In the high-dimensional regression we have too many parameters relative to the number of observations and then we can have the problem of the overfitting. A method to solve this problem is to use the Lasso (Least Absolute Shrinkage and Selection Operator) to estimate the regression's coefficients. This estimator has become very popular because, among other properties, it does variable selection, in the sense that some estimated coefficients are equal to zero.
We study the Lasso estimator proving its consistency and finding an oracle inequality in the case of squared error loss.
In this thesis we also talk about survival analysis: this branch of the statistic studies the failure times of an individual (or of a group of individuals) to conclude if for example a new treatment is effective, or if a certain group of individuals has more survival probability than another. We mainly focus on the Cox Proportional Hazard model and the Weibull Proportional Hazard model.
A natural question is: "Can we use the theory of the Lasso estimator in the survival analysis?"
We try to answer this question in the last chapter of this thesis (Chapter 5).
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