Examples

• \begin{equation}
  \hat{\theta}_{w_i} = \hat{\theta}(s(t,{\cal U}_{w_i})).
  \end{equation}
  

  gives
  

  $$\hat{\theta}_{w_i} = \hat{\theta}(s(t,{\cal U}_{w_i})).$$ (7)

  
  

• \begin{eqnarray}
  {\cal M}^2(\hat{\theta},\theta) &=& E[(\hat{\theta} - \theta)^2]
  \nonumber \\
  {\cal M}^2(\hat{\theta},\theta) &=& {\rm var}^2(\hat{\theta}) + 
  {\cal B}^2(\hat{\theta}).
  \end{eqnarray}
  

  gives
  

  $${\cal M}^2(\hat{\theta},\theta) E[(\hat{\theta} - \theta)^2]$$ =

  $${\cal M}^2(\hat{\theta},\theta) {\rm var}^2(\hat{\theta}) + {\cal B}^2(\hat{\theta}).$$ = (8)

  
  

• \begin{equation}
  \hat{W}_{s}(t,\omega;\phi) \bydefn 
  \int\limits_{-\infty}^{\infty} 
  {\hat{\cal R}_s(t,\tau;\psi) e^{-j\omega \tau}
  \, d \tau }
  \end{equation}
  

  gives
  

  $$\hat{W}_{s}(t,\omega;\phi) \stackrel{def}{=}\int\limits_{-\in... ...} {\hat{\cal R}_s(t,\tau;\psi) e^{-j\omega \tau} \, d \tau }$$ (9)

  
  

• \begin{eqnarray}
  {\cal B}(t,\omega) & \approx &
  {1 \over 4\pi}
  {\cal D}_t^2 W_{\bf S}(t, \omega)
  {{{\scriptstyle \infty} \atop
  {\displaystyle \int \! \int \!
  }}\atop {\scriptstyle -\infty}}
  t_1^2
  \phi(t_1,\omega_1) \, dt_1 d\omega_1
  \nonumber \\
  && +
  {1 \over 4\pi}
  {\cal D}_\omega^2 W_{\bf S}(t, \omega)
  {{{\scriptstyle \infty} \atop
  {\displaystyle \int \! \int \!
  }}\atop {\scriptstyle -\infty}}
  \omega_1^2
  \phi(t_1,\omega_1) \, dt_1 \, d\omega_1.
  \label{F4}
  \end{eqnarray}
  

  gives
  

   

  $${\cal B}(t,\omega) \approx {1 \over 4\pi} {\cal D}_t^2 W_{\bf S}(t, \omega) {{{\scriptstyle ... ...t \! }}\atop {\scriptstyle -\infty}} t_1^2 \phi(t_1,\omega_1) \, dt_1 d\omega_1$$

  $$+ {1 \over 4\pi} {\cal D}_\omega^2 W_{\bf S}(t, \omega) {{{\scrip... ...top {\scriptstyle -\infty}} \omega_1^2 \phi(t_1,\omega_1) \, dt_1 \, d\omega_1.$$ (10)

  
  

• \newsavebox{\DERIVBOXZLM}
  \savebox{\DERIVBOXZLM}[2.5em]{$\Longrightarrow\hspace{-1.5em}
  \raisebox{.2ex}{*}
  \hspace{-.7em}\raisebox{-.8ex}{\scriptsize lm}\hspace{.7em}$}
  \newcommand{\Deriveszlm}{\usebox{\DERIVBOXZLM}}
  
  \Deriveszlm
  

  gives