## Oracle property :

The Oracle property, also known as the Oracle estimator, refers to a statistical estimator that is assumed to have perfect knowledge of the true underlying distribution or population from which a sample is drawn. In other words, an Oracle estimator is a hypothetical statistical estimator that is able to make the most accurate predictions or estimates possible, as if it had access to all of the relevant information and data.

One example of an Oracle estimator is the Oracle MLE (Maximum Likelihood Estimator). The Oracle MLE is a hypothetical estimator that is able to calculate the maximum likelihood estimate (MLE) of a parameter, given that it has access to the true underlying distribution or population. The MLE is a well-known statistical estimate that is based on the principle of maximizing the likelihood function, which is a measure of how likely it is that a particular set of data would have been observed given a specific set of parameters.

For example, suppose we are trying to estimate the mean of a normal distribution based on a sample of data. The Oracle MLE would be able to calculate the exact MLE of the mean, because it has access to the true underlying distribution and all of the relevant data. In contrast, a non-Oracle estimator would have to make an approximation of the MLE based on the sample data that is available, which may not be as accurate.

Another example of an Oracle estimator is the Oracle Lasso, which is a hypothetical estimator that is able to perform Lasso regression with perfect knowledge of the true underlying model. Lasso regression is a statistical method that is used to identify the most important predictors in a linear regression model, by imposing a constraint on the model parameters that encourages sparsity (i.e., the selection of a small number of predictors). The Oracle Lasso is able to perform this task with perfect accuracy, because it has access to the true underlying model and all of the relevant data.

In both of these examples, the Oracle estimator is assumed to have perfect knowledge of the true underlying distribution or model, which allows it to make the most accurate predictions or estimates possible. While it is not possible to actually construct an Oracle estimator in practice, the concept of an Oracle estimator can be useful for understanding the limits of statistical estimation and for comparing the performance of different estimators.