Hotelling's T- square distribution

The Hotelling's T-square distribution is a probability distribution that was first described by Harold Hotelling in 1931 . It is a generalization of the Student's t-distribution .

definition

Hotelling's T-square distribution is defined as

${\displaystyle t^{2}=n({\mathbf {x} }-{\mathbf {\mu } })'{\mathbf {W} }^{-1}({\mathbf {x} }-{\mathbf {\mu } })}$

With

• ${\displaystyle n}$ a number of points
• ${\displaystyle {\mathbf {x} }}$is a column vector with elements ${\displaystyle p}$
• ${\displaystyle {\mathbf {W} }}$is a - covariance matrix . ${\displaystyle p\times p}$

properties

Let it be a random variable with a multivariate normal distribution and (independent of ) have a Wishart distribution with a non-singular variance matrix and with . Then, the distribution is : , Hotelling's T-squared distribution with parameters and . ${\displaystyle x\sim N_{p}(\mu ,{\mathbf {V} })}$ ${\displaystyle {\mathbf {W} }\sim W_{p}(m,{\mathbf {V} })}$ ${\displaystyle x}$ ${\displaystyle \mathbf {V} }$ ${\displaystyle m=n-1}$ ${\displaystyle t^{2}}$ ${\displaystyle T^{2}(p,m)}$ ${\displaystyle p}$ ${\displaystyle m}$

${\displaystyle F}$be the F-distribution . Then it can be shown that:

${\displaystyle {\frac {m-p+1}{pm}}T^{2}\sim F_{p,m-p+1}}$.

Assuming that

${\displaystyle {\mathbf {x} }_{1},\dots ,{\mathbf {x} }_{n}}$

${\displaystyle p\times 1}$Are column vectors with real numbers.

${\displaystyle {\overline {\mathbf {x} }}=(\mathbf {x} _{1}+\cdots +\mathbf {x} _{n})/n}$

be the mean. The positive definite matrix ${\displaystyle p\times p}$

${\displaystyle {\mathbf {W} }=\sum _{i=1}^{n}(\mathbf {x} _{i}-{\overline {\mathbf {x} }})(\mathbf {x} _{i}-{\overline {\mathbf {x} }})'/(n-1)}$

be your “sample variance” matrix. (The transpose of a matrix is denoted by). Let be a column vector (if used an estimator of the mean). Then the Hotellings T-squared distribution ${\displaystyle M}$ ${\displaystyle M'}$ ${\displaystyle \mu }$ ${\displaystyle p\times 1}$

${\displaystyle t^{2}=n({\overline {\mathbf {x} }}-{\mathbf {\mu } })'{\mathbf {W} }^{-1}({\overline {\mathbf {x} }}-{\mathbf {\mu } }).}$

${\displaystyle t^{2}}$has a close relationship with the squared Mahalanobis distance .

In particular, it can be shown that if are independent and and are as defined above, then Wishart has a distribution with degrees of freedom such that ${\displaystyle {\mathbf {x} }_{1},\dots ,{\mathbf {x} }_{n}\sim N_{p}(\mu ,{\mathbf {V} })}$ ${\displaystyle {\overline {\mathbf {x} }}}$ ${\displaystyle {\mathbf {W} }}$ ${\displaystyle {\mathbf {W} }}$ ${\displaystyle n-1}$

${\displaystyle \mathbf {W} \sim W_{p}(V,n-1)}$

and is independent of and ${\displaystyle {\overline {\mathbf {x} }}}$

${\displaystyle {\overline {\mathbf {x} }}\sim N_{p}(\mu ,V/n)}$.

It follows

${\displaystyle t^{2}=n({\overline {\mathbf {x} }}-{\mathbf {\mu } })'{\mathbf {W} }^{-1}({\overline {\mathbf {x} }}-{\mathbf {\mu } })\sim T^{2}(p,n-1).}$

Individual evidence

1. ^ H. Hotelling (1931). The generalization of student's ratio, Ann. Math. Statist., 2 (3), pp. 360-378, doi : 10.1214 / aoms / 1177732979 JSTOR 2957535 .
2. ^ KV Mardia, JT Kent, and JM Bibby (1979) Multivariate Analysis , Academic Press, ISBN 0-12-471250-9 .