442 Normal approximation

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In [15, 16], the bounded normal approximation (BNA) property is de ned, asserting that the Gaussian approximation on which the con dence interval, and thus the con dence interval coverage, is based remains uniformly bounded as tends to 0 It nds its roots in the Berry Esseen theorem which states that if is the third absolute moment of each of the n independently and identically distributed copies Xi of random variable X (with 2 its variance), the standard normal dis2 tribution, n = n 1 n Xi , n = n 1 n (Xi n )2 and Fn the distribution i=1 i=1 of the centered and normalized sum ( n )/ n , then there exists an absolute constant a > 0 such that, for each x and n, |Fn (x) (x)| a 3 n

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De nition 6 We say that n satis es the bounded normal approximation property if / 3 remains bounded as 0 When this property is satis ed, only a xed number of iterations are required to obtain a con dence interval having a xed error no matter the level of rarity We could also look at a stricter condition, by making sure that the variance satis es BRE This is a stricter condition than BNA because it means looking at

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the fourth moment divided by the square of the variance, and, from the Jensen inequality, BRE for the variance implies BNA [17] In [15], an example is given where BRE is satis ed, but not BNA, so the coverage of the con dence interval is not validated BRE is therefore not suf cient alone to guarantee the robustness of a rare event estimator Note that BNA is a suf cient condition for coverage certi cation, and not a necessary one [15] For instance, there exist more general versions of the Berry Esseen bound (see [10]) for which the moment of order 2 + is used (with > 0) instead of the third moment, being then less restrictive Note nonetheless that this is at the expense of the convergence rate to the Gaussian distribution, O(n /2 ) instead of O(n 1/2 ) A generalized version of BNA property could then be as follows: De nition 7 We say that n satis es bounded normal approximation if there exists > 0 such that E[|X |2+ ]/ 2+ remains bounded as 0

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443 Coverage function

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In order to more directly investigate the actual coverage of con dence intervals for small values of when the number of replications is xed, we can look at the so-called coverage function de ned by LW Schruben in [13] De ne n n R( , X) = n c , n + c n n as the con dence interval at con dence level obtained using data X = (Xi )1 i n (ie, c = 1 ((1 + )/2)) Under normality assumptions, it is easy to show that P[ R( , X)] = Now de ne the random variable = inf{ [0, 1] : R( , X)} should be uniformly distributed, that is, F ( ) = P[ ] = Not satisfying normal assumptions leads to two potential sources of error: F ( ) < may lead to wrong conclusions (lower coverage), while if F ( ) > the method is not ef cient because a smaller sample size could have been used to get the desired coverage

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In order to investigate the actual coverage function, one can consider independent blocks of data X = (Xi )1 i n , producing independent realizations of , from which its empirical distribution can be deduced Reproducing it for different values of and looking at deviations from the uniform distribution illustrates the robustness of the estimator This will be helpful below when discussing possible diagnostic-oriented approaches

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