3. Let X1...Xn be N(μX,σ) and Y1...Yn be iid N(μy,σ)
with the two samples X1...Xn, and Y1...Xn independent of each
other. Assume that the common population SD σ is known but the two
means are not. Consider testing the hypothesis null: μx = μy vs
alternative: μx ≠ μy.
d. Assume σ=1 and n=20. How large must δ be for the size
0.01 test to have power at least 0.99?
e. Assume σ=1and δ=0.2. How large must n be for...
3. Let X1...Xn be N(μX,σ) and Y1...Yn be iid N(μy,σ)
with the two samples X1...Xn, and Y1...Xn independent of each
other. Assume that the common population SD σ is known but the two
means are not. Consider testing the hypothesis null: μx = μy vs
alternative: μx ≠ μy.
a. Find the likelihood ratio test statistic Λ. Specify
which MLEs you are using and how you plug them in.
*Please show work and explain steps*
Assume Y1, ... , Yn are IID continuous variables with PDF f(yi;
θ), where f is dependent on a parameter θ.
Complete the following:
a) Derive the likelihood, L(θ), and the log-likelihood, l(θ), in
terms of the function f.
b) Find dl/d(theta) in terms of f(yi; θ) and df/d(theta). Note
that dl/d(θ) is usually referred to as the score function.
c) Show that E[dl/d(θ)]= 0. Hint: you can use without proof the
following: ∫...
Suppose that the random variable Y1,...,Yn
satisfy Yi = ?xi +
?i i=1,...,n.
where the set of xi are fixed constants and
?i are iid random variables following a normal
distributions of mean zero and variance ?2.
?a (with a hat on it) =
?i=1nYi
xi / ?i=1nx2i
is unbiased estimator for ?.
The variance is ?a (with a hat on it) =
?2/ ?i=1nx2i
. What is the distribation of this variance?
Let
Y1, ... , Yn be a random sample that follows normal distribution
N(μ,2σ^2)
i)get the mle for σ^2
ii)prove using i) that it is an efficient estimator
Suppose that Y1 ,Y2 ,...,Yn is
a random sample from distribution Uniform[0,2].
Let Y(n) and Y(1) be the order
statistics.
(a) Find E(Y(1))
(b) Find the density of (Y(n) − 1)2
(c) Find the density of Y(n) − Y (1)
Suppose that X1, X2, , Xm and Y1, Y2, , Yn are independent
random samples, with the variables Xi normally distributed with
mean μ1 and variance σ12 and the variables Yi normally distributed
with mean μ2 and variance σ22. The difference between the sample
means, X − Y, is then a linear combination of m + n normally
distributed random variables and, by this theorem, is itself
normally distributed.
(a) Find E(X − Y).
(b) Find V(X − Y).
(c)...
Suppose Y1, . . . , Yn are independent
random variables with common density fY(y) =
eμ−y y > μ.
Derive a 95% confidence interval for μ. Find the MLE for μ to
use as the estimator