Hotelling's T-squared distribution

In statistics Hotelling's T-squared distribution (T2) is a multivariate distribution proportional to the F-distribution and arises importantly as the distribution of a set of statistics which are natural generalizations of the statistics underlying Student's t-distribution. Hotelling's t-squared statistic (t2) is a generalization of Student's t-statistic that is used in multivariate hypothesis testing.To show this use the fact that x ¯ ∼ N p ( μ , Σ x ¯ ) {displaystyle {overline {mathbf {x} }}sim {mathcal {N}}_{p}({oldsymbol {mu }},{mathbf {Sigma } }_{ar {mathbf {x} }})} derive the characteristic function of the random variable y = n ( x ¯ − μ ) ′ Σ − 1 ( x ¯ − μ ) {displaystyle mathbf {y} =n({ar {mathbf {x} }}-{oldsymbol {mu }})'{mathbf {Sigma } }^{-1}({ar {mathbf {x} }}-{oldsymbol {mathbf {mu } }})} . This is done below: In statistics Hotelling's T-squared distribution (T2) is a multivariate distribution proportional to the F-distribution and arises importantly as the distribution of a set of statistics which are natural generalizations of the statistics underlying Student's t-distribution. Hotelling's t-squared statistic (t2) is a generalization of Student's t-statistic that is used in multivariate hypothesis testing. The distribution arises in multivariate statistics in undertaking tests of the differences between the (multivariate) means of different populations, where tests for univariate problems would make use of a t-test.The distribution is named for Harold Hotelling, who developed it as a generalization of Student's t-distribution. If the vector pd1 is Gaussian multivariate-distributed with zero mean and unit covariance matrix N(p01,pIp) and pMp is a p x p matrix with unit scale matrix and m degrees of freedom with a Wishart distribution W(pIp,m), then the Quadratic form m(1dT p M−1pd1) has a Hotelling T2(p,m) distribution with dimensionality parameter p and m degrees of freedom. If a random variable X has Hotelling's T-squared distribution, X ∼ T p , m 2 {displaystyle Xsim T_{p,m}^{2}} , then: where F p , m − p + 1 {displaystyle F_{p,m-p+1}} is the F-distribution with parameters p and m−p+1.