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Lecture

Lecture_Chap6_part2.pdf

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School
Department
Statistics
Course
STAT 330
Professor
Christine Dupont
Semester
Fall

Description
STAT 330 Lecture, July 12th, 2012 Last lecture: 1. Notation (a) θ and Θ; (b) Statistic; (c) Estimator and estimate of θ; (d) We use θ for both estimator and estimate. 2. Method of Moment (MM) (a) Idea: (b) MM estimator of τ(θ): 3. Maximum likelihood method (a) Likelihood function: 1 Maximum likelihood estimate/estimator ˆ ˆ The the maximum likelihood (ML) estimate of θ, denoted by θ = θ(x ,...,1 ) maxnmizes L(θ). That is, θ = θ(x 1...,x n = argmaxL(θ). θ∈Θ The corresponding ML estimator is θ = θ(X ,...,X ). 1 n Log-likelihood function The log-likelihood function is deﬁned as l(θ) = logL(θ), where log is the natural logarithmic function. The ML estimate of θ also maximizes l(θ). That is, θ = θ(x1,...,x n = argmaxL(θ) = argmaxl(θ). θ∈Θ θ∈Θ Estimating τ(θ) The ML estimator of τ(θ) is just τ(θ).ˆ Example 2. Suppose X ,...1X are ind rvs from (a) Poi(θ) (b) Exp(θ) (try by yourself) 2 (c) N(µ,σ ) (d) Unif(0,θ), ﬁnd the ML estimator of θ in each case.
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