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 \(u^2 \sim F_{1, m}(\delta^2/2)\).