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N is a sample of size n from fx l a statistic is any

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n is a sample of size n from f(x) l A statistic is any function of sample data l An estimator is a statistic whose purpose is to estimate a parameter or some function thereof l A point estimator is simply a formula (rule) for combining sample information to produce a single number to estimate

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Estimation . .. l Estimators are random variables (because they are a function of rv’s X 1, X 2, …, X n ) l Examples of point estimators l Sample mean is a point estimator for the population mean l Sample variance is an estimator of the population variance l Why does it make no sense to expect an estimator to always produce an estimate equal to the parameter of interest? 25
26 Properties of estimators l Sample mean is a ‘natural’ choice of estimator for the population mean l But there may be other (better?) estimators l Why not use (n-1) s 2/n as an estimator for σ 2? l Desirable properties of estimators l Unbiasedness: On average, does the estimator achieve the correct value? l Consistency: As sample size gets larger does the probability that the estimator deviates from the parameter by more than a ‘small’ amount get small? l Relative efficiency: If there are two competing estimators of a parameter, does one have less expected dispersion?

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27 Properties of estimators … ) ~ var( ) ˆ var( if efficient relatively is ˆ then unbiased both are ~ estimator e alternativ an ˆ If 1 ) | ˆ (| lim small any for if of estimator consistent a is ˆ ) ˆ if of estimator unbiased an is ˆ then of estimator an is ˆ Suppose θ ε < = < - = P n E(
28 Properties of estimators… 2 2 2 2 2 2 2 2 2 1 2 2 1 2 2 of estimator consistent a is ˆ small is bias the n large for however estimator biased a is thus 1 1 ) ˆ ( but of estimator unbiased an is thus ) ( ) ( ˆ than rather 1 ) ( as variance sample the of definition our Recall σ - = - = = - = - - = = = n n s n n E E s E n X X n X X s n i i n i i

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29 Progress report #4 l Have introduced a selection of distributions l l These enable us to model a range of phenomena l Normal also plays a key role in theory of estimation l Have introduced the basics of estimation l Need to understand better the notion of a point estimator as a rv l Leads us to sampling distributions l Role of Normal distribution in theory of estimation comes through the Central Limit Theorem l Can also use interval rather then point estimators l Leads us to confidence intervals
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n is a sample of size n from fx l A statistic is any...

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