 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2021

Paper 3

Under $H_0$, write the <singular value decomposition>
$$
X=UDV^T,\qquad D_{jj}=\sqrt{\lambda_j}.
$$
Then
$$
\widehat\beta
=VD^{-1}U^T\varepsilon.
$$
The coordinates of $U^T\varepsilon/\sigma$ are independent standard normal variables. Hence
$$
\frac{\|\widehat\beta\|^2}{\sigma^2}
\mathrel{\overset d=}
\sum_{j=1}^p\lambda_j^{-1}W_j,
\qquad
W_j\sim\chi^2_1
$$
independently. By parts (ii) and (iii),
$$
Z=\frac{n\widehat\sigma^2}{\sigma^2}\sim\chi^2_{n-p}
$$
is independent of all the $W_j$. Consequently
$$
\boxed{
T\mathrel{\overset d=}
\frac{\sum_{j=1}^p\lambda_j^{-1}W_j}{Z}
}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 18H / iv / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 18H / iv / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 18H / iv / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)