Choice of the hypothesis matrix for using the Wald-type-statistic

A widely used formulation for null hypotheses in the analysis of multivariate $d$-dimensional data is $\mathcal{H}_0: \boldsymbol{H} \boldsymbol{\theta} =\boldsymbol{y}$ with $\boldsymbol{H}$ $\in\mathbb{R}^{m\times d}$, $\boldsymbol{\theta}$ $\in \mathbb{R}^d$ and $\boldsymbol{y}\in\mathbb{R}^m$, where $m\leq d$. Here the unknown parameter vector $\boldsymbol{\theta}$ can, for example, be the expectation vector $\boldsymbol{\mu}$, a vector $\boldsymbol{\beta} $ containing regression coefficients or a quantile vector $\boldsymbol{q}$. Also, the vector of nonparametric relative effects $\boldsymbol{p}$ or an upper triangular vectorized covariance matrix $\textbf{v}$ are useful choices. However, even without multiplying the hypothesis with a scalar $\gamma\neq 0$, there is a multitude of possibilities to formulate the same null hypothesis with different hypothesis matrices $\boldsymbol{H}$ and corresponding vectors $\boldsymbol{y}$. Although it is a well-known fact that in case of $\boldsymbol{y}=\boldsymbol{0}$ there exists a unique projection matrix $\boldsymbol{P}$ with $\boldsymbol{H}\boldsymbol{\theta}=\boldsymbol{0}\Leftrightarrow \boldsymbol{P}\boldsymbol{\theta}=\boldsymbol{0}$, for $\boldsymbol{y}\neq \boldsymbol{0}$ such a projection matrix does not necessarily exist. Moreover, since such hypotheses are often investigated using a quadratic form as the test statistic, the corresponding projection matrices often contain zero rows; so, they are not even effective from a computational aspect. In this manuscript, we show that for the Wald-type-statistic (WTS), which is one of the most frequently used quadratic forms, the choice of the concrete hypothesis matrix does not affect the test decision. Moreover, some simulations are conducted to investigate the possible influence of the hypothesis matrix on the computation time.