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General concepts Properties of estimators Maximum Likelihood estimation Point estimation Sayan Mukherjee Sta. 113 Chapter 6 of Devore October 18, 2007 Sayan Mukherjee Point estimation
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General concepts Properties of estimators Maximum Likelihood estimation Table of contents 1 General concepts 2 Properties of estimators 3 Maximum Likelihood estimation Sayan Mukherjee Point estimation
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General concepts Properties of estimators Maximum Likelihood estimation A point estimate Defnition 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 . Sayan Mukherjee Point estimation
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General concepts Properties of estimators Maximum Likelihood estimation 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 . Sayan Mukherjee Point estimation
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General concepts Properties of estimators Maximum Likelihood estimation 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 0 i x i where n 0 are the observations not in the largest and smallest 10% Sayan Mukherjee Point estimation
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General concepts Properties of estimators Maximum Likelihood estimation Gaussian -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 estimators of mean mean median extreme trmmed mean Sayan Mukherjee Point estimation
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General concepts Properties of estimators Maximum Likelihood estimation Cauchy 0 500 1000 1500 0 0.002 0.004 0.006 0.008 0.01 mean 1.5 2 2.5 3 0 0.5 1 1.5 2 median 0 2 4 6 x 10 4 0 1 x 10 -4 extreme 2 3 4 5 6 0 0.5 1 1.5 trimmed Sayan Mukherjee Point estimation
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General concepts Properties of estimators Maximum Likelihood estimation Uniform 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0 10 20 30 40 50 60 estimators of mean mean median extreme trmmed mean Sayan Mukherjee Point estimation
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General concepts Properties of estimators Maximum Likelihood estimation Unbiased estimators Defnition 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. Sayan Mukherjee Point estimation
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General concepts Properties of estimators Maximum Likelihood estimation 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.
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lec6print - General concepts Properties of estimators...

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