This is related to the sensitivity curve take the influence function valued at

This is related to the sensitivity curve take the

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This is related to the sensitivity curve; take the influence function valued at F n - 1 and substitute epsilon1 for 1 n . In fact, SC n ( x ) converges to IF ( x ; T,F θ ) for many functionals T , as n → ∞ . The influence function has a number of nice properties that make it a useful inference tool. For instance, under appropriate regularity conditions we can show that the following relations hold: integraldisplay -∞ IF ( x ; T,F θ ) dF θ ( x ) = 0 ,
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9.2. ROBUST INFERENCE METHODS 97 T n = T ( F n ) = T ( F ) + 1 n n summationdisplay i =1 IF ( X i ; T,F ) + R n , where the remainder term R n = o p ( 1 n ). Also n [ T n - T ( F )] N [0 , V ( T,F )] , in distribution , where V ( T,F ) = integraldisplay IF ( x ; T,F ) 2 dF ( x ) , is the asymptotic variance of T n = T ( F n ). The latter has many useful ap- plications, including its use in computing relative efficiencies to compare estimators. Some influence measures are independent of the outlier value x , or con- tamination level. One example is gross–error sensitivity: Definition 9.3. The gross–error sensitivity of T at F θ is defined by γ * ( T,F θ ) = sup x | IF ( x ; T,F θ ) | . A robust estimator will be based on a functional T with finite γ * ( T,F θ ). The gross–error sensitivity is often re–scaled to be used in comparisons of different estimators: γ ** ( T,F θ ) = sup x | IF ( x ; T,F θ ) | radicalBig integraltext IF ( x ; T,F θ ) 2 dF θ ( x ) . Note that γ ** ( T,F ) 1, hence the closer to the lower bound, the more robust the estimator. It can be shown that for the median, T = F - 1 ( 1 2 ) we have γ ** ( T,F θ ) = 1, for any F θ . In a sense, this makes the median the most robust estimator. The goal of robust inference techniques is to find an estimator T n = T ( F n ), with a bounded gross–error sensitivity, but that also exhibits other interesting properties (consistency, unbiasedness, efficiency, ease of computation). Take for instance the maximum likelihood estimator for a sample X 1 ,X 2 ,...,X n , generated by F θ , or its density function f θ . It is defined to maximize n productdisplay i =1 f θ ( X i ) ,
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98 CHAPTER 9. ROBUST STATISTICS or, equivalently to minimize n summationdisplay i =1 [ - ln f θ ( X i )] = n summationdisplay i =1 ρ ( X i ) , with respect to θ . Assuming that ρ admits a derivative, ψ , with respect to θ then the definition is equivalent to the implicit solution T n = T ( F n ) of n summationdisplay i =1 ψ ( X i ,T n ) = 0 . Huber (1964) suggested to generalize this approach to define a class of “Maximum-Likelihood type” estimators. Definition 9.4. A functional T defined implicitly by integraldisplay ψ [ x,T ( F )] dF ( x ) = 0 , where ψ : R 2 R is called an M–functional, and T n = T ( F n ) an M–estimator defined implicitely by n summationdisplay i =1 ψ ( X i ,T n ) = 0 . Here are some M–estimator examples: (i) For the median, the M–estimating equation to solve is n summationdisplay i =1 ψ ( X i ,t ) = n summationdisplay i =1 sign( X i - t ) = 0 , where the sign function sign( x ) = - 1 , 0 , 1 if x < 0, x = 0 or x > 0, respectively.
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