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Linear regression and maximum likelihood estimates

  1. Jan 28, 2012 #1
    1. The problem statement, all variables and given/known data

    Suppose that data (x1,y1),(x2,y2),.?.,(xn,yn) is modelled 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
     
  2. jcsd
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