Generalized run tests for heteroscedastic time series

The problem of testing for nonhomogeneous white noise (i.e. independently but possibly nonidentically distributed observations, with a common, specified or unspecified, median) against alternatives of serial dependence is considered. This problem includes as a particular case the important problem of testing for heteroscedastic white noise. When the value of the common median is specified, invariance arguments suggest basing this test on a generalized version of classical runs: the generalized runs statistics. These statistics yield a run-based correlogram concept with exact (under the hypothesis of nonhomogeneous white noise) p-values. A run-based portmanteau test is also provided. The local powers and asymptotic relative efficiencies (AREs) of run-based correlograms and the corresponding run-based tests with respect to their traditional parametric counterparts (based on classical correlograms) are investigated and explicitly computed. In practice, however, the value of the exact median of the observations is seldom specified. For such situations, we propose two different solutions. The first solution is based on the classical idea of replacing the unknown median by its empirical counterpart, yielding aligned runs statistics. The asymptotic equivalence between exact and aligned runs statistics is established under extremely mild assumptions. These assumptions do not require that the empirical median consistently estimates the exact one, so that the continuity properties usually invoked in this context are totally helpless. The proofs we are giving are of a combinatorial nature, and related to the so-called Banach match box problem. The second solution is a finite-sample, nonasymptotic one, yielding (for fixed n) strictly conservative testing procedures, irrespectively of the underlying densities. Instead of the empirical median, a nonparametric confidence interval for the unknown median is considered. Run-based correlograms can be expected to play the same role in the statistical analysis of time series with nonhomogeneous innovation process as classical correlograms in the traditional context of second-order stationary ARMA series.