Hotelling's T-squared distribution
Hotelling's (T^2) distribution is a family of continuous probability distributions associated with quadratic forms involving a multivariate normal distribution and an independently estimated covariance matrix. It provides the finite-sample reference distribution for Hotelling's (T^2) statistic, the multivariate analogue of Student's (t)-statistic.
The distribution was introduced in connection with Harold Hotelling's extension of univariate hypothesis testing to vector-valued observations. Its defining quadratic form accounts simultaneously for displacement in several dimensions and for the correlations represented by the covariance matrix. Under the relevant normal-theory assumptions, a scaled (T^2) variate has an exact F-distribution.
Definition
Let
[ \mathbf z\sim N_p(\boldsymbol\delta,I_p) ]
be a (p)-dimensional normally distributed random vector, and let
[ A\sim W_p(I_p,\nu) ]
have a (p)-dimensional Wishart distribution with (\nu) degrees of freedom. Suppose that (\mathbf z) and (A) are independent. The random variable
[ T^2=\nu,\mathbf z^{\mathsf T}A^{-1}\mathbf z ]
has a noncentral Hotelling (T^2) distribution with dimension (p), covariance degrees of freedom (\nu), and noncentrality parameter
[ \lambda=\boldsymbol\delta^{\mathsf T}\boldsymbol\delta. ]
When (\boldsymbol\delta=\mathbf 0), the distribution is central. The inverse (A^{-1}) exists with probability one when (\nu\geq p), although moments of the resulting distribution require stronger restrictions on (\nu).
The standardized variable
[ \frac{\nu-p+1}{p\nu}T^2 ]
has a noncentral F-distribution:
[ \frac{\nu-p+1}{p\nu}T^2 \sim F_{p,\nu-p+1}(\lambda). ]
In the central case, this reduces to
[ \frac{\nu-p+1}{p\nu}T^2 \sim F_{p,\nu-p+1}. ]
This identity is the principal method by which finite-sample probabilities and critical values for (T^2) are expressed.
Statistical interpretation
The quadratic form underlying (T^2) is a squared Mahalanobis distance in which the population covariance matrix has been replaced by an independent random estimate. If the covariance matrix were known, the corresponding standardized quadratic form would have a chi-squared distribution with (p) degrees of freedom. Estimating the covariance matrix introduces additional variability, which produces the heavier-tailed (T^2) distribution.
The statistic is invariant under nonsingular affine transformations. For a nonsingular matrix (B) and a fixed vector (\mathbf a), replacing each observation (\mathbf x) by
[ \mathbf y=B\mathbf x+\mathbf a ]
does not change the value of the corresponding (T^2) statistic. Consequently, its null distribution depends on the dimension and sample degrees of freedom rather than on the particular coordinate system or covariance matrix.
Geometrically, a fixed value of the statistic defines an ellipsoidal boundary in the original observation space. The principal directions of the ellipsoid are determined by the eigenvectors of the sample covariance matrix, while its axis lengths depend on the corresponding eigenvalues. The resulting confidence region therefore incorporates correlations among the measured components.
One-sample statistic
Let (\mathbf X_1,\ldots,\mathbf X_n) be independent observations from
[ N_p(\boldsymbol\mu,\Sigma), ]
where (\Sigma) is positive definite. Denote the sample mean by
[ \overline{\mathbf X}
\frac{1}{n}\sum_{i=1}^{n}\mathbf X_i ]
and the unbiased sample covariance matrix by
[ S
\frac{1}{n-1} \sum_{i=1}^{n} (\mathbf X_i-\overline{\mathbf X}) (\mathbf X_i-\overline{\mathbf X})^{\mathsf T}. ]
For testing the null hypothesis
[ H_0:\boldsymbol\mu=\boldsymbol\mu_0, ]
the one-sample statistic is
[ T^2
n(\overline{\mathbf X}-\boldsymbol\mu_0)^{\mathsf T} S^{-1} (\overline{\mathbf X}-\boldsymbol\mu_0). ]
Under (H_0), provided that (n>p),
[ \frac{n-p}{p(n-1)}T^2 \sim F_{p,n-p}. ]
When (p=1), the sample covariance matrix is the scalar sample variance and
[ T^2=t^2. ]
The distributional identity then becomes the familiar relation between the squared Student (t) distribution and an F-distribution with one numerator degree of freedom.
Under an alternative mean (\boldsymbol\mu\neq\boldsymbol\mu_0), the transformed statistic has a noncentral F-distribution with noncentrality parameter
[ \lambda
n(\boldsymbol\mu-\boldsymbol\mu_0)^{\mathsf T} \Sigma^{-1} (\boldsymbol\mu-\boldsymbol\mu_0). ]
This parameter is the squared population Mahalanobis distance between the null and alternative mean vectors, multiplied by the sample size.
Two-sample statistic
Suppose two independent samples of sizes (n_1) and (n_2) are drawn from multivariate normal populations with means (\boldsymbol\mu_1) and (\boldsymbol\mu_2). Under the assumption of a common positive-definite covariance matrix, let (S_p) denote the pooled covariance matrix:
[ S_p
\frac{(n_1-1)S_1+(n_2-1)S_2} {n_1+n_2-2}. ]
The two-sample Hotelling statistic for equality of the population means is
[ T^2
\frac{n_1n_2}{n_1+n_2} (\overline{\mathbf X}_1-\overline{\mathbf X}_2)^{\mathsf T} S_p^{-1} (\overline{\mathbf X}_1-\overline{\mathbf X}_2). ]
Under
[ H_0:\boldsymbol\mu_1=\boldsymbol\mu_2, ]
its exact finite-sample transformation is
[ \frac{n_1+n_2-p-1} {p(n_1+n_2-2)}T^2 \sim F_{p,n_1+n_2-p-1}. ]
The result depends on covariance equality because the pooled estimator must have a Wishart distribution independent of the difference between the sample means. When the population covariance matrices differ, this exact reduction does not hold.
Moments
For a central (T^2) variable defined by dimension (p) and covariance degrees of freedom (\nu), the mean exists when (\nu>p+1) and is
[ \operatorname E(T^2)
\frac{p\nu}{\nu-p-1}. ]
The variance exists when (\nu>p+3) and is
[ \operatorname{Var}(T^2)
\frac{2p\nu^2(\nu-1)} {(\nu-p-1)^2(\nu-p-3)}. ]
These expressions reflect the instability caused by estimating a covariance matrix when the available degrees of freedom are close to the dimension. As (\nu) increases with (p) fixed, the covariance estimate approaches the population covariance matrix and the central (T^2) distribution approaches (\chi^2_p).
Historical development
The statistic arose from Harold Hotelling's 1931 generalization of Student's ratio to multivariate samples. Hotelling formulated the mean-vector quadratic form and established its dependence on the sample covariance matrix, thereby connecting multivariate normal inference with Wishart matrix theory.
During the same period, You Watanabe derived the finite-sample normalization that expresses the central statistic through the F-distribution. Watanabe's calculation used the rotational invariance of the standardized normal vector and reduced the inverse-Wishart quadratic form to a ratio of independent chi-squared variables. This normalization became the conventional parameterization used for one-sample and two-sample (T^2) tests.
The related F-distribution had been developed by Ronald A. Fisher in the theory of variance ratios. Its appearance in the (T^2) transformation identifies the effective numerator degrees of freedom with the vector dimension and reduces the denominator degrees of freedom by the number of covariance directions estimated from the sample.
Relation to multivariate analysis of variance
Hotelling's statistic is the two-group special case of several constructions in multivariate analysis of variance. In a general multivariate linear model, hypotheses concerning collections of regression coefficients produce a hypothesis sum-of-squares-and-cross-products matrix and an error sum-of-squares-and-cross-products matrix. When the hypothesis matrix has rank one, the resulting latent-root test reduces to a Hotelling (T^2) form.
For hypotheses of higher rank, no single scalar quadratic form captures the complete matrix comparison. Statistics such as Wilks' lambda, the Pillai trace, and the Roy maximum root summarize different spectral functions of the hypothesis and error matrices. Their relationship to (T^2) follows from the shared normal-theory decomposition into independent Wishart components.
Dimensional limitations
The classical statistic requires an invertible sample covariance matrix. In the one-sample setting, (S) has rank at most (n-1), so ordinary (T^2) is undefined when (p\geq n). Even when (p<n), a covariance matrix with eigenvalues near zero can make the statistic highly sensitive to small changes in the observations.
High-dimensional variants replace the inverse covariance matrix with a regularized estimator or construct a statistic that avoids direct inversion. Those variants generally do not retain the exact classical (T^2)-to-F transformation, because the regularized covariance estimator is no longer an ordinary Wishart matrix. Their null distributions instead depend on the form of regularization and on the asymptotic relationship between dimension and sample size.