Therefore, in order to calculate the variance of T, we need to calculate the covariance for each pair (j,k) which can be expressed as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\fontsize{8.5pt}{9.3pt}\selectfont{ \begin{aligned} {}\lefteqn{\text{cov}\{-2\log(p_{j}), -2\log(p_{k})\}}\\ {}&=& E\{[-2\log(p_{j})] [-2\log(p_{k})]\} - E\{-2\log(p_{j})\}E\{-2\log(p_{k})\}\\ {}&=& 4\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\log\{2\Phi(-|z_{j}|)\} \log\{2\Phi(-|z_{k}|)\}dF(z_{j},z_{k}) - 4, \end{aligned}}} $$ \end{document}cov{−2log(pj),−2log(pk)}=E{[−2log(pj)][−2log(pk)]}−E{−2log(pj)}E{−2log(pk)}=4∫−∞∞∫−∞∞log{2Φ(−|zj|)}log{2Φ(−|zk|)}dF(zj,zk)−4, where F is the standard bivariate Gaussian distribution. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\delta_{jk}=\text{cov}\{-2\log(p_{j}), -2\log(p_{k})\}. $$ \end{document}δjk=cov{−2log(pj),−2log(pk)}.
