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class06-3-handouts - PSTAT 120B Probability Statistics...

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PSTAT 120B - Probability & Statistics Class # 06-3- Suﬃciency Jarad Niemi University of California, Santa Barbara 7 May 2010 Jarad Niemi (UCSB) Suﬃciency 7 May 2010 1 / 15

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Class overview Announcements Announcements Homework Homework 5 is up, due Monday 10 May by 4pm in SH 5521 Mid-term II Friday 14 May Covers 8.5–8.8,9.1–9.5 Jarad Niemi (UCSB) Suﬃciency 7 May 2010 2 / 15
Class overview Goals (Minimal) suﬃcient statistics Rao-Blackwell Theorem Minimum-Variance Unbiased Estimators (MVUE) Jarad Niemi (UCSB) Suﬃciency 7 May 2010 3 / 15

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Suﬃcient statistics Intuition Data Suﬃcient statistics Jarad Niemi (UCSB) Suﬃciency 7 May 2010 4 / 15
Suﬃcient statistics Intuition Suppose X 1 , . . . , X n iid Ber ( p ). If we are interested in inference on p , what statistic do we need? What do I mean by inference ? Estimation, e.g. what is our best guess for p ? Conﬁdence intervals, e.g. what is a reasonable range for p ? Hypothesis testing, e.g. is p = 0 . 5? What statistic do we need? Generally we need Y = n i =1 X i If we know Y , then individual X i are irrelevant Somehow Y suﬃciently describes the data Jarad Niemi (UCSB) Suﬃciency 7 May 2010 5 / 15

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Suﬃcient statistics Deﬁnition Deﬁnition Let Y 1 , Y 2 , . . . , Y n be iid from a probability distribution with unknown parameter θ . Then the statistic U = g ( Y 1 , Y 2 , . . . , Y n ) is said to be suﬃcient for θ if the conditional distribution of Y 1 , Y 2 , . . . , Y n given U does not depend on θ . Theorem If f Y ( y 1 , . . . , y n | θ ) is the joint pdf or pmf of the data and f U ( u | θ ) is the pdf or pmf of U = g ( Y 1 , Y 2 , . . . , Y n ) , then U is a suﬃcient statistic for θ if the ratio f Y ( y 1 , . . . , y n | θ ) /
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class06-3-handouts - PSTAT 120B Probability Statistics...

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