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lec26 - Estimator(Contd Review What is estimator What is...

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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 2 so that if n → ∞ P ( | ¯ X µ | > ϵ ) 0 which means ¯ X is consistent with µ . 1
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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 k-th population moments is defined as µ k = E ( X k ) For example: Var ( X ) = µ 2 µ 2 1 The k-th 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
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The k-th population central moments is defined as µ k = E (( X µ ) k ) The k-th 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
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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 MoM ) of λ .
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