An Introduction to Probability and Statistical Inference, 2nd Edition

Exercises

3.1 Verify the result $\frac{d^2}{d{t}^2}\text{?}\log \left(y_1\dots ,y_n;\text{?}{\widehat{\beta }}_1,\text{?}{\widehat{\beta }}_2,\text{?}t\right)|{t=S{S}_{E/n}}_{}=-\frac{n^3}{2S{S}_E^2}$ as claimed in the proof of Theorem 4(ii), where t = σ ².

3.2 Consider Table 13.1 and leave out the "outlier" pairs (3.63, 447), (3.44, 692), and (2.44, 336). Then recalculate all quantities below:

$\begin{array}{cccc}\begin{array}{cccc}\begin{array}{cc}{\displaystyle \sum _i{x}_i,}& {\displaystyle \sum _i{y}_i,}\end{array}& {\displaystyle \sum _i{x}_i^2,}& {\displaystyle \sum _i{x}_i{y}_i,}& {\displaystyle \sum _i{y}_i^2,}\end{array}& S{S}_x,& S{S}_y,& S{S}_{xy}.\end{array}$

3.3 Use the calculations in Exercise 3.2 to compute the estimates ${\widehat{\beta }}_1,\text{?}{\widehat{\beta }}_2$ and the fitted regression line.

3.4 Refer to Exercise ...