Mathematical Statistics question

qwertykeyboards

Here it is,

Let the General Linear Model be represented by

y= Xß + u, u ~ N(0, (σ^2)I)

where X is n x p non-stochastic and of rank p and let ß = (X^t)y / [(X^t)X] be the OLS estimate of ß. Let c be a p x 1 vector, determine the distribution of (c^t)(ß), and hence devise a test statistic to test the hypothesis that (c^t)ß = d, for any number d.

If anyone could help me out with this it would be unreal. Totally lost.

Thomas_S_Hunterson

What exactly is it you don't understand? Is it the linear algebra? devising the test statistic? or just the whole question?

It's just a hypothesis test for a point estimate of y.

c is just the vector for an observation.
ß is the vector of coeficients

Multiplying them gives the estimate for y.

Have you access to a library? If so I'd recommend Applied Linear Statistical Models 5th ed, by Kutner, Nachtsheim, Neter & Li. IIRC chapter 5 covers the linear algebra of the models.

qwertykeyboards

thanks for the help. pretty much lost with all of it. im lookin through my lecturers notes but they're too long and cant seem to start the question. in library now just grabbed the 3rd edition of that book you said, its the latest copy here. gonna try read some of it and see if i can make sense of it. thanks

qwertykeyboards

read through all these notes now, is it common paractice to begin with 3 assumptions: 1. E[ui] = 0, 2. V[ui] = σ^2, 3. Cov[ui, uj] = 0

is that where i should start to do my test statistic?

Thomas_S_Hunterson

qwertykeyboards wrote: »

read through all these notes now, is it common paractice to begin with 3 assumptions: 1. E[ui] = 0, 2. V[ui] = σ^2, 3. Cov[ui, uj] = 0

Yea, all these assumptions are given in the question, albeit phrased differently:
u ~ N(0, (σ^2)I)

What it's saying is that observations will be normally distributed around the predicted value (for given X) with variance σ^2.

cbhoy

Just you wait til Gary Keogh sees this!:mad:

carlowboy

qwertykeyboards wrote: »

Here it is,

Let the General Linear Model be represented by

y= Xß + u, u ~ N(0, (σ^2)I)

where X is n x p non-stochastic and of rank p and let ß = (X^t)y / [(X^t)X] be the OLS estimate of ß. Let c be a p x 1 vector, determine the distribution of (c^t)(ß), and hence devise a test statistic to test the hypothesis that (c^t)ß = d, for any number d.

If anyone could help me out with this it would be unreal. Totally lost.

Is that for the assignment?