Homework 5

In this homework we will study the no-intercept model \(y_i=\beta x_i+\epsilon_i\), where \(\epsilon_i\sim N(0, \sigma^2)\), \(\sigma\) known. As data we have

set.seed(123)
x=1:10
y=round(10*x+rnorm(10, 0, 10), 1)
rbind(x, y)

##   [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## x  1.0  2.0  3.0  4.0  5.0  6.0  7.0  8.0  9.0  10.0
## y  4.4 17.7 45.6 40.7 51.3 77.2 74.6 67.3 83.1  95.5

Solve all the problems without using any of the results from the class.

Problem 1

Find the least squares estimator \(\hat{\beta}\) of \(\beta\).

Problem 2

Show that \(\hat{\beta}\) is an unbiased estimator of \(\beta\). Find its variance.

Problem 3

Find the distribution of \(\hat{\beta}\).

Problem 4

Use problem 3 to find a \(90\%\) confidence interval for \(\beta\) for the data.

Problem 5

Find that the maximum likelihood estimator of \(\beta\).

Problem 6

Now assume \(\sigma\) is not known. Show that

\[s^2=\frac{1}{n-1}\sum_{i=1}^n (y_i-\hat{\beta}x_i)^2\] is an unbiased estimator of \(\sigma^2\).