Inference

The Inference Ladder

Our estimator can be defined via the following components:

$$\begin{array}{rl}{\theta }_n& ::{\mathrm{\Omega }}_n\to {\mathcal{R}}^{d(n)}\\ \\ f_n& ::{\mathcal{R}}^{d(n)}\to \mathcal{X}\to \mathcal{R}\\ \\ {\gamma }_n& ::(\mathcal{X}\to \mathcal{R})\to {\mathrm{\Omega }}_n\to \mathcal{R}\end{array}$$

Our estimator is constructed by composing these elements as follows:

$$\begin{array}{rl}{\theta }_n& ::{\mathrm{\Omega }}_n\to {\mathcal{R}}^{d(n)}\\ \\ f_n\circ {\theta }_n& ::{\mathrm{\Omega }}_n\to \mathcal{X}\to \mathcal{R}\\ \\ {\gamma }_n\circ f_n\circ {\theta }_n& ::{\mathrm{\Omega }}_n\to {\mathrm{\Omega }}_n\to \mathcal{R}\end{array}$$

Random Function(als)

A random variable has the following type signature:

$$Z::\mathrm{\Omega }\to \mathcal{R}$$

A random function as the following type signature:

$$Z::\mathrm{\Omega }\to \mathcal{X}\to \mathcal{R}$$

A random funtional as the following type signature:

$$Z::\mathrm{\Omega }\to (\mathcal{X}\to \mathcal{R})\to \mathcal{R}$$

From this, we observe that $f_n\circ {\theta }_n$ is a random function and that ${\gamma }_n$ is a random functional

Other

$$\begin{array}{r}\hat{g}::{\mathrm{\Omega }}_n\to \mathcal{X}\to \mathcal{R}\end{array}$$

Partial Convergence