Linear regression and maximum likelihood estimates

[stukbv]

Homework Statement

Suppose that data (x1,y1),(x2,y2),.?.,(xn,yn) is modeled with xi being non random and Yi being observed values of random variables Y1,Y2,...Yn which are given by
Yi = a + b(xi-xbar) + σεi
Where a, b, σ are unknown parameters and εi are independent random variables each having the Gaussian distribution with mean 0 variance 1. xbar = 1/n * Ʃxi

Find the maximum likelihood estimate of a, b and σ



2. The attempt at a solution
Firstly I know I need to find the joint distribution of the random variables Yi
Yi is a linear combination of Gaussian random variables so Yi has a normal distribution too
E[Yi] = E[a + b(xi-xbar) + σεi] = a + b(xi-xbar)
Var[Yi] = var [a + b(xi-xbar) + σεi] = var[σεi] = σ²
All Yi's are mutually independent so the joint distribution of all the Yi's is just the product of n normally distributed random variables with means and variances as shown above.

So now we want MLE of a so we take logs of the joint distribution and differentiate wrt to a and set this equal to zero, we then rearrange to get the a= xbar is the MLE for a. Is this correct, if so I can go on to do the rest, if not why not?
Thanks

 

[mighty2000]

for your help!

Your approach is correct for finding the MLE of a. To find the MLE of b and σ, you will need to take partial derivatives with respect to b and σ and set them equal to zero. This will give you a system of equations that you can solve to find the MLE of b and σ.