[QuantEcon]MATLAB混编FORTRAN语言

With simple algebra the copula density with normal marginal distributions simplifies to
$$
c(u, ~v, ~\rho) = \frac{1}{\sqrt{1 - \rho^2}} ~exp\bigg(\frac{1}{2 ~(1 ~-~ \rho^2)}~ \bigg( \frac{2~\rho~(x - \mu_1)~(y - \mu_2)}{\sigma_1 ~\sigma_2} ~-~ \rho^2 ~\bigg (\frac{{x ~-~ \mu_1}^2}{\sigma_1} ~+~ \frac{{y ~-~ \mu_2}^2}{\sigma_2}\bigg)\bigg ) \bigg ). $$

The product of the copula density above, the normal marginal of $logit(\pi_{i1}$) and $logit(\pi_{i2}$) form a bivariate normal distribution which characterize the model by [@Reitsma], [@Arends], [@Chu], and [@Rileyb], the so-called bivariate random-effects meta-analysis (BRMA) model, recommended as the appropriate method for meta-analysis of diagnostic accuracy studies.
Study level covariate information explaining heterogeneity is introduced through the parameters of the marginal and the copula as follows
$$\boldsymbol{\mu}_j = \textbf{X}_j\textbf{B}_j^{\top}. $$                                                               
$\boldsymbol{X}_j$ is a $n \times p$ matrix containing the covariates values for the mean sensitivity($j = 1$) and specificity($j = 2$). For simplicity, assume that $\boldsymbol{X}_1 = \boldsymbol{X}_2 = \boldsymbol{X}$.   $\boldsymbol{B}_j^\top$ is a $p \times$ 1$ vector of regression parameters, and $p$ is the number of parameters.
By inverting the logit functions, we obtain
$$
\pi_{ij} = logit^{-1} (\mu_j + \varepsilon_{ij}).
$$                                                                       

$$
f'(x) = \lim\limits_{h \rightarrow 0} \dfrac{f(x + h) - f(x)}{h}
$$

$$
\begin{aligned} \frac{\partial Y}{\partial X_3} = & \beta_3 + \beta_5 X_2 + \\ & \beta_6 X_1 + \beta_7 X_1 X_2 \end{aligned}
$$

$$
f'(x) = \lim\limits_{h \rightarrow 0} \dfrac{f(x+h) - f(x-h)}{2h}
$$

$$
\begin{align}
ME(X_1) = \dfrac{\partial Y}{\partial X_1} = f_1'(X) = g_1(f(X)) \\
ME(X_2) = \dfrac{\partial Y}{\partial X_2} = f_2'(X) = g_2(f(X))
\end{align}
$$

$$
J = \begin{bmatrix}
        \frac{\partial g_1}{\partial \beta_0} &
        \frac{\partial g_1}{\partial \beta_1} &
        \frac{\partial g_1}{\partial \beta_2} &
        \dots &
        \frac{\partial g_1}{\partial \beta_K} \\

        \frac{\partial g_2}{\partial \beta_0} &
        \frac{\partial g_2}{\partial \beta_1} &
        \frac{\partial g_2}{\partial \beta_2} &
        \dots &
        \frac{\partial g_2}{\partial \beta_K} \\

        \dots \\

        \frac{\partial g_M}{\partial \beta_0} &
        \frac{\partial g_M}{\partial \beta_1} &
        \frac{\partial g_M}{\partial \beta_2} &
        \dots &
        \frac{\partial g_M}{\partial \beta_K} \\
\end{bmatrix}
$$

In first-year calculus, we define intervals such
as $(u, v)$ and $(u, \infty)$. Such an interval
is a \emph{neighborhood} of $a$
if $a$ is in the interval. Students should
realize that $\infty$ is only a
0 symbol, not a number. This is important since
we soon introduce concepts
such as $\lim_{x \to \infty} f(x)$.

When we introduce the derivative
$$\left[
\lim_{x \to a} \frac{f(x) - f(a)}{x - a},
\right]$$
we assume that the function is defined and
continuous in a neighborhood of $a$.

$$
\frac{1 + 2x}{x + y + xy}
$$

$$
\left( \frac{1 + x}{2 + y^{2}} \right)^{2}
$$

$$
\lim_{x \to 0} f(x) = 0
$$