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General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Point estimation Artin Armagan Sta. 113 Chapter 6 of Devore February 12, 2009 Artin Armagan Point estimation
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General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Table of contents 1 General concepts 2 Properties of estimators 3 Maximum Likelihood estimation 4 Bayesian inference Artin Armagan Point estimation
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General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference A point estimate Definition A point estimate of a parameter θ is a single number that is a reasonable value for θ . The point estimate is given by a suitable statistic and computing this statistic from data. This statistic is the point estimator of θ . Artin Armagan Point estimation
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General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Example A car manufacturer makes cars that explode or not explode with probability p . What is a point estimator of p ? If we observe cars 30 cars and 22 of them explode then ˆ p = 22 30 . Artin Armagan Point estimation
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General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Estimators of the mean Given x 1 , ..., x n the following are estimators of the population mean μ and assume n is odd. 1 sample mean ¯ x = 1 n i x i 2 sample median (order data) ˜ x = x ( n +1) / 2 3 average of extremes ˇ x = min( x i )+max( x i ) 2 4 trimmed mean ¯ x tr (10) = 1 n i x i where n are the observations not in the largest and smallest 10% Artin Armagan Point estimation
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General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Unbiased estimators Definition A point estimator ˆ θ is said to be an unbiased estimator of θ if E ( ˆ θ ) = θ for every possible value of θ . If ˆ θ is not unbiased then the bias is E ( ˆ θ ) θ. Artin Armagan Point estimation
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General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Gamma example X 1 , ..., X 200 drawn from a Gamma distribution with α = 6 and β = 2 p ( x ) = 1 2 6 Γ(2) x 5 e x / 2 , x 0 , the mean is μ = α × β = 12. Artin Armagan Point estimation
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General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Binomial Proposition If X is a binomial rv with parameters n and p then the sample proportion ˆ p = X n is an unbiased estimate of p. Artin Armagan Point estimation
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General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Principle of unbiased estimation (PUE) When choosing among several estimators of θ select one that is unbiased. Artin Armagan Point estimation
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General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Estimates of variance Proposition Let X 1 , ..., X n be a random sample from a distribution with mean μ and variance σ 2 . The estimator ˆ σ 2 = s 2 = i ( X i ¯ X ) 2 n 1 , is an unbiased estimator of σ 2 .
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  • Fall '08
  • Staff
  • Maximum likelihood, Estimation theory, maximum likelihood estimation, estimators Maximum Likelihood, Artin Armagan, estimation Bayesian inference

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