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Unformatted text preview: 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 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 General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference A point estimate Definition A point estimate of a parameter is a single number that is a reasonable value for . The point estimate is given by a suitable statistic and computing this statistic from data. This statistic is the point estimator of . Artin Armagan Point estimation 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 p = 22 30 . Artin Armagan Point estimation 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 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 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 General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Unbiased estimators Definition A point estimator is said to be an unbiased estimator of if E ( ) = for every possible value of . If is not unbiased then the bias is E ( ) . Artin Armagan Point estimation General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Gamma example X 1 , ..., X 200 drawn from a Gamma distribution with = 6 and = 2 p ( x ) = 1 2 6 (2) x 5 e x / 2 , x , the mean is = = 12. Artin Armagan Point estimation 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 p = X n is an unbiased estimate of p. Artin Armagan Point estimation General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Principle of unbiased estimation (PUE) When choosing among several estimators of select one that is unbiased....
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This note was uploaded on 01/17/2010 for the course STA 113 taught by Professor Staff during the Fall '08 term at Duke.
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