# Regression model

1. Jun 5, 2016

### squenshl

1. The problem statement, all variables and given/known data
Suppose that we observe $i=1,2,\ldots,n$ independent observations that can be modeled as follows:
$$Y_i = i(\theta+\epsilon_i) \quad \text{where} \; \; \epsilon_i \sim N(0,\sigma^2).$$

1. Write the above as a regression model, $E(Y) = X\theta$, $\text{Cov}(Y) = \sigma^2W$ for matrices $X$ and $W$.

2. Show that $X^TW^{-1}X = n$.

3. Show that the least squares estimate for $\theta$ is given by
$$\hat{\theta} = \frac{1}{n}\sum_{i=1}^{n} \frac{Y_i}{i}.$$

Consider the following transformation: $Z_i = \frac{Y_i}{i}.$

4. Show the transformed model can be written as a regression model,
$$E(Z) = 1_n\theta, \quad \text{Cov}(Z) = \sigma^2I_n$$
where $1_n$ is a column vector of $1$s and $I_n$ is an identity matrix of dimension $n$.

5. Show that the least squares estimate from this model is exactly the same as the solution from part c).

2. Relevant equations

3. The attempt at a solution

$$X = \begin{bmatrix} 1 \\ 2 \\ \vdots \\ n \end{bmatrix}$$ but not sure on $W$. Once I can get this I can pretty much do the rest.

2. Jun 6, 2016

### squenshl

Never mind I got it but as usual your help was appreciated!!!!!!