Lecture 5- Unbiasedness

# Lecture 5- Unbiasedness - Let ˆ θ n be an estimator of θ...

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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
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
Asymptotically unbiased

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Unformatted text preview: Let ˆ θ n be an estimator of θ based on a sample size of n . The estimator is asymptotically unbiased if lim n →∞ E ( ˆ θ n ) = θ An estimator is (weakly) consistent if, for all δ > 0 and any θ , lim n →∞ Pr [ | ˆ θ n-θ | > δ ] = 0 A suﬃcient (although not necessary) condition for consistency is that the estimator is asymptotically unbiased and Var ( ˆ θ ) → 0. 5 / 7 Mean Squared Error (MSE) The mean squared error (MSE) of an estimator is MSE ( ˆ θ ) = E [( ˆ θ-θ ) 2 ] Alternatively, it is written as MSE ( ˆ θ ) = [ bias ( ˆ θ )] 2 + Var ( ˆ θ ) 6 / 7 Example (Asymptotic unbiasedness in Uniform) A rv is uniformly distributed on the interval (0 , θ ) . Consider the estimtor ˆ θ = max( X 1 , ..., X n ) . Show that ˆ θ is asymptotically unbiased Show that ˆ θ is a consistent estimator of θ Evaluate the MSE of ˆ θ 7 / 7...
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Lecture 5- Unbiasedness - Let ˆ θ n be an estimator of θ...

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