Question

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

Answer #1

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,...

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 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...

X1,
X2,...Xn are iid random variables from a U(0,b) distribution. Which
estimator is an unbiased estimator for b?
2 X bar n
X bar n
1/n (X1squared + X2squared +....Xnsquared)
1/n2(X1squared + X2squared +....Xnsquared)

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 i.i.d. with N(theta, 1)
a) find the CR Rao lower-band for the variance of an
unbiased estimator of theta
b)------------------------------------of theta^2
c)-----------------------------------of P(X>0)

Let X1,...,Xn be iid
exp(θ) rvs.
(a) Compute the pdf of Xmin.
(b) Create an unbiased estimator for θ based on Xmin.
Compute the variance of the resulting estimator.
(c) Perform a Monte Carlo simulation of N= 10,0000 samples of
your unbiased estimator from part (b) using θ = 2 and n = 100 to
validate your answer. Include a histogram of the samples.
(d) Which is more efficient: your estimator from part (b) or the
MLE for θ?
(e)...

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 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.

Let n ≥ 2 and x1, x2, ..., xn > 0 be such that x1 + x2 + · ·
· + xn = 1. Prove that √ x1 + √ x2 + · · · + √ xn /√ n − 1 ≤ x1/ √
1 − x1 + x2/ √ 1 − x2 + · · · + xn/ √ 1 − xn

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