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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
\[\begin{align*} \hat{g} :: \Omega_n \to \mathcal{X} \to \mathcal{R}\end{align*}\]

Partial Convergence