Since the end of the last ice age the earth's climate has enjoyed a period of relative stability. As the earth is now in a period of rising global temperatures a number of authors have considered the stochastic properties of time series of both atmospheric and oceanic temperatures from instrumental and proxy records on time scales of a few decades to several millenia in an effort to estimate the natural variability of the earth's climate. These series almost universally exhibit the property of statistical long memory. Long memory time series were brought to prominence by H. E. Hurst in 1951 in his study of river flows. Since then the physical cause or causes of the so-called Hurst phenomena have remained elusive. Two sets of competing models have been proposed. The fractional Gaussian noises (FGNs) and their discrete time counter-parts, the fractionally integrated processes of order d (FI(d)), possess genuine long memory in the sense that the present state of a system is temporally dependent on all past states. The alternative are models with a non-stationary mean. In these models the long memory is merely an artifact of the method of analysis. Some authors have proposed multifractals as a potential model. These are FGNs or FI(d) series in which the self-similarity parameter, H, or fractional integration order, d, is allowed to change with time. A number of authors have attempted to develop statistical tests to distinguish between true long memory and other types of processes displaying statistical long memory. Most of these tests exploit, in some fashion, the fact that the self-similarity parameter, H, in the FGNs is required to be constant across the whole series. It is known that structural break location methods tend to report breaks in simulated long memory series where no breaks exist. We have combined established methods for estimating H and/or d with a computationally fast structural break location method, Atheoretical Regression Trees (ART), to obtain empirical bivariate distributions of H or d and regime length for simulated FGNs. These bivariate distributions are then compared with a 2649 year warm season temperature reconstruction using data from a stalagtite from Shihua Cave near Beijing, China and with an existing test for fit to an FGN due to Beran. We find the time series is not H-self-similar. We further compared several other empirically determined bivariate distributions from the simulated data with the Shihua Cave data. In all but one case (mean vs regime length) the Shihua Cave data did not fit the empirical distributions for FGNs. We can discount the FGNs and FI(d)s as appropriate models for the Shihua Cave data. However, we could not establish statistical primacy between multifractals and multiple regimes of short memory processes. The implications for the climate change debate are minimal. There seems little doubt the current rising global temperatures are occurring because of increases in greenhouse gas concentrations as a result of human activity. Discounting of the H-self-similar and the FI(d) models for this data leaves the doubters with one less argument to support their case.
