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# Corollary 6

As a corollary of the collage Theorem 4 we can anwser to the question: how many quantiles are needed to approximate a distribution function with a given precision, say ε ? The answer is: take the first integer N such that This value of N is in fact the one that guarantees that the sup-norm distance between the true F and the fixed point of TF is less than ε. In general, this distance could be considerably smaller as shown in Table 1. Proof. To estimate d1(TFF, F) we can slipt the interval [0,1] as  In each of the interval [qi, qi+1) the distance between TF F ( x ) and F ( x ) is most 1/N. So, by the collage theorem, we have since So, given ε > 0, it is suffcient to choose To investigate the asymptotic behaviour of TF it is worth to show the relation between this functional on the space of distribution function and the one proposed by Forte and Vrscay (1995) on the space of measures. Assuming that for any we have and then TF can be rewritten as letting Stefano M. Iacus, Davide La Torre

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