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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 estimator is. In the preceding example, the mean square error of the estimate can be written as:

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Mean squared error
Mean Squared Error

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


Mean squared error and maximum likelihood


Mean squared error and maximum likelihood


Mses of each estimator
MSEs of Each Estimator estimate can be written as:


Mean squared error and maximum likelihood

  • Definition 7.2.1. Let estimate can be written as: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.


Mean squared error and maximum likelihood

  • 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.


Strategies for choosing an estimator
Strategies for Choosing an Estimator: dominated estimator is inadmissable.

  • 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:


Mean squared error and maximum likelihood

  • Definition 7.2.3: Let dominated estimator is inadmissable.q^ be an estimator of q. It is a minimax estimator if for any other estimator of q~ , we have:


Best linear unbiased estimator
Best Linear Unbiased Estimator: dominated estimator is inadmissable.

  • Definition 7.2.4: q^ is said to be an unbiased estimator of q if

    for all q in Q. We call

    bias


Mean squared error and maximum likelihood

  • In our previous discussion dominated estimator is inadmissable.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


Mean squared error and maximum likelihood

  • Theorem 7.2.11 Let { dominated estimator is inadmissable.Xi} i=1,2,…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


Mean squared error and maximum likelihood

Then dominated estimator is inadmissable.

for all ai satisfying the unbiasedness condition. Further, this condition holds with equality only for ai=1/n.


Mean squared error and maximum likelihood


Mean squared error and maximum likelihood


Mean squared error and maximum likelihood

The solution to this problem is given by dominated estimator is inadmissable.


Asymptotic properties
Asymptotic Properties dominated estimator is inadmissable.

  • Definition 7.2.5. We say that q^ is a consistent estimator of q if


Maximum likelihood
Maximum Likelihood dominated estimator is inadmissable.

  • 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 X={5,6,7,8,10}. The probability of each of these points based on the unknown mean, m, can be written as


Mean squared error and maximum likelihood


Mean squared error and maximum likelihood

  • The value of 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: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: