We focus on RPMs having a bounded support set and prescribed values for the first four moments.
Assume that for any (varepsilon >0) and any bounded subset (B subset M) there exists a bounded measurable function f of bounded support such that (fequiv 1) on B and (mathrm {Var}_{mathbb {P}} S_f < varepsilon ).
where λ i for i = 1, 2 is a real number, | h i | H atb 1, ∞, 0 = λ 1 + λ 2, a i for i = 1, 2 is a bounded function supported on some cubes Q i ⊂ R and it satisfies ∥ a i ∥ L ∞ ≤ [ μ ( 4 Q i ) S Q i, R ] − 1. (2.2).
where λ i for i = 1, 2, is a real number, | h | H atb 1, ∞ = | λ 1 | + | λ 2 |, a i for i = 1, 2, is a bounded function supported on some cube Q i ⊂ R and it satisfies ∥ a i ∥ L ∞ ≤ [ μ ( 4 Q i ) S Q i, R ] − 1. (2.2).
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We also find the minimum of the -probes, i.e and the maximum of the -probes, i.e. We point out that in fact all of these extrema represent robust values, because they do not represent extrema on the bounded (0,1) support, i.e. the values and are not close to 0 or 1. Next, we define distances We want the new normalized maximum and minimum values of -probes to satisfy so that.
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