2009-74.md

course	course_year	question_number	tags	title	year
Statistics	IB	74	IB 2009 Statistics	Paper 1, Section I, $\mathbf{7 H} \quad$	2009

What does it mean to say that an estimator $\hat{\theta}$ of a parameter $\theta$ is unbiased?

An $n$-vector $Y$ of observations is believed to be explained by the model

$$Y=X \beta+\varepsilon$$

where $X$ is a known $n \times p$ matrix, $\beta$ is an unknown $p$-vector of parameters, $p<n$, and $\varepsilon$ is an $n$-vector of independent $N\left(0, \sigma^{2}\right)$ random variables. Find the maximum-likelihood estimator $\hat{\beta}$ of $\beta$, and show that it is unbiased.