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Unformatted text preview: Estimator (Cont’d) Review: What is estimator? What is estimates? What are the properties we used to compare estimators? Example: The sample mean ¯ x is consistent for µ . That means that if the sample size is getting large, then ¯ X is getting very closed to µ in the sense of probability. Derivation : using Chebyshev’s inequality, P (  ¯ X − µ  > ϵ ) ≤ Var ( ¯ X ) ϵ 2 = σ 2 nϵ 2 so that if n → ∞ P (  ¯ X − µ  > ϵ ) → which means ¯ X is consistent with µ . 1 Estimating parameters Method of moments: is one of the methods to estimate the parameters. The basic idea is to match the sample moments with the population moments based on the sample ( x 1 , ··· ,x n ) , where x i is the value of X i . ♣ The kth population moments is defined as µ k = E ( X k ) ♢ For example: Var ( X ) = µ 2 − µ 2 1 ♣ The kth sample moments is defined as m k = n − 1 n ∑ i =1 x k i , x i is the realization/sample value of X i ♢ For example: m 1 = ¯ x 2 ♣ The kth population central moments is defined as µ ′ k = E (( X − µ ) k ) ♣ The kth sample central moments is defined as m ′ k = n − 1 n ∑ i =1 ( x i − ¯ x ) k , x i is the realization/sample value of X i , where ¯ x is the sample mean ♠ To estimate k parameters, equate the first k population and sample moments µ i = m i , i = 1 , 2 , ··· ,k So we have k equations to solve 3 Estimating parameters: Examples Example 1: To estimate the parameter λ of Poisson distribution, we need µ 1 = E ( X ) = λ = m 1 = ¯ x Only one unknown is there, solving it for λ we obtain ˆ λ = ¯ x which is the method of moment estimator (short as...
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This note was uploaded on 02/01/2012 for the course STAT 330B taught by Professor Zhou during the Spring '11 term at Iowa State.
 Spring '11
 Zhou

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