Question

VERY URGENT !!!! Suppose that X and Y are random samples of
observations from a population with mean μ and variance
σ^{2}.

Consider the following two unbiased point estimators of μ.

A = (7/4)X - (3/4)Y _{ } B = (1/3)X +
(2/3)Y

**[Give your answers as ratio (eg: as number _{1} /
number_{2} ) and DO NOT make any
cancellation]**

**1.** Find variance of A. Var(A)
=(Answer)*σ^{2}

**2.** Find variance of B. Var(B)
=(Answer)*σ^{2}

**3.** Efficient and unbiased point estimator for μ
is (Answer)

Answer #1

Suppose that X and Y
are random samples of observations from a population with mean μ
and variance σ2.
Consider the following
two unbiased point estimators of μ.
A = (2/3)X + (1/3)Y B
= (5/4)X - (1/4)Y
Find variance of A.
Var(A) and Var(B)
Efficient and unbiased
point estimator for μ is = ?

Suppose that E[X]= E[Y] = mu, where mu is a fixed unknown
number. We have independent simple random samples of size n each
from the distribution of X and Y, respectively. Suppose that Var[X]
= 2*Var[Y]. Consider the following estimators of mu:
m1 = bar{X}
m2 = bar{Y}/2
m3 = 3*bar{X}/4 + 2*bar{Y}/8
where bar{X} and bar{Y} are the sample mean of X and Y values,
respectively. Which of the estimators are unbiased?

Suppose an investor can invest in two stocks, whose returns are
random variables X and Y, respectively. Both are assumed to have
the same mean returns E(X) = E(Y) = μ; and they both have the same
variance Var(X) = Var(Y) = σ2. The correlation between X and Y is
some valueρ.
The investor is considering two invesment portfolios: (1)
Purchase 5 shares of the first stock (each with return X ) and 1 of
the second (each with return...

Problem 3. Let Y1, Y2, and Y3 be independent, identically
distributed random variables from a population with mean µ = 12 and
variance σ 2 = 192. Let Y¯ = 1/3 (Y1 + Y2 +
Y3) denote the average of these three random
variables.
A. What is the expected value of Y¯, i.e., E(Y¯ ) =? Is Y¯ an
unbiased estimator of µ?
B. What is the variance of Y¯, i.e, V ar(Y¯ ) =?
C. Consider a different estimator...

Suppose X1 is from a population with mean µ and variance 1 and
X2 is from a population with mean µ and variance 4 (X1, X2 are
independent). Construct an estimator of ? as ?̂=??1+(1−?)?2. Show
that ?̂ is unbiased for ?. Find the most efficient estimator in
this class, that is, find the value of a such that the estimator
has the smallest variance.

Suppose X1 is from a population with mean µ and variance 1 and
X2 is from a population with mean µ and variance 4 (X1, X2 are
independent). Construct an estimator of ? as ?̂=??1+(1−?)?2. Show
that ?̂ is unbiased for ?. Find the most efficient estimator in
this class, that is, find the value of a such that the estimator
has the smallest variance.

Suppose X_1, X_2, … X_n is a random sample from a population
with density f(x) = (2/theta)*x*e^(-x^2/theta) for x greater or
equal to zero.
Determine if Theta-Hat_1 (MLE) is a minimum variance unbiased
estimator for thet
Determine if Theta-Hat_2 (MOM) is a minimum variance unbiased
estimator for theta.

1) Suppose a random variable, x, arises from a binomial
experiment. Suppose n = 6, and p = 0.11.
Write the probability distribution. Round to six decimal places,
if necessary.
x
P(x)
0
1
2
3
4
5
6
Find the mean.
μ =
Find the variance.
σ2 =
Find the standard deviation. Round to four decimal places, if
necessary.
σ =
2) Suppose a random variable, x, arises from a binomial
experiment. Suppose n = 10, and p =...

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, X2, . . . , Xn be iid exponential random variables with
unknown mean β.
(1) Find the maximum likelihood estimator of β.
(2) Determine whether the maximum likelihood estimator is
unbiased for β.
(3) Find the mean squared error of the maximum likelihood
estimator of β.
(4) Find the Cramer-Rao lower bound for the variances of
unbiased estimators of β.
(5) What is the UMVUE (uniformly minimum variance unbiased
estimator) of β? What is your reason?
(6)...

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