Inference of seasonal long-memory aggregate time series

TL;DR

This paper investigates the spectral and autocorrelation properties of aggregated seasonal long-memory time series, deriving a limiting model and establishing the consistency and normality of the Whittle likelihood estimator, with applications to internet traffic data.

Contribution

It introduces a limiting spectral density model for aggregated seasonal long-memory processes and proves the asymptotic properties of the Whittle likelihood estimator for such models.

Findings

Derived the limit of the spectral density function for aggregated processes.

Proved the consistency and asymptotic normality of the Whittle likelihood estimator.

Demonstrated the approach's effectiveness on real internet traffic data.

Abstract

Time-series data with regular and/or seasonal long-memory are often aggregated before analysis. Often, the aggregation scale is large enough to remove any short-memory components of the underlying process but too short to eliminate seasonal patterns of much longer periods. In this paper, we investigate the limiting correlation structure of aggregate time series within an intermediate asymptotic framework that attempts to capture the aforementioned sampling scheme. In particular, we study the autocorrelation structure and the spectral density function of aggregates from a discrete-time process. The underlying discrete-time process is assumed to be a stationary Seasonal AutoRegressive Fractionally Integrated Moving-Average (SARFIMA) process, after suitable number of differencing if necessary, and the seasonal periods of the underlying process are multiples of the aggregation size. We derive the limit of the normalized spectral density function of the aggregates, with increasing aggregation. The limiting aggregate (seasonal) long-memory model may then be useful for analyzing aggregate time-series data, which can be estimated by maximizing the Whittle likelihood. We prove that the maximum Whittle likelihood estimator (spectral maximum likelihood estimator) is consistent and asymptotically normal, and study its finite-sample properties through simulation. The efficacy of the proposed approach is illustrated by a real-life internet traffic example.