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Unformatted text preview: 1 Parameter estimation Population Random sample: Independent variables Same probability distribution Statistic – any function of random sample Point estimator – any statistic Point estimate – value of some point estimator 2 Examples of statistics Given random sample Sample mean Sum of squares Log of ratio of first two observations Any function of you might think of… ) / log( 2 1 X X n X X X ,..., , 2 1 n X X X X n + + + = ... 2 1 2 2 2 2 1 ... n X X X + + + n X X X ,..., , 2 1 3 Examples of parameters Mean of a single population Variance of a single population Difference in means of two population μ 2 σ 2 1 μ μ 4 Unbiased estimators An estimator Θ of some unknown quantity (parameter) θ is said to be unbiased if the procedure that yields Θ has the property that, were it used repeatedly the longterm average of these estimates would approach θ . That is to say, E( Θ ) = θ . 5 Example 1 Want to find an estimator for population mean The Sample Mean The Sample Median The sample mean of first and last observations This estimator X X ~ 2 1 , 1 n n X X X + = 2 1 2 1 n n X X X X + + 6 Example 2 Want an estimator for population variance Sample variance The estimator Expectation: can show that 1 ) ( 1 2 2 = ∑ = n X X S n i i 2 1 2 ) 1 ( ) ( σ =  ∑ = n X X E n i i n X X S n i i ∑ = = 1 2 2 ) ( ~ 7 Variance of a Point Estimator MVUE – Smallest variance unbiased estimator. (Not always easy to find.) Theorem If X 1 ,…X n is a random sample of size n form a normal distribution with mean and variance , then the sample mean is the MVUE for μ 2 σ X μ 8 Example 1 (continued) 2 2 2 2 1 σ = + X X V ? ) ~ ( = X V n X V 2 ) ( σ = 9...
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This note was uploaded on 02/26/2008 for the course IE 121 taught by Professor Perevalov during the Spring '08 term at Lehigh University .
 Spring '08
 Perevalov

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