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09_02_09 - STAT 409 Examples for Fall 2009 Def An estimator...

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Unformatted text preview: STAT 409 Examples for 09/02/2009 Fall 2009 Def An estimator θ ˆ for θ is said to be consistent if θ θ ˆ P → , i.e., for all ε > 0, P ε θ θ ˆ → ≥- as n → ∞ . The ( Weak ) Law of Large Numbers: Let X 1 , X 2 , … be i.i.d. with mean μ and standard deviation σ . Let n n n X ... X X 1 + + = , n = 1, 2, … . Then μ X P n → . That is, for all ε > 0, ( 29 X P lim ε μ = ≥- ∞ → n n . 1. If the random variable Y denotes an individual’s income, Pareto’s law claims that P ( Y ≥ y ) = θ y k , where k is the entire population’s minimum income. It follows that f Y ( y ) = 1 θ θ 1 θ + y k , y ≥ k ; θ ≥ 1. Assume k is known. a) Recall that the method of moments estimator θ ~ of θ , is k- = Y Y θ ~ . Show that θ ~ is a consistent estimator of θ . b) Recall that the maximum likelihood estimator θ ˆ of θ , is k n n n i i ln 1 Y ln θ ˆ ⋅- = ∑ = ....
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09_02_09 - STAT 409 Examples for Fall 2009 Def An estimator...

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