Problem set 9: Linear Regression

Problem 1

There are two pieces of cheese with weights \(a\) and \(b\). Using the same scale, the first piece, the second piece, and then both pieces together are weighed.

Find the least squares estimators of \(a\) and \(b\), as well as an unbiased estimator of the variance of the measurement errors.

Problem 2

In the Gaussian linear model, find confidence intervals with confidence level \(\gamma\) for the parameters \[ \theta_i,\qquad i\in\{1,\ldots,k\}, \] and for \(\sigma^2\).

Problem 3

Let \[ X_i=\beta_1+i\beta_2+\varepsilon_0+\cdots+\varepsilon_i, \qquad i=0,1,\ldots,n, \] where \(\beta_1,\beta_2\) are unknown parameters and \[ \varepsilon_0,\ldots,\varepsilon_n \] are independent random variables distributed according to \(\mathcal N(0,\sigma^2)\).

Reduce the problem to a linear model and find the least squares estimators of \(\beta_1\) and \(\beta_2\), as well as an unbiased estimator of \(\sigma^2\).

Problem 4

Let \(X_1,\ldots,X_n\) be a sample from a normal distribution with parameters \((a,\sigma^2)\).

Prove that the statistics \(\overline{X}\) and \(S^2\) are independent, and find the distribution of the statistic \[ nS^2. \]

Problem 5

Let \[ X_1,\ldots,X_n \] be a sample from \(\mathcal N(a,\sigma^2)\), where both parameters are unknown.

Construct exact confidence intervals for each of the parameters \(a\) and \(\sigma^2\).

Problem 6

Let \(X_i\), \(i\in\{1,2,\ldots,n\}\), be independent random variables distributed according to \[ \mathcal N(a+bi,\sigma^2). \]

Construct exact confidence intervals for the parameters \[ a,\qquad b,\qquad \sigma^2. \]

Problem 7

There are two objects with weights \(a\) and \(b\). We weigh, with measurement errors, the first object, the second object, and then both objects together. The variance of the measurement error in the last case is four times larger than in the first two cases.

Reduce the problem to a linear regression model and find the optimal estimators of \(a\) and \(b\).

Problem 8

Three objects with masses \(a\), \(b\), and \(c\) are weighed using the same scale as follows:

the second and third objects are weighed together \(n_1\) times;
the first and third objects are weighed together \(n_2\) times;
the first and second objects are weighed together \(n_3\) times.

Assuming that all measurement errors have distribution \(\mathcal N(0,\sigma^2)\), reduce the problem to a linear regression model and find the optimal estimators of \(a\), \(b\), and \(c\).