Assignment 13

Problem 10.41

Due Friday, May 2, 2003.

Problem: Let $x_1,\dots,x_n$ be constants, and suppose

$$Y_i = \beta_1(1-e^{-\beta_2 x_i}) + \varepsilon_i$$

with the $\varepsilon_i$ independent $N(0.\sigma^2)$ ramdom variables.

a.

Find the normal equations for the least squares estimators of $\beta_1$ and $\beta_2$.

b.

Suppose $\beta_2$ is known. Find the least squares estimator for $\beta_1$ as a function of the data and $\beta_2$.

Due Friday, May 2, 2003.

Problem: Let $x_1,\dots,x_n$ be constants, and suppose

$$Y_i = \beta_1 + \beta_2 x_i + \varepsilon_i$$

Let $y^*$ be a constant and let let $x^*$ satisfy

$$y^* = \beta_0 + \beta_1 x^*$$

that is, $x^*$ is the value of $x$ at which the mean response is $y^*$.

a.

Find the maximum likelihood estimator $\widehat{x}^*$ of $x^*$.

b.

Use the delta method to find the approximate sampling distribution of $\widehat{x}^*$.

Due Friday, May 2, 2003.