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4. Properties of Estimators (Jan10)

# 4. Properties of Estimators (Jan10) - Properties of...

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Properties of Estimators We study estimators as random variables. In this setting we suppose X 1 , X 2 , . . . , X n are random variables observed from a statistical model F with parameter space Θ. In our usual setting we also then assume that X i are iid with pdf (or pmf) f ( · ; θ ) for some θ Θ. Also in our usual setting Θ R d for some finite d , that is a finite dimensional parameter model. In this case then X 1 , X 2 , . . . , X n has joint pdf (or pmf) given by the function f n ( x 1 , x 2 , . . . , x n ) = n i =1 f ( x i ; θ ) . In our statistical inference setting the specific value of the parameter θ is not known and is the object to be estimated form observable data. If the statistical model is correct then there is one special value of the parameter that is the true value of the parameter, say θ 0 , so that f n ( x 1 , x 2 , . . . , x n ) = n i =1 f ( x i ; θ 0 ) . Aside : Unless we need this extra notation we do not the subscript 0 to designate the true value θ 0 of the parameter. Definition 1 Consider an experiment with random data (r.v.s) X 1 , X 2 , . . . , X n from a statistical model F and parameter space Θ . Consider a random variable T = h ( X 1 , X 2 , . . . , X n ) for some function h . We say T is a statistic if T can be calculated from the observable data only, and does not require knowing which parameter value θ is the true value of the parameter. Examples of Statistics : X i are iid from a distribution f . In some the examples below we need n 2. The following are statistics 1. ¯ X n = 1 n n i =1 X i 2. ˆ μ k,n = 1 n n i =1 X k i 1

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Parametric Estimation Properties 2 3. S 2 n = 1 n - 1 n i =1 ( X i - ¯ X n ) 2 4. If P ( X i > 0) = 1, let T n = 1 n n i =1 log( X i ) W n = 1 ¯ X n 5. T n = n i =1 X i 6. median( X 1 , X 2 , . . . , X n ) End of Example Example of a RV that is not a statistic The following is not a statistic : Suppose X i are iid exponential, parameter λ . Notice that the population median is given by x which solves F ( x ) = 1 2 Therefore x solves 1 2 = 1 - e - λx and hence x = 1 λ log(2). For convenience of notation let m be the population median. Notice it changes depending on the value of the parameter.
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4. Properties of Estimators (Jan10) - Properties of...

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