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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 - where q is the true parameter value between zero

and one.

- This expected value is conditioned on the

probability of T at each level value of q.

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MSEs of Each Estimator

- 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.

- 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

- 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

- 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

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

- 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

- 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

- Then
- for all ai satisfying the unbiasedness

condition. Further, this condition holds with

equality only for ai1/n.

- To prove these points note that the ais must sum

to one for unbiasedness - The final condition can be demonstrated through

the identity

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- Theorem 7.2.12 Consider the problem of

minimizing - with respect to ai subject to the condition

- The solution to this problem is given by

Asymptotic Properties

- Definition 7.2.5. We say that q is a consistent

estimator of q if

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

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- 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

- 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

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