Speaker
Description
Linear hypotheses Hp = y regarding a parameter vector p arise in a wide range of scientific fields, including life sciences, psychology, economics, environmental sciences, and other areas of applied statistics, due to their ability to encode a wide variety of scientific questions using a simple algebraic framework. The unknown parameter vector p can represent, for example, an expectation vector, a vector of regression coefficients, a quantile vector, the vector of nonparametric relative effects, or an upper-triangular vectorised covariance matrix. This general formulation allows both classical parametric models and modern semi- or nonparametric frameworks to be analysed.
There exist numerous ways to express the same null hypothesis through different matrices H and corresponding vectors y. While for y = 0 there exists a unique projection matrix P representing the same hypothesis, such a unique matrix does not necessarily exist otherwise. Furthermore, practical implementations often use matrices that are not of full rank, which can increase computational cost and reduce numerical stability, particularly in high-dimensional settings.
Linear hypotheses are typically tested using quadratic forms, resulting in univariate test statistics; such quadratic forms possess numerous desirable properties. Among the most widely used are the Wald-type statistic (WTS) and the ANOVA-type statistic (ATS), which are commonly employed and considered fundamental tools for testing linear hypotheses. This raises important methodological questions: To what extent does the value of the quadratic form based test statistic depend on the specific choice of the pair (H,y)? How can one obtain a representation that is both unique and minimal in dimension, avoiding redundancy? And how can additional structure be imposed to improve computational and interpretational properties?
In this contribution, we show that for WTS, ATS and quadratic forms, a companion hypothesis matrix with the minimal number of rows can be constructed for each hypothesis, that formulates the same null hypothesis, while always yielding identical test decisions. Moreover, these minimal matrices can be derived constructively, with explicit formulas available for key matrices such as the centering matrix. Finally, our approach yields a procedure for selecting the hypothesis matrix when y is not equal to zero, thereby improving reproducibility.
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