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A new bound for discrete distributions based on Maximum Entropy † Gzyl H., ‡ Novi Inverardi P.L. and ‡ Tagliani A. † USB and IESA - Caracas (Venezuela) ‡ Department of Computer and Managements Sciences University of Trento (Italy) CNRS, Paris, France, July 8-13, 2006 MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 2 Our aim is to compare some classical bounds for nonnegative integer-valued random variables for estimating the survival probability and a new tighter bound obtained through the Maximum Entropy method constrained by fractional moments given by We will show the superiority of this last bound with respect to the others. Aim of the paper MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 3 Most used candidates as an upper bound of are: the Chernoff’s bound (Chernoff (1952)) the Chernoff’s bound (Chernoff (1952)) the moment bound (Philips and Nelson (1995)) the moment bound (Philips and Nelson (1995)) the factorial moment boundthe factorial moment bound These three bounds come from the Markov Inequality and involve only integer moments or moment generating function ( mgf ). Some classical distribution bounds MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 4 Given two distribution F and G sharing the first 2Q moments, the following distribution bound is well known in literature (Akhiezer (1965)), where, and and is the Hankel matrix. For any real value of x, Q (x) represents the maximum mass can be concentrated at the point x under the condition that 2Q moments are met; being Q (x) the reciprocal of a polynomial of degree 2Q in x, the bound goes to zero at the rate x -2Q as x goes to infinity: this behavior gives relatively sharp tail information but no much on the central part of the distribution. Akhiezer’s bound MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 5 Improving Akhiezer’s bound The main question is how to choose how to choose the approximant distribution sharing the same 2Q moments of F(x) which allows the bound improvement. A key role in achieving this improvement is played by the MaxEnt technique. MaxEnt techniqueMaxEnt technique constrained by integer moments MaxEnt MaxEnt constrained by fractional moments MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 6 Improving Akhiezer’s bound:integer moments Improving Akhiezer’s bound: integer moments If X is a discrete r.v. with pmf with assigned is the MaxEnt M-approximant of P based on the first M integer moments the first M integer moments then combining and and we prove that the Akhiezer’s bound can be replaced by the following uniform uniform bound This bound is tighter than Akhiezer’s bound because. Further, H[P] is unknown but, it can be estimated through Aitken D 2 - method. Aitken D 2 - methodAitken D 2 - method MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 7 Improving Akhiezer’s bound:fractional moments Improving Akhiezer’s bound: fractional moments If X is a non negative discrete r.v. with pmfX is a non negative discrete r.v. with pmf a sequence of M+1 fractional moments a sequence of M+1 fractional momentsfractional momentsfractional moments with chosen according to is the MaxEnt M-approximant of P, based on is the MaxEnt M-approximant of P, based on the previous M fractional moments then the chain of inequalities, similarly in the case of integer moments, gives Numerical evidence proves that, even for small M, then the Akhiezer’s bound can be replaced by the following new uniform and computable bound MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 8 Improving Akhiezer’s bound: a new bound The proposed bound is sharper than Akhiezer’s bound in the central part of the distribution and vice versa, the Akhiezer’s bound is sharper than it in the tails. Combining Akhiezer’s and our MaxEntbound, we have a sharper upper The proposed bound is sharper than Akhiezer’s bound in the central part of the distribution and vice versa, the Akhiezer’s bound is sharper than it in the tails. Combining Akhiezer’s and our MaxEnt bound, we have a sharper uppersharper bounds, valid for X ≥ 0 and M which guarantees : bounds, valid for X ≥ 0 and M which guarantees : where is obtained from through a convergence accelerating process, so that may be assumed. Here the maximum value allowed of Q stems from the number of given moments or from numerical stability requirements. where is obtained from through a convergence accelerating process, so that may be assumed. Here the maximum value allowed of Q stems from the number of given moments or from numerical stability requirements. MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 9 How to calculate fractional moments? As seen, the fractional moments play the role of building blocks of the proposed procedure. But, how to calculate them? Several scenarios will be analyzed, depending on the available information on the distribution of. This latter is assumed given by a finite or infinite sequence of moments and/or by the mgf. Here we present three cases: a)both and are known; b) assigned and existing mgf ; c) known, R finite or infinite MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 10 Case a): both and are known MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 11 Case b) assigned and existing mgf MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 12 Case c) known, R finite or infinite MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 13 A numerical example (1) MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 14 A numerical example (1) MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 15 A numerical example (1) MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 16 A numerical example (2) MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 17 A numerical example (2) MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 18 THANK YOU for your attention!

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A new bound for discrete distributions based on Maximum Entropy 19 Density reconstruction via ME technique MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 20 Density reconstruction via ME technique But it is not the unique choice! MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 21 An uniform bound MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 22 Aitken D 2 -method MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 23 How to choose fractional moments? MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 24 Fractional moments Fractional moments MaxEnt 2006

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A new bound for discrete distributions based on Maximum Entropy 25 Comparing and combining bounds MaxEnt 2006

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