NMST539  Lab Session 4Wishart Distribution(application for confidence bands and statistical tests)LS 2017  Tuesday 21/03/2017Rmd file (UTF8 encoding)The Rsoftware is available for download from the website: https://www.rproject.org A userfriendly interface (one of many): RStudio. Manuals and introduction into R (in Czech or English):
Wishart distributionThe Wishart distribution is, as an analogy to the \(\chi^2\) distribution and the variance inference in the univariate case, a very useful tool for the analysis of the variancecovariance matrix of same random sample \(X_{1}, \dots, X_{n}\) for \(n \in \mathbb{N}\) where each \(X_{i}\) in the sample is a \(p\)dimensional random vector (\(p\) different covariates recorded on each subject). Thus, it is probability distributions defined over symmetric and nonnegativedefinite random matrices. In general, the Wishart distribution is a multivariate generalization of the \(\chi^2\) distribution: for instance, for a normaly distributed random sample \(\boldsymbol{X}_{1}, \dots, \boldsymbol{X}_{n}\) drawn from some multivariate distribution \(N_{p}(\boldsymbol{0}, \Sigma)\), with the zero mean vector and \(\Sigma\) being some variancecovariance matrix we have that \[ \mathbb{X}^{\top}\mathbb{X} \sim W_{p}(\Sigma, n), \] for \(\mathbb{X} = (\boldsymbol{X}_{1}, \dots, \boldsymbol{X}_{n})^{\top}\). Similarly, for a random sample \(X_{1}, \dots, X_{n}\) drawn from \(N(0,1)\) we have that \[ \boldsymbol{X}^{\top}\boldsymbol{X} \sim \chi_{n}^2, \] where \(\boldsymbol{X} = (X_{1}, \dots, X_{n})^\top\). Thus, for a sample of size \(n \in \mathbb{N}\) drawn from \(N(0,1)\) the corresponding distribution of \(\mathbb{X}^\top\mathbb{X}\) is \(W_{1}(1, n) \equiv \chi_{n}^2\). The corresponding density function of the Wishart distribution takes the form \[ f(\mathcal{X}) = \frac{1}{2^{np/2} \Sigma^{n/2} \Gamma_{p}(\frac{n}{2})} \cdot \mathcal{X}^{\frac{n  p  1}{2}} e^{(1/2) tr(\Sigma^{1}\mathcal{X})}, \] for \(\mathcal{X}\) being a \(p\times p\) random matrix and \(\Gamma_{p}(\cdot)\) is a multivariate generalization of the Gamma function \(\Gamma(\cdot)\) In the R software there are various options (packages) on how to use and apply the Wishart distribution. To Do by Yourself (Theoretical and Practical)
A simple random sample from Wishart distribution \(W_{1}(\Sigma = 1, n = 10)\) (equivalently a \(\chi^2\) distribution with \(n = 10\) degrees of freedom) can be obtained, for instance, as
Using a standard approach for generating a univariate random sample we would use the R function
To Do by Yourself
Hotelling’s \(\boldsymbol{T^2}\) DistributionIn a similar manner as we define a classical \(t\)distribution in an univariate case (i.e. standard normal \(N(0,1)\) variable devided by a square root of a \(\chi^2\) variable normalized by its degrees of freedom) we define a multivariate generalization (Hotelling’s \(T^{2}\) distribution) as \[ n \boldsymbol{Y}^{\top} \mathbb{M}^{1} \boldsymbol{Y} \sim T^{2}(n, p), \] to be a random variable with the Hotelling’s \(T^2\) distribution with \(p \in \mathbb{N}\) to be the dimension of \(Y \sim N_{p}(0, \mathbb{I})\) and \(n \in \mathbb{N}\) being the parameter of the Wishart distribution of \(\mathbb{M} \sim W_{p}(\mathbb{I}, n)\). A special case for \(p = 1\) gives the standard Fisher distribution with one and \(n\) degrees of freedom (equivalently a square of the \(t\)distribution with \(n\) degrees of freedom). The Hotelling’s \(T^2\) distribution with parameters \(p, n \in \mathbb{N}\) can be identivied with the Fisher distribution using the following expression \[ T^{2}(p, n) \equiv \frac{n p}{n  p + 1}F_{p, n  p + 1}. \] In general, the relationship betwenn the Hotelling’s \(T^2\) distribution and the Fisher distribution is given by the same expression \[ T^{2}(p, n) \equiv \frac{n p}{n  p + 1}F_{p, n  p + 1}. \] The effect of different parameter settings can be (a little) visualized, for instance, by the following figure:
Moreover, using the relationship between the Hotelling’s \(T^2\) distribution and the Fisher’s F distriubtion we can effectively use the \(F\)distribution to draw the corresponding critical values when needed for some statistical tests. For some multivariate random sample \(\boldsymbol{X}_{1}, \dots, \boldsymbol{X}_{n} \sim N_{p}(\boldsymbol{\mu}, \Sigma)\) for some unknown mean vector \(\boldsymbol{\mu} \in \mathbb{R}^{p}\) and some variancecovariance matrix \(\Sigma\), we have that \[ (n  1)\Big(\overline{\boldsymbol{X}}  \boldsymbol{\mu}\Big)^\top \mathcal{S}^{1} \Big(\overline{\boldsymbol{X}}  \boldsymbol{\mu}\Big) \sim T^2(p, n  1), \] which can be also equivalently expressed as \[ \frac{n  p}{p} \Big(\overline{\boldsymbol{X}}  \boldsymbol{\mu}\Big)^\top \mathcal{S}^{1} \Big(\overline{\boldsymbol{X}}  \boldsymbol{\mu}\Big) \sim F_{p, n  p}. \] This can be now used to construct confidence regions for the unknown mean vector \(\boldsymbol{\mu}\) of for testing hypothesis about the true value of the vector of parameters \(\boldsymbol{\mu} = (\mu_{1}, \dots, \mu_{p})^{\top}\). However, rather than construction a confidence region for \(\boldsymbol{\mu}\) (which can be impractical in higher dimensions for even slightly larger values of \(p\)) one focusses on construction confidence intervals for the elements of \(\boldsymbol{\mu}\) such that the mutual coverage is under control (usually we require a simultaneous coverage of \((1  \alpha)\times 100~\%\) for some small \(\alpha \in (0,1)\)). For a hypothesis test \[ H_{0}: \boldsymbol{\mu} = \boldsymbol{\mu}_{0} \in \mathbb{R}^{p} \] \[ H_{1}: \boldsymbol{\mu} \neq \boldsymbol{\mu}_{0} \in \mathbb{R}^{p} \] we can use the following test statistic: \[ (n  1)\Big(\overline{\boldsymbol{X}}  \boldsymbol{\mu}_{0}\Big)^\top \mathcal{S}^{1} \Big(\overline{\boldsymbol{X}}  \boldsymbol{\mu}_{0}\Big), \] which, under the null hypothesis, follows the \(T^2(p, n  1)\) distribution. Equivalently we also have that \[ \frac{n  p}{p} \Big(\overline{\boldsymbol{X}}  \boldsymbol{\mu}_{0}\Big)^\top \mathcal{S}^{1} \Big(\overline{\boldsymbol{X}}  \boldsymbol{\mu}_{0}\Big) \] follows the Fisher \(F\)distribution with \(p\) and \(n  p\) degrees of freedom. In the R software we can use the library
and we can use the function Using the same approach we can also construct the confidence elipsoid for \(\boldsymbol{\mu} \in \mathbb{R}^p\)  it holds that \[ \frac{n  p}{p} \Big(\overline{\boldsymbol{X}}  \boldsymbol{\mu}_{0}\Big)^\top \mathcal{S}^{1} \Big(\overline{\boldsymbol{X}}  \boldsymbol{\mu}_{0}\Big) \sim F_{p, n  p}, \] and therefore, the following set \[ \left\{\boldsymbol{\mu} \in \mathbb{R}^p;~ \Big( \overline{\boldsymbol{X}}  \boldsymbol{\mu}\Big)^\top \mathcal{S}^{1} \Big( \overline{\boldsymbol{X}}  \boldsymbol{\mu}\Big) \leq \frac{p}{n  p} F_{p, n p}(1  \alpha) \right\} \] is a confidence region at the confidence level of \(\alpha = 1  \alpha\) for the vector of parameters \(\boldsymbol{\mu} \in \mathbb{R}^p\)  an interior of isodistance ellipsoid in \(\mathbb{R}^p\). A brief example on how it works (example from the lecture notes):
