\begin{array} { r l r } { E \left[ \boldsymbol { Y } _ { \tau } ( \boldsymbol { \theta } ) \right] } & { = } & { a \mathrm { ~ } \frac { \tau } { 2 } \left\vert \boldsymbol { \Sigma } \left( \boldsymbol { \theta } \right) \right\vert ^ { - \tau / 2 } E \left[ - t r a c e \left( \boldsymbol { \Sigma } \left( \boldsymbol { \theta } \right) ^ { - 1 } \frac { \partial \boldsymbol { \Sigma } \left( \boldsymbol { \theta } \right) } { \partial \boldsymbol { \theta } } \right) \right. } \\ & { } & { \left. \left. \exp \left\{ - \frac { \tau } { 2 } \left( \boldsymbol { Y } - \boldsymbol { \mu } \left( \boldsymbol { \theta } \right) \right) ^ { T } \boldsymbol { \Sigma } \left( \boldsymbol { \theta } \right) ^ { - 1 } \left( \boldsymbol { Y } - \boldsymbol { \mu } \left( \boldsymbol { \theta } \right) \right) \right\} + b \mathrm { ~ } t r a c e \left( \boldsymbol { \Sigma } \left( \boldsymbol { \theta } \right) ^ { - 1 } \frac { \partial \boldsymbol { \Sigma } \left( \boldsymbol { \theta } \right) } { \partial \boldsymbol { \theta } } \right) \right. \right. } \\ & { } & { \left. \left. + \mathrm { ~ } \exp \left\{ - \frac { \tau } { 2 } \left( \boldsymbol { Y } - \boldsymbol { \mu } \left( \boldsymbol { \theta } \right) \right) ^ { T } \boldsymbol { \Sigma } \left( \boldsymbol { \theta } \right) ^ { - 1 } \left( \boldsymbol { Y } - \boldsymbol { \mu } \left( \boldsymbol { \theta } \right) \right) \right\} \right. \right. } \\ & { } & { \left. \left[ - 2 \left( \frac { \partial \boldsymbol { \mu } \left( \boldsymbol { \theta } \right) } { \partial \boldsymbol { \theta } } \right) ^ { T } \boldsymbol { \Sigma } \left( \boldsymbol { \theta } \right) ^ { - 1 } \left( \boldsymbol { Y } - \boldsymbol { \mu } \left( \boldsymbol { \theta } \right) \right) \right. \right. } \\ & { } & { \left. \left. + \left( \boldsymbol { Y } - \boldsymbol { \mu } \left( \boldsymbol { \theta } \right) \right) ^ { T } \left( \boldsymbol { \Sigma } \left( \boldsymbol { \theta } \right) ^ { - 1 } \frac { \partial \boldsymbol { \Sigma } \left( \boldsymbol { \theta } \right) } { \partial \boldsymbol { \theta } } \boldsymbol { \Sigma } \left( \boldsymbol { \theta } \right) ^ { - 1 } \right) \left( \boldsymbol { Y } - \boldsymbol { \mu } \left( \boldsymbol { \theta } \right) \right) \right] \right] } \\ & { = } & { a \mathrm { ~ } \frac { \tau } { 2 } \left\vert \boldsymbol { \Sigma } \left( \boldsymbol { \theta } \right) \right\vert ^ { - \tau / 2 } \left\{ - t r a c e \left( \boldsymbol { \Sigma } \left( \boldsymbol { \theta } \right) ^ { - 1 } \frac { \partial \boldsymbol { \Sigma } \left( \boldsymbol { \theta } \right) } { \partial \boldsymbol { \theta } } \right) \frac { 1 } { \left( \tau + 1 \right) ^ { m / 2 } } \right. } \\ & { } & { \left. + \frac { \tau } { \left( \tau + 1 \right) ^ { \frac { m } { 2 } + 1 } } t r a c e \left( \boldsymbol { \Sigma } \left( \boldsymbol { \theta } \right) ^ { - 1 } \frac { \partial \boldsymbol { \Sigma } \left( \boldsymbol { \theta } \right) } { \partial \boldsymbol { \theta } } \right) \right. } \\ & { } & { \left. + \frac { 1 } { \left( \tau + 1 \right) ^ { \frac { m } { 2 } + 1 } } t r a c e \left( \boldsymbol { \Sigma } \left( \boldsymbol { \theta } \right) ^ { - 1 } \frac { \partial \boldsymbol { \Sigma } \left( \boldsymbol { \theta } \right) } { \partial \boldsymbol { \theta } } \right) \right\} } \\ & { = } & { \boldsymbol { 0 } _ { d } . } \end{array}