Linear Statistical Models Quiz 1: 29 October 2023 (September 2023 term)

1. Let $x$ and $y$ be vectors of integers.

(i) Write an R code to check if the given two vectors, say $x$ and $y$, have the same length or not. [2 Marks]

(ii) Write an R code to compute the dot product of two numeric vectors $x$ and $y$. [2 Marks]

(iii) Write an R code to create a new vector containing only the even numbers from the vector $x$. [3 Marks]

1. Suppose $\vec{Y} = X\vec{\beta} + \vec{\epsilon}$ be a linear model, where

$\begin{aligned} \vec{Y} &= (y_1, y_2, y_3, \ldots, y_n)^T, \\ \vec{\beta} &= (\beta_1, \beta_2, \beta_3, \ldots, \beta_m)^T, \\ X &= ((x_{ij})) \;; 1 \leq i \leq n, 1 \leq j \leq m \end{aligned}$

and

$\vec{\epsilon} = (\epsilon_1, \epsilon_2, \epsilon_3, \ldots, \epsilon_n)^T$

where, $\vec{\epsilon}$ has mean $\vec{0}$ and variance covariance matrix $\sigma^2 I_{n \times n}$.

(a) Define what is meant by $\hat{\vec{\beta}}$ is the least square estimate of $\vec{\beta}$? [3 Marks]

(b) Show that any least square estimate $\hat{\vec{\beta}}$ of $\vec{\beta}$ satisfies:

$(X^T X)\hat{\vec{\beta}} = X^T \vec{Y}$

[5 Marks]

1. An analyst wanted to predict the number of goals scored by a Canadian football club, for which he/she collected the following data:

(a) Draw a scatter plot of $(\vec{x}, \vec{y})$. [2 Marks]

(b) Let

$X = \begin{pmatrix} 1 & -1 \\ 1 & 0 \\ 1 & 1 \\ 1 & 2 \end{pmatrix}$

Suppose we assume the linear model $(\vec{y}, X\vec{\beta}, I_{5\times5})$ for the data. Find the least square estimate for $\vec{\beta}$. [4 Marks]

(c) For the values of $\vec{\beta}$ obtained, draw the least square line on the graph above. [2 Marks]

(d) Another analyst claim that the least square estimate for $\vec{\beta}$ is given by:

$\tilde{\vec{\beta}} = \hat{\vec{\beta}} + \begin{pmatrix} 1 \\ -1 \end{pmatrix}$

where, $\hat{\vec{\beta}}$ is the least square estimate obtained in (c).

Whose claim for least square estimate for $\vec{\beta}$ is correct and why? Elaborate [4Marks]

(e) Estimate the value of $y$ when $x = 5, -2$. [3 Marks]