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Regression model

  1. Jun 5, 2016 #1
    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

    I have no idea about this question. I get the matrix
    $$X = \begin{bmatrix}
    1 \\
    2 \\
    \vdots \\
    n
    \end{bmatrix}$$ but not sure on ##W##. Once I can get this I can pretty much do the rest.

    Please help!!!!
     
  2. jcsd
  3. Jun 6, 2016 #2
    Never mind I got it but as usual your help was appreciated!!!!!!
     
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