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# post13 - Properties of SS Theorem X1 Xn is a random sample...

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Properties of SS Theorem X 1 , · · · , X n is a random sample from f ( x ; θ ) , θ Ω. If a sufficient statistic Y 1 = u 1 ( X 1 , · · · , X n ) for θ exists and the MLE ˆ θ of θ exists uniquely, then ˆ θ is a function of Y 1 . Example : X 1 , · · · , X n : a random sample from Bernoulli( θ ), 0 < θ < 1 X i is sufficient for θ . The unique MLE is ¯ X . Example : X 1 , · · · , X n : a random sample N ( θ, σ 2 ), σ 2 > 0 known, -∞ < θ < ¯ X is sufficient for θ , ¯ X is the unique MLE of θ . 1 Example : X 1 , · · · , X n : a random sample from Poisson( θ ). f ( x 1 , · · · , x n ; θ ) = n Y i =1 θ x i e - θ x i ! = θ x i e - × n Y i =1 x i ! - 1 By factorization theorem, Y 1 = X i is suffi- cient for θ . Is ¯ X also sufficient for θ ? 2 SS’s are not unique Any invertible function of a sufficient statis- tic is also a sufficient statistic. Why? Is ¯ X/θ or ¯ X + θ a sufficient statistic? Any statistic from which you can calcu- late a sufficient statistic is also a sufficient statistic.

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post13 - Properties of SS Theorem X1 Xn is a random sample...

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