Refining the central limit theorem approximation via extreme value theory
Introduction
Consider approximations to the distribution of the sum of independent mean-zero random variables with distribution function . If exists, then is asymptotically normal by the central limit theorem. The quality of this approximation is poor if is not much smaller than , since then a single non-normal random variable has non-negligible influence on . Extreme value theory provides large sample approximations to the behavior of the largest observations, suggesting that it may be fruitfully employed in the derivation of better approximations to the distribution of .
For simplicity, consider the case where has a light left tail and a heavy right tail. Specifically, assume and for , so that the right tail of is approximately Pareto with shape parameter and scale parameter . Let be the order statistics. For a given sequence , , split into two pieces Note that conditional on the th order statistic , has the same distribution as , where are i.i.d. from the truncated distribution with for and otherwise. Let and be the mean and variance of . Since is less skewed than , one would expect the distributional approximation (denoted by “”) of the central limit theorem, to be relatively accurate. At the same time, extreme value theory implies that under (1), Combining (3), (4) suggests with independent of .
If , the approximate Pareto tail (1) and imply and for large. From , this further yields which depends on only through the unconditional variance and the two tail parameters . Note that and , so the right-hand side of (6) is the sum of a mean-zero right skewed random variable, and a (dependent) random-scale mean-zero normal variable.
Our main Theorem 1 provides an upper bound on the convergence rate of the error in the approximation (6). The proof combines the Berry–Esseen bound for the central limit theorem approximation in (3) and the rate result in Corollary 5.5.5 of Reiss (1989) for the extreme value approximation in (4). If the tail of is such that the approximation in (4) is accurate, then for both fixed and diverging the error in (6) converges to zero faster than the error in the usual mean-zero normal approximation. The approximation (6) thus helps illuminate the nature and origin of the leading error terms in the first order normal approximation, as derived in Chapter 2 of Hall (1982), for such . We also provide a characterization of the bound minimizing choice of .
If , then the distribution of converges to a one-sided stable law with index . An elegant argument by LePage et al. (1981) shows that this limiting law can be written as . The approximation (5) thus remains potentially accurate under also for infinite variance distributions. To obtain a further approximation akin to (6), note that (1) implies for large . Let . Then which depends on only through the tail parameters and the sequence of truncated variances . The approximation (7) could also be applied to the case , so that one obtains a unifying approximation for values of both smaller and larger than . Indeed, for mean-centered Pareto of index , the results below imply that for suitable choice of , this approximation has an error that converges to zero much faster than the error from the first order approximation via the normal or non-normal stable limit for close to . The approach here thus also sheds light on the nature of the leading error terms of the non-normal stable limit, such as those derived by Christoph and Wolf (1992).
For , the idea of splitting up as in (2) and to jointly analyze the asymptotic behavior of the pieces is already pursued in Csörgö et al. (1988). The contribution here is to derive error rates for resulting approximation to the distribution of the sum, especially for , and to develop the additional approximation of the truncated mean and variance induced by the approximate Pareto tail.
The next section formalizes these arguments and discusses various forms of writing the variance term and the approximation for the case where both tails are heavy. Section 3 contains the proofs.
Section snippets
Assumptions and main results
The following condition imposes the right tail of to be in the -neighborhood of the Pareto distribution with index , as defined in Chapter 2 of Falk et al. (2004).
Condition 1 For some and , admits a density for all of the form with uniformly in .
As discussed in Falk et al. (2004), Condition 1 can be motivated by considering the remainder in the von Mises condition for extreme value theory. It is also closely related to the
Proofs
Let . The proof of Theorem 1 relies heavily on Corollary 5.5.5 of Reiss (1989) (also see Theorem 2.2.4 of Falk et al. (2004)), which implies that under Condition 1, where the supremum is over Borel sets in .
Without loss of generality, assume , and . We first prove two elementary lemmas. Let denote a generic positive constant that does not depend on or
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Funding via NSF grant SES-1627660 is gratefully acknowledged.