Lecture 5- Unbiasedness

Lecture 5- Unbiasedness - Let n be an estimator of based on...

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Parameter estimation – suuplement note ActSCi 432/832 1 / 7
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Review: Unbiased estimation Consider an iid random sample X = ( X 1 , ..., X n ) from a model specified by its pdf (or pmf) f ( x | θ ) Here parameter θ is a vector or scalar to be estimated from X An estimator ˆ θ n = ˆ θ n ( X ) is a function of X . Sometimes subscript n is suppressed. ˆ θ is unbiased of θ if E ( ˆ θ ) = θ The bias is defined as bias ( ˆ θ ) = E ( ˆ θ ) - θ . Remarks: Unbiasedness is not preserved under parameter transformation. E.g., ¯ X is unbiased for the mean of X , but 1 / ¯ X is biased for 1 / E ( X ) Some unbiased estimators are nonsensical. 2 / 7
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Unbiased estimation Example (Unbiasedness of sample mean and variance) Consider an iid random sample X = ( X 1 , ..., X n ) from a model specified by its pdf f ( x | μ, σ ) with mean μ and variance σ 2 . Show that ¯ X is an unbiased estimator of μ ( n - 1) - 1 n i =1 ( X i - ¯ X ) 2 is an unbiased estimator of σ 2 3 / 7
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Unbiased estimation Example (Nonsensical estimator in Poisson) Let X be Poisson distributed with mean λ . Show that ( - 1) X is an unbiased estimator for e - 2 λ . Is this sensible? 4 / 7
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Asymptotically unbiased
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Unformatted text preview: Let n be an estimator of based on a sample size of n . The estimator is asymptotically unbiased if lim n E ( n ) = An estimator is (weakly) consistent if, for all > 0 and any , lim n Pr [ | n- | > ] = 0 A sucient (although not necessary) condition for consistency is that the estimator is asymptotically unbiased and Var ( ) 0. 5 / 7 Mean Squared Error (MSE) The mean squared error (MSE) of an estimator is MSE ( ) = E [( - ) 2 ] Alternatively, it is written as MSE ( ) = [ bias ( )] 2 + Var ( ) 6 / 7 Example (Asymptotic unbiasedness in Uniform) A rv is uniformly distributed on the interval (0 , ) . Consider the estimtor = max( X 1 , ..., X n ) . Show that is asymptotically unbiased Show that is a consistent estimator of Evaluate the MSE of 7 / 7...
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Lecture 5- Unbiasedness - Let n be an estimator of based on...

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