Inference
The Inference Ladder
Our estimator can be defined via the following components:
\[\begin{align*}\theta_n &:: \Omega_n \to \mathcal{R}^{d(n)} \\ \\
f_n &:: \mathcal{R}^{d(n)} \to \mathcal{X} \to \mathcal{R} \\ \\
\gamma _n &:: (\mathcal{X} \to \mathcal{R}) \to \Omega_n \to \mathcal{R} \end{align*}\]
Our estimator is constructed by composing these elements as follows:
\[\begin{align*}\theta_n &:: \Omega_n \to \mathcal{R}^{d(n)} \\ \\
f_n \circ \theta_n &:: \Omega_n \to \mathcal{X} \to \mathcal{R} \\ \\
\gamma _n \circ f_n \circ \theta_n &:: \Omega_n \to \Omega_n \to \mathcal{R} \end{align*}\]
Random Function(als)
A random variable has the following type signature:
\[Z :: \Omega \to \mathcal{R}\]
A random function as the following type signature:
\[Z :: \Omega \to \mathcal{X} \to \mathcal{R}\]
A random funtional as the following type signature:
\[Z :: \Omega \to (\mathcal{X} \to \mathcal{R}) \to \mathcal{R}\]
From this, we observe that \(f_n \circ \theta_n\) is a random function and that \(\gamma _n\) is a random functional
Other
\[\begin{align*}
\hat{g} :: \Omega_n \to \mathcal{X} \to \mathcal{R}\end{align*}\]