Does the Error Distribution Affect Least Squares Estimation in Simple Models?

[cutesteph]

Suppose we have an observation y = c+ e where c is an unknown constant and e is error with the pdf = exp(-e-1) for e >-1 . We want to determine the least square estimator of c given by the c* which minimizes the error cost function E(c) = .5(y-c)^2

Minimizing the error cost is done by taking the derivative wrt c so y=c. Shouldnt it take into account the distribution of the error?

I understand in the matrix case E(c) = T(e)e where T( ) is the transpose . where y=Hc+e

= T(y-Hc) (y-Hc) = T(y)y -T( x)Hy -T(y)Hx +T(x)T(H)Hx .
The derivative wrt x is -2T(y)H+2T(x)T(H)H= 0 => x=inverse(T(H)H)*T(H)y

I guess I am just confused on the scalar case on what to do.

 

[mfb]

Minimizing the error cost is done by taking the derivative wrt c so y=c. Shouldnt it take into account the distribution of the error?

This would fix e to 0. You cannot fix your random variable to minimize your error.
What is the expected value for E(c)? Afterwards you can try to minimize this.