Erratum to: A Wald test with enhanced selectivity properties in homogeneous environments

After publication of our work [1] we noticed the following mistakes:

Equation (3) should be

$$ \boldsymbol{I}\left(\boldsymbol{\theta} \right)=\mathrm{E}\left[\left(\frac{\partial \ln f\left(\boldsymbol{x},\boldsymbol{X};\boldsymbol{\theta} \right)}{\partial {\boldsymbol{\theta}}^{*}}\right)\left(\frac{\partial \ln f\left(\boldsymbol{x},\boldsymbol{X};\boldsymbol{\theta} \right)}{\partial {\boldsymbol{\theta}}^T}\right)\right] $$

(3)

Equation (5) should be

$$ {t}_{\mathrm{Wald}}={\left({\widehat{\boldsymbol{\theta}}}_{r_1}-{\boldsymbol{\theta}}_{r_0}\right)}^H{\left\{{\left[{\boldsymbol{I}}^{-1}\left({\widehat{\boldsymbol{\theta}}}_1\right)\right]}_{{\boldsymbol{\theta}}_r{\boldsymbol{\theta}}_r^{*}}\right\}}^{-1}\left({\widehat{\boldsymbol{\theta}}}_{r_1}-{\boldsymbol{\theta}}_{r_0}\right) $$

(5)

The notation “--“ in equations (6), (11), and (12) should be replaced by “-“, i.e., the minus sign.

Equations (17), (18), and (19) should be

$$ {\tilde{t}}_{\mathrm{ACE}}=\frac{{\tilde{\boldsymbol{x}}}^H{\boldsymbol{P}}_{\tilde{\boldsymbol{s}}}\tilde{\boldsymbol{x}}}{{\tilde{\boldsymbol{x}}}^H{\boldsymbol{P}}_{\tilde{\boldsymbol{s}}}^{\perp}\tilde{\boldsymbol{x}}}=\frac{{\left\Vert {\boldsymbol{P}}_{\tilde{\boldsymbol{s}}}\tilde{\boldsymbol{x}}\right\Vert}^2}{{\left\Vert {\boldsymbol{P}}_{\tilde{\boldsymbol{s}}}^{\perp}\tilde{\boldsymbol{x}}\right\Vert}^2} $$

(17)

$$ {t}_{\mathrm{Wald}}=\frac{1}{1+{\left\Vert \tilde{\boldsymbol{x}}\right\Vert}^2{\left\Vert {\boldsymbol{P}}_{\tilde{\boldsymbol{s}}}^{\perp}\tilde{\boldsymbol{x}}\right\Vert}^2/{\left\Vert {\boldsymbol{P}}_{\tilde{\boldsymbol{s}}}\tilde{\boldsymbol{x}}\right\Vert}^2} $$

(18)

and

$$ {\tilde{t}}_{\mathrm{Wald}}=\frac{{\left\Vert {\boldsymbol{P}}_{\tilde{\boldsymbol{s}}}\tilde{\boldsymbol{x}}\right\Vert}^2}{{\left\Vert {\boldsymbol{P}}_{\tilde{\boldsymbol{s}}}^{\perp}\tilde{\boldsymbol{x}}\right\Vert}^2{\left\Vert \tilde{\boldsymbol{x}}\right\Vert}^2} $$

(19)

respectively.