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2 Applicationes Mathematicae

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On convergence of the empirical mean method for non-identically distributed random vectors

100%

Gordienko E. , Ruiz de Chávez J. , Zaitseva E.

Applicationes Mathematicae

2014

tom 41

nr 1

1-12

We consider the following version of the standard problem of empirical estimates in stochastic optimization. We assume that the underlying random vectors are independent and not necessarily identically distributed but that they satisfy a "slow variation" condition in the sense of the definition given in this paper. We show that these assumptions along with the usual restrictions (boundedness and equicontinuity) on a class of functions allow one to use the empirical mean method to obtain a consistent sequence of estimates of infimums of the functional to be minimized. Also, we provide certain estimates of the rate of convergence.

Note on stability estimation in sequential hypothesis testing

100%

Gordienko E. , Ruiz de Chávez J. , García A.

Applicationes Mathematicae

2013

tom 40

nr 1

109-116

We introduce a quantitative measure Δ of stability in optimal sequential testing of two simple hypotheses about a density of observations: f=f₀ versus f=f₁. The index Δ represents an additional cost paid when a stopping rule optimal for the pair (f₀,f₁) is applied to test the hypothesis f=f₀ versus a "perturbed alternative" f=f̃₁. An upper bound for Δ is established in terms of the total variation distance between f₁(X)/f₀(X) and f̃₁(X)/f₀(X) with X∼f₀.