How to Find Least Square Estimates for Object Weights with Normal Error?

[bonfire09]

Homework Statement

Suppose that object 1 weighs \theta_1 and object two weights \theta_2. Then each object is weighed once and then together getting three observations y_1,y_2,y_3. The scale gives unbiased weights with normally distributed error (constant variance) Find the least square estimates for \theta_1 and \theta_2.

Homework Equations

The ls estimates for theta 1 and theta 2 is $\hat{\theta}=(X^TX)^{-1}X^TY$

The Attempt at a Solution

I wrote the full mode as such
y_1=\theta_1+\epsilon_1
y_2=\theta_2+\epsilon_2
y_3=\theta_1+\theta_2+\epsilon_3
Then it follows that in matrix form we get $\begin{bmatrix} y_{1} \\ y_{2} \\ y_{3} \end{bmatrix} = \begin{bmatrix} \ \theta_1 \\ \theta_2 \\ \theta_1+\theta_2 \end{bmatrix}+\begin{bmatrix} \epsilon_1 \\ \epsilon_2 \\ \epsilon_3 \end{bmatrix}$ $=\begin{bmatrix} y_{1} \\ y_{2} \\ y_{3} \end{bmatrix} = \begin{bmatrix} 1 &0 \\ 0 & 1 \\1 & 1 \end{bmatrix} \begin{bmatrix} \theta_1 \\ \theta_2 \end{bmatrix}+\begin{bmatrix} \epsilon_1 \\ \epsilon_2 \\ \epsilon_3 \end{bmatrix}$.

Then from here I got as my final answer
$\hat{\theta}=(X^TX)^{-1}X^TY=\begin{bmatrix} \hat{\theta_1} \\ \hat{\theta_2} \end{bmatrix} =\begin{bmatrix} y_1-y_2 \\ \frac{-1}{2}y_1+y_2+\frac{1}{2}y_3 \end{bmatrix}$ The thing the I am not sure about is if I wrote my full model correctly. I was thinking that the observations are just the true weight of the object with some random error. But I am not sure.

 

[Ray Vickson]

Homework Statement

Suppose that object 1 weighs \theta_1 and object two weights \theta_2. Then each object is weighed once and then together getting three observations y_1,y_2,y_3. The scale gives unbiased weights with normally distributed error (constant variance) Find the least square estimates for \theta_1 and \theta_2.

Homework Equations

The ls estimates for theta 1 and theta 2 is $\hat{\theta}=(X^TX)^{-1}X^TY$

The Attempt at a Solution

I wrote the full mode as such
y_1=\theta_1+\epsilon_1
y_2=\theta_2+\epsilon_2
y_3=\theta_1+\theta_2+\epsilon_3
Then it follows that in matrix form we get $\begin{bmatrix} y_{1} \\ y_{2} \\ y_{3} \end{bmatrix} = \begin{bmatrix} \ \theta_1 \\ \theta_2 \\ \theta_1+\theta_2 \end{bmatrix}+\begin{bmatrix} \epsilon_1 \\ \epsilon_2 \\ \epsilon_1+\epsilon_2 \end{bmatrix}$ $=\begin{bmatrix} y_{1} \\ y_{2} \\ y_{3} \end{bmatrix} = \begin{bmatrix} 1 &0 \\ 0 & 1 \\1 & 1 \end{bmatrix} \begin{bmatrix} \theta_1 \\ \theta_2 \end{bmatrix}+\begin{bmatrix} \epsilon_1 \\ \epsilon_2 \\ \epsilon_1+\epsilon_2 \end{bmatrix}$.

Then from here I got as my final answer
$\hat{\theta}=(X^TX)^{-1}X^TY=\begin{bmatrix} \hat{\theta_1} \\ \hat{\theta_2} \end{bmatrix} =\begin{bmatrix} y_1-y_2 \\ \frac{-1}{2}y_1+y_2+\frac{1}{2}y_3 \end{bmatrix}$ The thing the I am not sure about is if I wrote my full model correctly. I was thinking that the observations are just the true weight of the object with some random error. But I am not sure.

According to your model you are dealing with a very smart scale. It says to itself "I remember the errors $\epsilon_1$ and $\epsilon_2$ that I made when the guy weighed objects 1 and 2 separately. I see now that he is weighing the two objects together, so I had better add up the two errors I made before". As I said, a very smart scale indeed.

 

[mfb]

I was thinking that the observations are just the true weight of the object with some random error.

Right, but the random error applies to the measurements, not to the weights.

 

[bonfire09]

Oops now I fixed that. I don't know why I placed that in the first place. So I changed $\epsilon_1+\epsilon_2$ to $\epsilon_3$. I think it looks correct now.

 

[Ray Vickson]

Oops now I fixed that. I don't know why I placed that in the first place. So I changed $\epsilon_1+\epsilon_2$ to $\epsilon_3$. I think it looks correct now.

Yes, I would say so.