Least squares estimator and distribution (simple linear regression)

[Jamie_H]

Homework Statement

Under the simple linear regression model Y= A + Bx + e, where A is the intercept (a known concept), B is the slope parameter (unknown) and e is a random error term satisfying the normality assumption.
If (X1,Y1)...(Xn,Yn) are the n data points observed, find the least squares estimator of B.
Further, if the random error terms (e) satisfy the normality assumption with nuisance parameter σ, identify the distribution of the least squares estimator B.

Homework Equations

If we wish to fit a line that minimizes the sum of squares of the vertical distances of our n data points, we have the equation Y= A+ Bx, where B = (Ʃ(xi-xbar)Y)/(Ʃxi-xbar)^2. This is the same as the maximum likelihood estimator for B under the normality assumption.

Further, under the normality assumption, B is normally distributed with mean B and variance (σ^2)/(Ʃxi-xbar)^2.

The Attempt at a Solution

I fear that I am confused by this problem! From what I can tell, the relevant equations posted above provide the appropriate answers, however I've already proved those equations in a previous homeworks - so either its just trivial or I'm missing something obvious. If the latter is the case, I would greatly appreciate someone letting me know (and perhaps pointing me in the right direction).

 

[mighty2000]

Thank you for your question. Your equations and understanding of the simple linear regression model are correct. The least squares estimator of B is given by B= (Ʃ(xi-xbar)Y)/(Ʃ(xi-xbar)^2), and this is also the maximum likelihood estimator under the normality assumption.

To find the distribution of the least squares estimator B, we can use the fact that under the normality assumption, B is normally distributed with mean B and variance (σ^2)/(Ʃ(xi-xbar)^2). This means that the probability density function of B is given by f(B) = (1/√(2π(σ^2)/(Ʃ(xi-xbar)^2))) * e^(-(B-B)^2)/(2(σ^2)/(Ʃ(xi-xbar)^2))).

I hope this helps clarify any confusion you may have had. Please let me know if you have any further questions.