# DoubleML Coverage

The website contains coverage simulations for the DoubleML-package.

Most simulations are based on the dataset generators provided by the package, but sometimes contain slight deviations. You can find the code for the simulations in the GitHub repository.

## Coverage Simulations

Generally, the DoubleML package solves a moment equation

\[ E[\psi(W,\theta_0,\eta_0)] = 0 \]

where \(W\) denotes the observed data, \(\theta_0\) the parameter of interest, and \(\eta_0\) a vector of nuisance functions/elements. Using the `confint()`

method, the package provides confidence intervals \([\hat{\theta}_{\text{lower}}, \hat{\theta}_{\text{upper}}]\) for the parameter of interest \(\theta_0\).

This repository verifies coverage properties of the confidence intervals provided by the package. Usually, the coverage is assessed by simulating \(n_{\text{sim}}\) datasets and calculating the proportion of confidence intervals that cover the true parameter value

\[ \text{Coverage} = \frac{1}{n_{\text{sim}}} \sum_{i=1}^{n_{\text{sim}}} \mathbb{1}(\hat{\theta}_{\text{lower},i} \leq \theta_0 \leq \hat{\theta}_{\text{upper},i}) \]

for a nominal coverage level is \(1-\alpha\). The corresponding coverage results are highlighted according to the following color scheme:

- Green if the deviation to the nominal level is below \(5\%\)
- Yellow if the deviation to the nominal level is above \(5\%\) and below \(10\%\)
- Red if the deviation to the nominal level is above \(10\%\)

For simulations with multiple parameters of interest, usually pointwise and uniform coverage is assessed.

Furthermore, the simulations provide the average length of the confidence intervals and the average absolute bias of the point estimates

\[ \begin{align*} \text{Bias} &= \frac{1}{n_{\text{sim}}} \sum_{i=1}^{n_{\text{sim}}} |\hat{\theta}_i - \theta_0|\\ \text{CI Length} &= \frac{1}{n_{\text{sim}}} \sum_{i=1}^{n_{\text{sim}}} (\hat{\theta}_{\text{upper},i} - \hat{\theta}_{\text{lower},i}). \end{align*} \]