Chapter 1 Preliminaries

• Idempotent: A matrix $A$ is idempotent if $AA = A$

  • The rank of an idempotent matrix is equal to its trace, i.e. $rank(A) = trace(A)$

• Generalized Inverse: $G$ is a generalized inverse of $A$ if $AGA = A$

• Quadratic form, positive definite, non-negative definite

  • $A$ is PD $\Rightarrow$ $A^{-1}$ exists

• linear transformation

• multivariate normal distribution

  • if $y \sim N(\mu, \Sigma)$, then $y \stackrel{d}{=} Az + \mu$ where $z\sim N(0, I_{n\times n})$ and $A$ is an $m\times n$ matrix of rank $n$ such that $AA' = \Sigma$

  • if $y\sim N(\mu, I_{n\times n})$, then $w = y'y \sim \chi_n^2(\mu'\mu/2)$

  • Suppose $\Sigma$ is an $n\times n$ positive definite matrix, $A$ is an $n\times n$ symmetric matrix of rank $m$ such that $A\Sigma$ is idempotent ($A\Sigma A\Sigma = A\Sigma$). Then $$y \sim N(\mu, \Sigma) \Rightarrow y'Ay\sim \chi_m^2(\mu'A\mu/2)$$

  • If $w\sim \chi_m^2(\theta)$ then $E(w) = m + 2\theta$ and $Var(w) = 2m + 8\theta$

• Non-central t distribution: $y\sim N(\delta, 1)$, $w \sim \chi_m^2$, $y$ and $w$ are independent then $\frac{y}{\sqrt{w/m}}$ has a non-central $t$ distribution with $m$ degrees of freedom and non-centrality parameter $\delta$

• Non-central F distribution: $w_1 \sim \chi_{m_1}^2(\theta)$ and $w_2\sim \chi_{m_2}^2$. $w_1$ and $w_2$ are independent then $\frac{w_1/m_1}{w_2/m_2}$ has a non-central F distribution with $m_1$ numerator degrees of freedom, $m_2$ denominator degrees of freedom and non-centrality parameter $\theta$: $\frac{w_1/m_1}{w_2/m_2} \sim F_{m_1, m_2}(\theta)$.

  • if $u\sim t_m(\delta)$, then $u^2 \sim F_{1, m}(\delta^2/2)$.