Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat 312: Lecture 03 Minimum Variance Unbiased Estimator Moo K. Chung [email protected] September 9, 2004 1. Population of interest is a collection of measur- able objects we are studying. Let X 1 , · · · , X n be a random sample from the population. Then sam- ple mean ¯ X and sample variance S 2 are unbiased estimators of population mean μ and population variance σ 2 respectively. Proof. Note that S 2 = 1 n - 1 X i =1 ( X i - ¯ X ) 2 = 1 n - 1 h n X i =1 X 2 i - n ¯ X 2 i . Then using the fact E ( ¯ X ) 2 = V ¯ X + ( E ¯ X ) 2 = σ 2 /n + μ 2 , it can be shown that E S 2 = σ 2 . 2. There may be many unbiased estimators of θ . Given two unbiased estimators ˆ θ 1 and ˆ θ 2 of θ . We choose one that gives less variance. If V ( ˆ θ 1 ) V ( ˆ θ 2 ) , ˆ θ 1 is called more efficient than ˆ θ 2 . An effi- cient estimator has less variability so we are more likely to make an estimate close to the true param- eter value. The following coin flipping example clearly demonstrate this.
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