Problem 1

Consider the following weighted no-intercept regression model: \(y_i=\beta x_i+\epsilon _i\), where \(\epsilon_i\sim N(0, \sigma^2 x_i)\) and \(cov(y_i,y_j)=0\) if \(i\ne j\).

  1. Find the maximum likelihood estimators of \(\beta\) and \(\sigma^2\) directly, without the use of theorem (6.5.2).

  2. Assuming we know \(\sigma\), find a \((1-\alpha)100\%\) confidence interval for \(\beta\).

  3. For the case \(x=0:100/100\), \(\beta=\sigma=2\) and a 95% confidence interval, write a simulation in R to check whether the interval from part b has correct coverage.

Problem 2

Below are the values of the response variable for the following model: \(y_i= \alpha i/10 + \beta i^2/100 +\epsilon_i\), i=10,..,100.

  1. Test at the 5% level

\[H_0:\beta=0\text{ vs }H_a:\beta\ne 0\] that is we want to test whether a quadratic term is needed.

  1. If in fact \(\beta=0.015\), what is the power of this test if the value of the F test statistic is the same as in part a? (Use \(\sigma=1/2\))
1.1 0.9 1.1 0.2 1.3 1.6 0.9 1.2 1.3 1.7 1.9 1.9 3.1 2.5 2.8 1.7 2.6 2.6 2.2 3.1 3.2 3.3 3.3 3.3 3.6 3.9 2.7 5.1 4 3.3 2.5 3.7 5 3.6 3.4 5.2 4.6 5 4.6 4.4 4.5 4.7 4.7 5.7 5.9 5.7 5.9 6.2 7.2 7.1 6.3 7.1 6.8 6.7 6.7 6.3 7.3 7.8 7.1 7.1 7.1 8.1 7.7 8 8.4 7.6 8 7.6 8.3 8.9 8.8 8.7 8.2 8.7 8.6 9.3 8.7 9 9.7 9.5 8.8 9.1 9.8 9.5 9.2 10.5 10.3 9.5 10 10.7 11.5