slope test in linear regression with known intercept and known error variance

[elodie....@gmail.com]

Dear Forum,

Suppose the following model
Y_ij=1+beta*X_i+eps_ij
with j=1,2,...n_i and i=1,2,3,4.
the eps_ij are iid standard normal rvs

The goal is to test
Ho:beta =<0
vs. H1:beta >0

Question 1:
Consider the first following procedure:
Run a two-sample test based only on Y_1j, j=1,...,100 (sample1) and
Y_4j, j=1,...,100 (sample 2). Here, n1=n4=100, n2=n3=0.
Find the 0.05-level test of this procedure.

I am thinking that I can use a t test. the test would reject if

(betaols-0)/s(betahat)>t_{200-1}(0.95)

where betaols=SSY/SSX is the OLS estimate of beta and s(betaols) is
computed as usual with the MSE. Should I be using a t test? it is
given that the eps_ij are standard normal, so I could use a normal
test, right? In this case, the test would be

(betaols-0)*sqrt{SSX}/1>z(0.95)

Question 2:
Take n replications at each x_i (n1=n2=n3=n4=n). Obtain the MLE and
base the test of that estimator.
Find the 0.05-level test of this procedure.

I am thinking that the distribution of betamle is Normal(beta, sigma^2/
SSX). So I would run the following the normal test
(betamle-0)*sqrt(SSX)/1>z(0.95)

Where betamle is SSY/SSX, but this time the SSX and SSY are different
from those of question 1.

I greatly appreciate your help. In a third question, I need to compare
the power of each of the tests, so I will need to express
SSY(question2) as a function of SSY(question1), and SSX(question2) as
a function of SSX(question1). Can anybody help me do that?

[Jack Tomsky]

YOu can use all the data. Let Zij = Yij-1.

Then the LS estimate of beta is

betahat = Sum[(Xi)Sum(Zij)]/Sum(Ni*Xi^2)

where the sums go from i = 1, ..., 4 and j = 1, ..., Ni.

The varaince of betahat is

Var(betahat) = 1/Sum(Ni*Xi^2)

Then

betahat/Sqrt(Var(betahat)) ~ N(0,1).

Jack

[elodie....@gmail.com]

Many thanks for your reply. Is there a book or a paper that would be a
good reference for this problem?

Many thanks for your help.

[Jack Tomsky]

All I did was to put it into the form of a general linear model, Z = X*beta, where z is a column vector of Z_11, ..., Z_4,N4, X is a vector of X1, ..., X1, ...,X4, ..., X4 and beta is a scalar.

Then the LS estimate of beta is

betahat = X'Z/X'X.

Algebraic manipulations reduce it to the form I gave.

Since the covariance matrix of Z is given as the identity matrix,

Var(betahat) = X'X/(X'X)^2 = 1/(X'X).

Hope this helps.

Jack

[elodie....@gmail.com]

I understand, many thanks!

Question 2 asks for a test based on the MLE. I can say that the MLE is
just the same as the OLS, right?

[Jack Tomsky]

Yes, under normality, the MLE and OLS of the means are the same.

Jack