diff --git a/index.html b/index.html index e9fd9b9..4fd1bde 100644 --- a/index.html +++ b/index.html @@ -26411,20 +26411,20 @@ w_{t,k}^{\text{Naive}} = \frac{1}{K}\label{eq:naive_combination}

First, rewrite \(QL_p\) using the check function:

\[\begin{align} - \rho_p(z) = z(1(0 < z) - p) \label{eq:check} + \rho_p(z) = z(\mathbb{1}(0 < z) - p) \label{eq:check} \end{align}\]

\[\begin{align} - QL_p(x, y) &= (\mathbf1(y < x) - p)(x - y) \\ + QL_p(x, y) &= (\mathbb{1}(y < x) - p)(x - y) \\ &= \rho_p(x-y) \end{align}\]

Now we can express the quantile risk as:

\[\begin{align} - \mathcal{Q}_p(x) = E[\rho_p(x-y)] = ∫ \rho_p(x-y)f(y)dy + \mathcal{Q}_p(x) = \mathbb{E}[\rho_p(x-y)] = ∫ \rho_p(x-y)f(y)dy \end{align}\]

This integral form is where the convolution becomes apparent. A convolution of functions is defined as:

\[\begin{align} (g * h)(x) &= ∫ g(z)h(x - z)dz \\ - &= ∫ h(x - z)g(z)dz + &= ∫ g(x - z)h(z)dz \end{align}\]

They are commutative.

@@ -26449,7 +26449,7 @@ w_{t,k}^{\text{Naive}} = \frac{1}{K}\label{eq:naive_combination} \end{align}\]

The function \(\rho'_p(z)\) jumps from \(-p\) to \(1-p\) at \(0\). So:

\[\begin{align} - \rho''_p(x) = \delta_0(z) \quad \text{Dirac Delta} + \rho''_p(x) = \delta_0(z) \quad \text{(Dirac Delta)} \end{align}\]

Now the magical part :

\[\begin{align} diff --git a/index.qmd b/index.qmd index aba0b72..65b37d7 100644 --- a/index.qmd +++ b/index.qmd @@ -1422,25 +1422,25 @@ $$\widehat{\mathcal{R}}_{t,\min} = 2\overline{\widehat{\mathcal{R}}}^{\text{QL}} First, rewrite $QL_p$ using the check function: \begin{align} - \rho_p(z) = z(1(0 < z) - p) \label{eq:check} + \rho_p(z) = z(\mathbb{1}(0 < z) - p) \label{eq:check} \end{align} \begin{align} - QL_p(x, y) &= (\mathbf1(y < x) - p)(x - y) \\ + QL_p(x, y) &= (\mathbb{1}(y < x) - p)(x - y) \\ &= \rho_p(x-y) \end{align} Now we can express the quantile risk as: \begin{align} - \mathcal{Q}_p(x) = E[\rho_p(x-y)] = ∫ \rho_p(x-y)f(y)dy + \mathcal{Q}_p(x) = \mathbb{E}[\rho_p(x-y)] = ∫ \rho_p(x-y)f(y)dy \end{align} This integral form is where the convolution becomes apparent. A convolution of functions is defined as: \begin{align} (g * h)(x) &= ∫ g(z)h(x - z)dz \\ - &= ∫ h(x - z)g(z)dz + &= ∫ g(x - z)h(z)dz \end{align} They are commutative. @@ -1477,7 +1477,7 @@ To find $\mathcal{Q}''_p(x)$ we rewrite \eqref{eq:check}: The function $\rho'_p(z)$ jumps from $-p$ to $1-p$ at $0$. So: \begin{align} - \rho''_p(x) = \delta_0(z) \quad \text{Dirac Delta} + \rho''_p(x) = \delta_0(z) \quad \text{(Dirac Delta)} \end{align} Now the magical part :