NotesNov12 - Sampling statistics and estimators Recall a...

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Sampling statistics and estimators Recall: a sample is a collection of observa- tions X 1 = x 1 , X 2 = x 2 , . . . , X n = x n . Most of statistics is based on assuming that the observations forming a sample are iid random variables . Recall: a statistic is any function of the sam- ple, i.e. a function of X 1 , . . . , X n .
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Examples of statistics : ¯ X n = X 1 + X 2 + . . . + X n n , M n = max( X 1 , . . . , X n ) . Since a statistic is a function of random obser- vations, it is itself a random variable.
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A statistic is typically used to estimate an unknown parameter of the distribution of the observations X 1 , X 2 , . . . , X n , a popula- tion parameter . Such statistic is called an estimator of the population parameter. Example : The sample mean ¯ X n is often used as an estimator of the (population) mean μ = EX i .
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Recall: an estimator d = d ( X 1 , . . . , X n ) of a population parameter θ is called unbiased if E ( d ( X 1 , . . . , X n )) = θ. Otherwise, the estimator is biased . Example : The sample mean ¯ X n is an unbi- ased estimator of the population mean μ be- cause E ¯ X n = μ.
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Example :
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