Mathematical Statistics, Lecture 2 Statistical Models
Statistical Models Statistical Models MIT Dr. Kempthorne Spring 2016 1MIT Statistical Models Statistical Models Definitions Examples Modeling Issues Regression Models Time Series Models Outline 1 Statistical Models Definitions Examples Modeling Issues Regression Models Time Series Models 2MIT Statistical Models Statistical Models Definitions Examples Modeling Issues Regression Models Time Series Models Statistical Models : Definitions Def: Statistical Model Random experiment with sample space . Random vector X = (X1, X2,..., Xn) defined on . : outcome of experiment X ( ): data observations Probability distribution of X X : Sample Space = {outcomes x}FX : sigma-field of measurable events P( ) defined on (X , FX ) Statistical Model P = {family of distributions } 3MIT Statistical Models Statistical Models Definitions Examples Modeling Issues Regression Models Time Series Models Statistical Models : Definitions Def: Parameters / Parametrization Parameter identifies/specifies distribution in P.
[·]: Expectation under the assumption X ∼ P. θ. For a. measurable function. g(X ), E. θ [g(X )] = X. g(x)dF (x | θ). p(x | θ) = p(x; θ): density or probability-mass function of X Assumptions: Either All of the P. θ are continuous with densities p(x | θ), Or All of the P θ are discrete with pmf’s p(x | …
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