The least favorable noise. (English) Zbl 1498.60070

Authors’ abstract: Suppose that a random variable \(X\) of interest is observed perturbed by independent additive noise \(Y\). This paper concerns the “the least favorable perturbation” \(\hat{Y}_{\epsilon}\), which maximizes the prediction error \(E(X-E(X|X+Y))^2\) in the class of \(Y\) with \(\mathrm{var}(Y )\leq \epsilon\). We find a characterization of the answer to this question, and show by example that it can be surprisingly complicated. However, in the special case where \(X\) is infinitely divisible, the solution is complete and simple. We also explore the conjecture that noisier \(Y\) makes prediction worse.


60E07 Infinitely divisible distributions; stable distributions
60E10 Characteristic functions; other transforms
60E05 Probability distributions: general theory
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[1] Bain, A. and Crisan, D.: Fundamentals of Stochastic Filtering (Vol. 60). Springer Science & Business Media, New York, 2008.
[2] Bryc, W. and Smolenski, W.: On the stability problem for conditional expectation. Statist. Probab. Lett. 15, (1992), 41-46. · Zbl 0764.62014
[3] Bryc, W., Dembo, A., and Kagan, A.M.: On the maximum correlation coefficient. Theory Probab. Appl. 49, (2005), 132-138. · Zbl 1089.62066
[4] Dembo, A., Kagan, A.M. and Shepp, L.A.: Remarks on the maximum correlation coefficient. Bernoulli 7, (2001), 343-350. · Zbl 0981.62051
[5] Lukacs, E.: Characteristic Functions. Charles Griffin & Co., Ltd., Glasgow, 1970. · Zbl 0201.20404
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