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Myungin Jae - One of the best experts on this subject based on the ideXlab platform.
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Tutorial on maximum Likelihood Estimation
Journal of Mathematical Psychology, 2003Co-Authors: Myungin JaeAbstract:In this paper, I provide a tutorial exposition on maximum Likelihood Estimation (MLE). The intended audience of this tutorial are researchers who practice mathematical modeling of cognition but are...
Jose Israel Rodriguez - One of the best experts on this subject based on the ideXlab platform.
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Numerical algebraic geometry for maximum Likelihood Estimation
2014Co-Authors: Jose Israel RodriguezAbstract:Author(s): Rodriguez, Jose Israel | Advisor(s): Sturmfels, Bernd | Abstract: Numerical algebraic geometry provides numerical descriptions of solution sets of polynomial systems of equations in several unknown. Such sets are called algebraic varieties. In algebraic statistics, a statistical model is associated to an algebraic variety to study its geometric structure. This thesis contains my work at UC Berkeley that uses numerical algebraic geometry for the algebraic statistics problem of maximum Likelihood Estimation.In Chapter 2 we study the maximum Likelihood Estimation problem on manifolds of matrices with bounded rank. These represent mixtures of distributions of two independent discrete random variables. We determine the maximum Likelihood degree for a range of determinantal varieties, and we apply numerical algebraic geometry to compute all critical points of their Likelihood functions. In Chapter 3 we prove a bijection between critical points of the Likelihood function on the complex variety of matrices of rank r and critical points on the complex variety of matrices of corank r-1. From the perspective of statistics, we show that maximum Likelihood Estimation for matrices of rank r is the same problem as minimum Likelihood Estimation for matrices of corank r-1,and vice versa.In Chapter 4, a description of the maximum Likelihood Estimation problem in terms of dual varieties and conormal varieties is given. With this description, we define the dual Likelihood equations. We show how solving these dual Likelihood equations give solutions to the maximum Likelihood Estimation problem without having the defining equations of the model itself. In Chapter 5, discrete algebraic statistical models are considered and solutions to the Likelihood equations when the data contain zeros are studied. Focusing on sampling and model zeros, we show that the solutions of the Likelihood equations in these cases are contained in a previously studied variety, the Likelihood correspondence. The number of solutions give a lower bound on the ML degree, and the problem of finding critical points to the Likelihood function can be partitioned into computationally easier problems involving sampling and model zeros. In Chapter 6the Macaulay2 package Bertini.m2 is introduced. Macaulay2 is a software system designed to support research in algebraic geometry, and Bertini is a popular software system for numerical algebraic geometry. The package Bertini.m2 provides an interface to Bertini via Macaulay2.
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Numerical algebraic geometry for maximum Likelihood Estimation - eScholarship
2014Co-Authors: Jose Israel RodriguezAbstract:Numerical algebraic geometry provides numerical descriptions of solution sets of polynomial systems of equations in several unknown. Such sets are called algebraic varieties. In algebraic statistics, a statistical model is associated to an algebraic variety to study its geometric structure. This thesis contains my work at UC Berkeley that uses numerical algebraic geometry for the algebraic statistics problem of maximum Likelihood Estimation.In Chapter 2 we study the maximum Likelihood Estimation problem on manifolds of matrices with bounded rank. These represent mixtures of distributions of two independent discrete random variables. We determine the maximum Likelihood degree for a range of determinantal varieties, and we apply numerical algebraic geometry to compute all critical points of their Likelihood functions. In Chapter 3 we prove a bijection between critical points of the Likelihood function on the complex variety of matrices of rank r and critical points on the complex variety of matrices of corank r-1. From the perspective of statistics, we show that maximum Likelihood Estimation for matrices of rank r is the same problem as minimum Likelihood Estimation for matrices of corank r-1,and vice versa.In Chapter 4, a description of the maximum Likelihood Estimation problem in terms of dual varieties and conormal varieties is given. With this description, we define the dual Likelihood equations. We show how solving these dual Likelihood equations give solutions to the maximum Likelihood Estimation problem without having the defining equations of the model itself. In Chapter 5, discrete algebraic statistical models are considered and solutions to the Likelihood equations when the data contain zeros are studied. Focusing on sampling and model zeros, we show that the solutions of the Likelihood equations in these cases are contained in a previously studied variety, the Likelihood correspondence. The number of solutions give a lower bound on the ML degree, and the problem of finding critical points to the Likelihood function can be partitioned into computationally easier problems involving sampling and model zeros. In Chapter 6the Macaulay2 package Bertini.m2 is introduced. Macaulay2 is a software system designed to support research in algebraic geometry, and Bertini is a popular software system for numerical algebraic geometry. The package Bertini.m2 provides an interface to Bertini via Macaulay2.
In Jae Myung - One of the best experts on this subject based on the ideXlab platform.
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Tutorial on maximum Likelihood Estimation
Journal of Mathematical Psychology, 2003Co-Authors: In Jae MyungAbstract:In this paper, I provide a tutorial exposition on maximum Likelihood Estimation (MLE). The intended audience of this tutorial are researchers who practice mathematical modeling of cognition but are unfamiliar with the Estimation method. Unlike least-squares Estimation which is primarily a descriptive tool, MLE is a preferred method of parameter Estimation in statistics and is an indispensable tool for many statistical modeling techniques, in particular in non-linear modeling with non-normal data. The purpose of this paper is to provide a good conceptual explanation of the method with illustrative examples so the reader can have a grasp of some of the basic principles.
Jason P. Fine - One of the best experts on this subject based on the ideXlab platform.
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Likelihood Estimation after nonparametric transformation
Statistics & Probability Letters, 2001Co-Authors: Ying Kuen Cheung, Jason P. FineAbstract:We propose a two-step Likelihood Estimation procedure for the coefficients in a semiparametric transformation model. A simple nonparametric estimator for the unknown transformation is substituted into the Likelihood. The resulting maximiser is shown to be consistent and asymptotically normal. Numerical studies indicate that the estimator may be as precise as an efficient semiparametric procedure.
Eckhard Platen - One of the best experts on this subject based on the ideXlab platform.
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APPLICATION OF MAXIMUM Likelihood Estimation TO STOCHASTIC SHORT RATE MODELS
Annals of Financial Economics, 2015Co-Authors: Kevin Fergusson, Eckhard PlatenAbstract:The application of maximum Likelihood Estimation is not well studied for stochastic short rate models because of the cumbersome detail of this approach. We investigate the applicability of maximum Likelihood Estimation to stochastic short rate models. We restrict our consideration to three important short rate models, namely the Vasicek, Cox–Ingersoll–Ross (CIR) and 3/2 short rate models, each having a closed-form formula for the transition density function. The parameters of the three interest rate models are fitted to US cash rates and are found to be consistent with market assessments.