This is my first question on Cross Validated so please bear with me if my question is lagging in any dimension. My question regards how to evaluate a Jacobian matrix when one variable is binary. I have googled quite extensively but found no answer.


I have three variables $p$, $r$ and $s$. $p$ and $r$ come from a copula and thus have uniform marginal distributions and covariance $E[(p-\bar p)(r-\bar p )]=\sigma_{pr}$. The binary variable $s\in\{0,1\}$ is independent of $p$ and $r$ and $E[s]=w$.

I define the vector $\mathbf{x}$ with mean $\mathbf{\bar x}$ and variance$\mathbf{\Sigma}_\mathbf{x}$ : $$ \mathbf{x}=\begin{pmatrix} p\\ r\\ s \end{pmatrix}, \mathbf{\bar x}=\begin{pmatrix} \bar p\\ \bar r\\ w \end{pmatrix},\mathbf{\Sigma}_\mathbf{x}=\begin{pmatrix} \sigma_p^2&\sigma_{pr}&0\\ \sigma_{pr}&\sigma_p^2&0\\ 0 &0&w(1-w) \end{pmatrix} $$

The variables $r$ and $s$ are used to construct a new variable with the function $\Lambda(r,s)$ defined as: $$ \Lambda(r,s)=wG\left(s{G}^{-1}(r)+(1-s){F}^{-1}(r)\right)+(1-w)F\left(s{G}^{-1}(r)+(1-s){F}^{-1}(r)\right) $$ $F$ and $G$ are continuous distributions with support on the positive real line.

I want to know the covariance between $p$ and the new variable created by $\Lambda(r,s)$. My approach has been to use the Delta method.

I define a new vector: $$ \mathbf{y}=q(\mathbf{x})=\begin{pmatrix} p\\ \Lambda(r,s) \end{pmatrix} $$ In the case where all variables where continuous I would use the delta method to find an expression for the covariance matrix of $\mathbf{y}$: $$ \mathbf{\Sigma}_\mathbf{y}=\mathbf{D}\mathbf{\Sigma}_\mathbf{x}\mathbf{D}^T $$ where $\mathbf{D}$ is the Jacobian evaluated at $\mathbf{\bar x}$, $\mathbf{D}=\mathbf{J}\Big|_{\mathbf{x}=\mathbf{\bar x}}$.


The last part is my central issue: how (if possible) can I calculate $\mathbf{D}$ when $s$ is a binary variable?

My own Suggestion

I have been toying around with central differences. But I don't know if this approach works for the delta method.

The Jacobian of $q(\mathbf{x})$ in this case could be defined as: $$ \mathbf{J}=\begin{pmatrix} 1&0 &0\\ 0 &\frac{\partial\Lambda(r,s)}{\partial r} &\frac{\Lambda(r,1)-\Lambda(r,0)}{2} \end{pmatrix} $$ But how do I get from $\mathbf{J}$ to $\mathbf{D}$? It is not obvious to me how to evaluate the $\frac{\partial\Lambda(r,s)}{\partial r}$ and $\frac{\Lambda(r,1)-\Lambda(r,0)}{2}$, which is necessary for knowing $\mathbf{D}$.


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