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

This is related to the sensitivity curve take the

• 17

This preview shows page 4 - 7 out of 17 pages.

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 ,

Subscribe to view the full document.

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 ) ,
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.

Subscribe to view the full document.

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes