Asymptotic Distributions of Some Scale Estimators in Nonlinear Models With Long Memory Errors Having Infinite Variance

Abstract

To have scale‐invariant M estimators of regression parameters in regression models, there is a need for a robust, scale‐invariant estimator of a scale parameter. Two such estimators are the median of the absolute residuals, s1, and the median of the absolute differences of pairwise residuals, s2. The asymptotic distributions of these estimators in regression models when errors have finite variances are known in case the errors are either i.i.d. or form a long‐memory stationary process. Since M estimators are robust against heavy‐tailed error distributions, it is natural to know whether these scale estimators are consistent under heavy‐tailed error distribution assumptions. This article derives their limiting distributions when errors form a linear, long‐memory, stationary process with α‐stable (1 < α < 2) innovations and moving average coefficients decaying as jd−1,0 < d < 1−1/α. We prove that s2 has an α∗‐stable limit distribution with α∗ = α(1−d) < α, while the convergence rate of s1 is generally worse than that of s2. The proof is based on the second‐order asymptotic expansion of the empirical process of the stated infinite‐variance stationary sequence derived in this article.