 2+ 1 = 8.64 and
a random sample, Y1, Y2, ..., Yn of size n = 10 from a distribution
that is N(μ2, σ2 ) give y ̄ = 5.6 and s 2 2 = 7.88. Find a 95%
confidence interval for μ1 − μ2.

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