Hotelling's T-squared distribution

Hotelling's T-squared distribution is a multivariate statistical distribution that generalizes the Student's t-distribution for the case of multiple variables. It is named after the American statistician Harold Hotelling, who introduced this distribution in the context of multivariate hypothesis testing and confidence intervals. The Hotelling's T-squared distribution is particularly useful in the field of multivariate analysis, where it is applied in procedures such as the analysis of variance (ANOVA) for multiple measurements and in the design of experiments.

Definition

The Hotelling's T-squared distribution arises when a sample mean vector from a normally distributed multivariate population is compared to a known mean vector, under the assumption that the population covariance matrix is known. The T-squared statistic is defined as:

\[ T^2 = n(\mathbf{\bar{x}} - \mathbf{\mu})' \mathbf{S}^{-1} (\mathbf{\bar{x}} - \mathbf{\mu}) \]

where \(n\) is the sample size, \(\mathbf{\bar{x}}\) is the sample mean vector, \(\mathbf{\mu}\) is the population mean vector, \(\mathbf{S}^{-1}\) is the inverse of the sample covariance matrix, and \('\) denotes the transpose of a vector or matrix.

Distribution

The distribution of the T-squared statistic under the null hypothesis (that the sample mean vector is equal to the population mean vector) follows a Hotelling's T-squared distribution. This distribution is closely related to the F-distribution, and the T-squared statistic can be transformed into an F-statistic:

\[ F = \frac{n-p}{(n-1)p} T^2 \]

where \(p\) is the number of variables. This transformation allows for the use of F-distribution tables to determine the critical values for hypothesis testing.

Applications

Hotelling's T-squared distribution is widely used in various fields such as biostatistics, psychometrics, and econometrics. Its applications include:

• Testing the equality of mean vectors in two or more groups
• Constructing confidence intervals for the difference between two population mean vectors
• Multivariate quality control, such as in the monitoring of manufacturing processes

See also

• Multivariate statistics
• F-distribution
• Analysis of variance
• Covariance matrix