Robust Inference for the Mean in the Presence of Serial Correlation and Heavy-Tailed Distributions

Abstract

The problem of statistical inference for the mean of a time series with possibly heavy tails is considered. We first show that the self-normalized sample mean has a well-defined asymptotic distribution. Subsampling theory is then used to develop asymptotically correct confidence intervals for the mean without knowledge (or explicit estimation) either of the dependence characteristics, or of the tail index. Using a symmetrization technique, we also construct a distribution estimator that combines robustness and accuracy: it is higher-order accurate in the regular case, while remaining consistent in the heavy tailed case. Some finite-sample simulations confirm the practicality of the proposed methods.

Randomization Tests Under an Approximate Symmetry Assumption

“…Note in particular that the rate of convergence in this case may depend on the tail index of the distribution. See, for example, McElroy and Politis (2002) and Ibragimov and Müller (2010). Following the discussion in Remarks 4.3 and 4.4, the test described above remains valid in such situations.…”

Large sample theory for statistics of stable moving averages

“…The absence of such a firmer result has promulgated a deeper investigation of the stochastic structure. As a preliminary remark, note that this theorem is well-known for independent random variables (i.e., when ψ is supported in the unit interval) (see Logan et al, 1973); and also for linear combinations of such (i.e., when ψ is a step function) (see McElroy and Politis, 2002). The basic concept of this proof is to cut apart the stochastic process until a k-dependent sequence is revealed, for which joint convergence of sample mean and sample variance is valid, and then to paste the process back up again.…”

“…(3) is continuous almost-everywhere and appropriately summable (we require that j |ψ j | δ < ∞), this example is subsumed by the model (1). Extensive work on examples of this type (with heavy-tailed input random variables) has been done in Resnick (1985, 1986); see also McElroy and Politis (2002).…”

“…In fact, there is equality in distribution to an α-stable for each n; we have recourse to asymptotics only in the more general case that our data are drawn from the α-stable Domain of Normal Attraction, see McElroy and Politis (2002) for a discussion. The major difference in our situation is the lack of independence.…”

“…Estimation of the characteristic exponent is a difficult practical problem -the ubiquitous Hill estimator for α has a bandwidth selection difficulty, namely the number of order statistics used in the computation is determined by the practitioner. In McElroy and Politis (2002), this issue was addressed in the context of mean estimation via normalizing the sample mean with the sample standard deviation, under an infinite order moving average model with heavy-tailed inputs. The resulting self-normalized ratio's rate of convergence no longer depended upon the hidden α, which provided a fortuitous circumvention of the Hill estimator.…”