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STAT 330
Christine Dupont

STAT 330 Lecture, July 10th, 2012 Chapter 6: Point estimation Section 6.1: Introduction Background: Suppose X ,...,X are iid random variables from pf (discrete case) or pdf 1 n (continuous case), f(x;θ). Here θ is unknown; it can be scalar or vector θ = (θ1,...,θk) . Purpose: We would like to estimate θ based on X ,..1,X . n For example, if X 1...,X ∼nN(µ,1), then θ = 2 For example, if X 1...,X ∼nN(µ,σ ), then θ = Some notations: • Θ: parameter space; all possible values of θ. For example, if X1,...,X n N(µ,1), then Θ = For example, if X ,...,X ∼ N(µ,σ ), then Θ = 1 n • Data: (X ,...,X ). 1 n • Observed data: observed value of (X ,...,X ); denoted by (x ,...,x ). 1 n 1 n • Statistic: function of data & does not depend on θ; denoted by T = T(X ,...,X ). For 1 n ▯ √ example, if X 1...,X ∼nN(µ,1), then X = ¯n 1 n Xiis a statistic; n(X n µ)/σ n i=1 is NOT a statistic. Estimator & Estimate (1) If the statistic T = T(X ,1..,X ) ns used to estimate θ, then T = T(X ,...,1 ) is n called the estimator of θ. (2) The observed value of T, t = t(x1,...,x n is called the estimate of θ. 1 Example Suppose X ,1..,X ∼nN(µ,1) and the observed value of (1 ,...nX ) i1 (x ,.n.,x ). Then 1 ▯ X n X i is n i=1 n 1 ▯ ¯n= xi is n i=1 Comments: 2 Section 6.2: Method of moments Problem of interest: Suppose X ,...,1 are iin random variables from pf (discrete case) T or pdf (continuous case), f(x;θ). Our purpose is to estimate θ with θ = (θ ,.1.,θ ) .k Population moments: j Let µ j E(X ), 1 = 1,...,k: the first k population moments. Sometimes, µ is a funjtion of θ, and therefore we also write it as µj(θ1,...,θk). Sample moments: Let M = 1 ▯ n X , j = 1,...,k: the first k sample moments. Note that E(M ) = µ . j n i=1 i j j Method of moments (MM): The idea of method of moments is to choose the estimators θ ,...1θ such khat the first k population moments are equal to the first k sample moments. That is, ▯n µ (θ ,...,θ ) = M = 1 X , j = 1,...,k. j 1 k j n i i=1 ˆ ˆ ˆ T The estimator θ = (θ ,1..,θ )kis called the Method of moment (
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