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Mean Squared Error and Maximum Likelihood

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Mean Squared Error and Maximum Likelihood Lecture XVIII Mean Squared Error As stated in our discussion on closeness, one potential measure for the goodness of an ... – PowerPoint PPT presentation

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Title: Mean Squared Error and Maximum Likelihood


1
Mean Squared Error and Maximum Likelihood
  • Lecture XVIII

2
Mean Squared Error
  • As stated in our discussion on closeness, one
    potential measure for the goodness of an
    estimator is

3
  • In the preceding example, the mean square error
    of the estimate can be written as
  • where q is the true parameter value between zero
    and one.

4
  • This expected value is conditioned on the
    probability of T at each level value of q.

5
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6
MSEs of Each Estimator
7
  • Definition 7.2.1. Let X and Y be two estimators
    of q. We say that X is better (or more
    efficient) than Y if E(X-q)2 ? E(Y-q) for all q
    in Q and strictly less than for at least one q in
    Q.

8
  • When an estimator is dominated by another
    estimator, the dominated estimator is
    inadmissable.
  • Definition 7.2.2. Let q be an estimator of q. We
    say that q is inadmissible if there is another
    estimator which is better in the sense that it
    produces a lower mean square error of the
    estimate. An estimator that is not inadmissible
    is admissible.

9
Strategies for Choosing an Estimator
  • Subjective strategy This strategy considers the
    likely outcome of q and selects the estimator
    that is best in that likely neighborhood.
  • Minimax Strategy According to the minimax
    strategy, we choose the estimator for which the
    largest possible value of the mean squared error
    is the smallest

10
  • Definition 7.2.3 Let q be an estimator of q.
    It is a minimax estimator if for any other
    estimator of q , we have

11
Best Linear Unbiased Estimator
  • Definition 7.2.4 q is said to be an unbiased
    estimator of q if
  • for all q in Q. We call
  • bias

12
  • In our previous discussion T and S are unbiased
    estimators while W is biased.
  • Theorem 7.2.10 The mean squared error is the sum
    of the variance and the bias squared. That is,
    for any estimator q of q

13
  • Theorem 7.2.11 Let Xi i1,2,n be independent
    and have a common mean m and variance s2.
    Consider the class of linear estimators of m
    which can be written in the form
  • and impose the unbaisedness condition

14
  • Then
  • for all ai satisfying the unbiasedness
    condition. Further, this condition holds with
    equality only for ai1/n.

15
  • To prove these points note that the ais must sum
    to one for unbiasedness
  • The final condition can be demonstrated through
    the identity

16
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17
  • Theorem 7.2.12 Consider the problem of
    minimizing
  • with respect to ai subject to the condition

18
  • The solution to this problem is given by

19
Asymptotic Properties
  • Definition 7.2.5. We say that q is a consistent
    estimator of q if

20
Maximum Likelihood
  • The basic concept behind maximum likelihood
    estimation is to choose that set of parameters
    that maximize the likelihood of drawing a
    particular sample.
  • Let the sample be X5,6,7,8,10. The
    probability of each of these points based on the
    unknown mean, m, can be written as

21
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22
  • Assuming that the sample is independent so that
    the joint distribution function can be written as
    the product of the marginal distribution
    functions, the probability of drawing the entire
    sample based on a given mean can then be written
    as

23
  • The value of m that maximize the likelihood
    function of the sample can then be defined by
  • Under the current scenario, we find it easier,
    however, to maximize the natural logarithm of the
    likelihood function

24
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