Download e-book for kindle: Probability Theory and Its Applications in China by Yan S.J., Wang J.G. (eds.)

By Yan S.J., Wang J.G. (eds.)

ISBN-10: 0821851268

ISBN-13: 9780821851265

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Extra resources for Probability Theory and Its Applications in China

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K ≤ x < 0 ⎪ ⎪ ⎪ ⎨ ˜ 0 ... |x| > k ψ(x) = ⎪ ⎪ ⎪ ⎩ 1 ... 23) The redescending functions are not monotone, and their corresponding primitive functions ρ are not convex. Besides the global minimum, the function n i=1 ρ(Xi −θ) can have local extremes, inducing further roots of the equation n i=1 ψ(Xi − θ) = 0. Moreover, the functions ψ generating the skipped mean and the skipped median have jump discontinuities, and hence the equation n i=1 ψ(Xi − θ) = 0 generally has no solution; the corresponding M -estimator n must be calculated as a global minimum of the function i=1 ρ(Xi − θ).

1 Show that both the sample mean and the sample median of the random sample X1 , . . , Xn are nondecreasing in each argument Xi , i = 1, . . , n. 25) for any fixed c ∈ R. Show that this class contains distributions of Type 1 and Type 2. 3 Let X1 , . . 25). Let Tn (X1 , . . , Xn ) be a translation equivariant estimator of θ, nondecreasing in each argument 42 BASIC CHARACTERISTICS OF ROBUSTNESS Xi , i = 1, . . , n. Then Tn has a universal breakdown point m∗n = m∗n (Tn ) and there exists a constant A such that Xn:m∗n − A ≤ Tn (X1 , .

Instead of the empirical functional, Tn is often called the (sample) statistic. Hampel’s definition of the (qualitative) robustness is based on the Prohorov metric dP on the system P of probability measures on the sample space. 6) where LP (Tn ) and LQ (Tn ) denote the probability distributions of Tn under P and Q, respectively. This robustness is only qualitative: it only says whether it is or is not the functional robust, but it does not numerically measure a level of this characteristic. Because such robustness concerns only the behavior of the functional in a small neighborhood of P0 , it is in fact infinitesimal.

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Probability Theory and Its Applications in China by Yan S.J., Wang J.G. (eds.)


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