Alternative Equation Formats

$$\begin{array}{rcl}{P}_U(d_{\ast <!--\; \ast \; -->})& =& \underset{p_2}{sup}(\underset{A,T(a)\ge <!--\; \ge \; -->t_0}{\sum <!--\; \sum \; -->}f(n_{11},n_{21};n_1,n_2,d_{\ast <!--\; \ast \; -->},p_2))\\ P_L(d_{\ast <!--\; \ast \; -->})& =& \underset{p_2}{sup}(\underset{A,T(a)\le <!--\; \le \; -->t_0}{\sum <!--\; \sum \; -->}f(n_{11},n_{21};n_1,n_2,d_{\ast <!--\; \ast \; -->},p_2))\end{array}$$

$$\begin{array}{rcl}{P}_U(d_{\ast <!--\; \ast \; -->})& =& \underset{p_2}{sup}(\underset{A,T(a)\ge <!--\; \ge \; -->t_0}{\sum <!--\; \sum \; -->}f(n_{11},n_{21};n_1,n_2,d_{\ast <!--\; \ast \; -->},p_2))\\ P_L(d_{\ast <!--\; \ast \; -->})& =& \underset{p_2}{sup}(\underset{A,T(a)\le <!--\; \le \; -->t_0}{\sum <!--\; \sum \; -->}f(n_{11},n_{21};n_1,n_2,d_{\ast <!--\; \ast \; -->},p_2))\end{array}$$

\begin{eqnarray*} P_ U(d_\ast ) & = & \sup _{p_2} ~ \bigl ( \sum _{A, T(a) \geq t_0} f( n_{11}, n_{21}; n_1, n_2, d_\ast , p_2 ) ~ \bigr ) \\[0.10in] P_ L(d_\ast ) & = & \sup _{p_2} ~ \bigl ( \sum _{A, T(a) \leq t_0} f( n_{11}, n_{21}; n_1, n_2, d_\ast , p_2 ) ~ \bigr ) \end{eqnarray*}