Consistency of Lasso Estimator

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Hamidi, Nima | 2015

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 47373 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Haji Mirsadeghi, Mir Omid
Abstract:
In this thesis, we are going to study the consistency properties of one of the most well-known linear estimators – LASSO estimator. First, we bring up some necessary background for introducing this estimator. Then, we are able to introduce primary and secondary forms of LASSO estimator. After that, we prove that primary form of the LASSO estimator is weakly consistent under certain weak conditions. We, also, prove a much stronger statement about the secondary form. We prove that the secondary form is strongly consistent. What’s more, with extra assumptions, we specify the convergence rate of the estimator. Finally, we propose a method for determining free parameter of primary form of LASSO, using cross-validation. Then, the estimator uses this parameter for giving the required estimate. In addition, we give an upper bound for mean squared error of the estimator
Keywords:
Group Lasso ; Cross-Validation ; Least Absolute Shrinkage and Selection Operator (LASSO) Estimator ; Weak Consistency ; Strong Consistency