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slide2_1 - Parameter estimation Population Random sample...

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1 Parameter estimation Population Random sample: Independent variables Same probability distribution Statistic – any function of random sample Point estimator – any statistic  Point estimate – value of some point  estimator
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2 Examples of statistics Given random sample  Sample mean  Sum of squares Log of ratio of first two observations Any function of                  you might think  of… ) / log( 2 1 X X n X X X ,..., , 2 1 n X X X X n + + + = ... 2 1 2 2 2 2 1 ... n X X X + + + n X X X ,..., , 2 1
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3 Examples of parameters Mean      of a single population Variance       of a single population Difference in means of two population μ 2 σ 2 1 μ - μ
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4 Unbiased estimators An estimator  Θ  of some unknown  quantity (parameter)  θ  is said to be  unbiased  if the procedure that yields  Θ   has the property that, were it used  repeatedly the long-term average of  these estimates would approach  θ . That is to say, E( Θ ) =  θ .
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5 Example 1 Want to find an estimator for population mean The Sample Mean   The Sample Median The sample mean of first and last observations This estimator X X ~ 2 1 , 1 n n X X X + = 2 1 2 1 n n X X X X - + + -
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6 Example 2 Want an estimator for population variance Sample variance The estimator  Expectation: can show that  1 ) ( 1 2 2 - - = = n X X S n i i 2 1 2 ) 1 ( ) ( σ - = - = n X X E n i i n X X S n i i = - = 1 2 2 ) ( ~
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7 Variance of a Point Estimator MVUE – Smallest variance unbiased  estimator. (Not always easy to find.) Theorem    If X 1 ,…X n  is a random sample of size  n form a normal distribution with  mean     and variance    , then the  sample mean      is the MVUE for       μ 2 σ X μ
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8 Example 1   (continued) 2 2 2 2 1 σ = + X X V ? ) ~ ( = X V n X V 2 ) ( σ =
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