Changes

We further derive the variance-covariance matrix of these statistics as

We further derive the variance-covariance matrix of these statistics as

                      <math> \mathbf{V}=(\mathbf{G}-\bar{\mathbf{G}})^T (\boldsymbol{\Omega}^(-1)-\boldsymbol{\Omega}^(-1) \mathbf{X}(\mathbf{X^T}\boldsymbol{\Omega}^(-1)\mathbf{X})^(-1) \mathbf{X^T} \boldsymbol{\Omega}^(-1))(\mathbf{G}-\bar{\mathbf{G}}) </math>.

<math> \mathbf{V}=(\mathbf{G}-\bar{\mathbf{G}})^T (\hat{\boldsymbol{\Omega}}^(-1)-\hat{\boldsymbol{\Omega}}^(-1) \mathbf{X}(\mathbf{X^T}\hat{\boldsymbol{\Omega}}^(-1)\mathbf{X})^(-1) \mathbf{X^T} \hat{\boldsymbol{\Omega}}^(-1))(\mathbf{G}-\bar{\mathbf{G}}) </math>.

Under the null, test statistics T_i=(U_i^2)/V_ii  is asymptotically distributed as chi-squared with one degree of freedom.

Under the null, test statistics T_i=(U_i^2)/V_ii  is asymptotically distributed as chi-squared with one degree of freedom.