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Math Help - Biasedness & Consistency

  1. #1
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    Biasedness & Consistency

    Q: Is the estimator (1/n) ∑ (X_i - X bar)^2] consistent or inconsistent for σ^2 ?

    I showed that the above estimator is biased, does this automatically imply that it is not consistent?

    Can a biased esimtator ever be consistent?

    Thanks!
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  2. #2
    MHF Contributor matheagle's Avatar
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    You have an example of a bias estimator that's consistent right here.
    Constistency usually refers to convergence in probability (MOO)
    Some people talk about strong (almost sure) consistency
    and weak constistency (convergence in probability).
    But most just say an estimator is consistent if it converges in probability to the parameter it's estimating.
    That's consistent with Wackerly (pun intented).
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  3. #3
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    Quote Originally Posted by matheagle View Post
    You have an example of a bias estimator that's consistent right here.
    [snip]
    Indeed.

    Another 'counter-example':

    Let X_1, \, X_2, \, .... X_n be a sample taken from the uniform distribution on (0, \theta] for some positive \theta and consider the maximum likelihood estimator of \theta: \hat{\theta}_n = X_{(n)}.

    E\left(\hat{\theta}_n\right) = \frac{n \theta}{n + 1} and is therefore biased for any finite sample size n.

    It's not difficult to prove that the sequence of random variables  \hat{\theta}_1, \, \hat{\theta}_2, \, .... converges to \theta in probability and so is consistent.

    Note: I'm not going to be inconsistent and make more work for myself by typing the proof. The interested reader can work through it him/herself or do some research.
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  4. #4
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    Quote Originally Posted by matheagle View Post
    You have an example of a bias estimator that's consistent right here.
    Constistency usually refers to convergence in probability (MOO)
    Some people talk about strong (almost sure) consistency
    and weak constistency (convergence in probability).
    But most just say an estimator is consistent if it converges in probability to the parameter it's estimating.
    That's consistent with Wackerly (pun intented).
    But how can (1/n) ∑ (X_i - X bar)^2] and σ^2 stay "close" as n->infinity when it is a BIASED estimator?
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  5. #5
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by kingwinner View Post
    But how can (1/n) ∑ (X_i - X bar)^2] and σ^2 stay "close" as n->infinity when it is a BIASED estimator?

    read about asymptotically UNBIASED.
    because this is a perfect example of that.
    in the limit it is unbiased
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  6. #6
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    OK, so since it is asymptotically unbiased and the variance tends to zero, it is a consistent estimator.

    Another question: If it is NOT asymptotically unbiased (and not unbiased), does this automatically imply that it is not consistent?

    Thanks!
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