sta 2006 0809_lecturenotes_4

sta 2006 0809_lecturenotes_4 - Sufficient Statistics...

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Unformatted text preview: Sufficient Statistics Definition : A statistic T ( X ) is a sufficient statistic for θ if the conditional distribution of the sample X given the value of T ( X ) does not depend on θ . Example Let X 1 , X 2 be independent Poisson variables with common expectation λ , then T = X 1 + X 2 is a sufficient statistic for λ . 36 Theorem : If p ( x | θ ) is the joint pdf or pmf of X , and q ( t | θ ) is the pdf or pmf of T ( X ), then T ( X ) is a sufficient statistic for θ if, and only if, for every x in the sample space the ratio p ( x | θ ) /q ( T ( x ) | θ ) is constant as a function of θ . Example Let X 1 , ..., X n be iid Bernoulli random variables with parameter θ, < θ < 1, then T ( X ) = X 1 + ··· + X n is a sufficient statistic for θ . 37 Factorization Theorem : Let f ( x | θ ) denote the joint pdf or pmf of a sample X . A statistic T ( X ) is a sufficient statistic for θ if and only if there exist functions g ( t | θ ) and h ( x ) such that, for all sample points x and all parameter points θ , f ( x | θ ) = g ( T ( x ) | θ ) h ( x ) . 38 Examples 1. Normal( μ, σ 2 ), σ 2 known 2. Normal( μ, σ 2 ), σ 2 unknown 3. Discrete Uniform [1 , θ ] where x = 1 , 2 , ..., θ 39 Theorem : Let X 1 , ..., X n be iid observations from a pdf or pmf, f ( x | θ ). Sup- pose f ( x | θ ) belongs to an exponential class given by f ( x | θ...
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sta 2006 0809_lecturenotes_4 - Sufficient Statistics...

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