Aligned Rank Tests for Linear Models with Autocorrelated Error Terms

Linear models in which the unobserved error constitutes a realization of some stationary ARMA process or, equivalently, ARMA processes with a linear regression trend, are considered under unspecified innovation densities. Due to serial dependence among the observations, the classical rank-based techniques, which have been developed for linear models with independent observations and unspecified error densities, do not apply; nor do the existing rank-based procedures for serial dependence problems, where the observations are assumed to be stationary in the mean (or the median). Moreover, all problems of practical interest (testing the significance of a subset of regression coefficients, identifying the orders p and q of the ARMA (p, q) dependence, overall diagnostic checking of the model,...) involve nuisance parameters. Typically, one is interested either in the regression trend, and the serial dependence parameters are nuisance parameters; or the serial dependence structure is the main concern, and trend somehow has to be removed. A rank-based approach to such problems thus not only requires extending the classical Hájek-type theory of rank tests to serially dependent situations, it also requires a generalized theory of aligned rank tests. This is the purpose of the present paper. The key result is a local asymptotic normality (LAN) result involving a rank-measurable central sequence; depending on the model considered (with symmetric or totally unspecified innovation densities), the ranks to be used are either signed or unsigned. This LAN result, along with a particular local asymptotic linearity property, implies the local asymptotic sufficiency of (aligned) ranks for a broad class of testing problems-mainly, testing linear restrictions on the parameters of the model. Asymptotically invariant aligned rank tests which are locally asymptotically most stringent similarly are derived. Unlike former results on aligned rank tests for linear models with independent observations, the present ones are entirely obtained from the LAN structure of the underlying problem, and provide a remarkable illustration of the power and appropriateness of LeCam's approach in nonparametric inference. They generalize and reinforce most previous results on rank-based tests for general linear models, as well as the few available ones on rank-based inference for time series analysis. The resulting testing procedures are invariant or asymptotically invariant, hence (asymptotically) distribution-free. They also are as powerful as (often, strictly more powerful than) their classical, correlogram-based counterparts. In addition, they are considerably more robust, and remain fully reliable in the presence of outliers, heavy-tailed distributions, atypical startup behavior, etc.