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Lecture9_412 - AMS412 Order Statistics Let X1 X2 be a...

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1 AMS412 Order Statistics Let X 1 , X 2 , …, be a random sample from a population with p.d.f. f(x). Then, p.d.f.’s for W.L.O.G. (W thout Loss of Ge er l ty), let’s assume X is continuous. f f = f f Example 1. Let exp( ) where , i =1,…,n Please 1. Derive the MLE of , is it unbiased? 2. Derive the p.d.f. of 3. Derive the p.d.f. of Solutions. 1. L f e e l l L l l Is an unbiased estimator of ? t t t t
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2 f y e y Let y y e y y y e y y s ot u se 2. = f f f e f u u e u u e u e f e e e e e ,x>0 3. = f f e e ,x>0 Example 2. Let X be a random variable with pdf. f f other se Derive the MLE of . Solution. Uniform Distribution important!! L f f ll other se MLE : max lnL -> max L
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3 e s Order statistics are useful to derive MLE. Therefore, If the L Therefore, any is an MLE for . Example 3. Let be a random sample from an exponential distribution with pdf: , Please derive (1) The maximum likelihood estimator (MLE) for θ . (2) Is the above MLE unbiased for θ ? (3) The method of moment estimator (MOME) for θ .
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