Two standard tools for measuring long memory in data come out biased

The whole argument turns on a single letter. In time series statistics, d measures how long a process remembers itself: how much of what happened a hundred steps back is still faintly audible in today's value. Set d near zero and the past fades quickly. Push it up and the influence of old shocks decays so slowly that the series never really settles down.

Chang Liu and Han Lin Shang, in a preprint posted to arXiv on 30 July 2026, make a blunt claim about that letter. The usual ways of estimating it are biased, and the bias is worst exactly where researchers are most likely to run into it.

Their subject is a family known as fractionally integrated processes, which come in stationary and nonstationary versions, with d as the one parameter placing a series somewhere on that scale. Two estimators dominate everyday practice. The local Whittle estimator inspects the low-frequency end of a series' spectrum, the slow sweeping components, and reads d off how steeply power piles up there. Detrended fluctuation analysis takes a different route, cutting a series into windows of many lengths and tracking how the wiggle grows as the windows widen. Both are workhorses. Both, the authors report, can miss.

The culprit they single out is short-range dependence, the ordinary autoregressive kind where today leans heavily on yesterday and on little else. To an estimator squinting at the slowest frequencies, strong short-range dependence can be hard to tell apart from genuine long memory. The stronger it is, the more it leaks into the estimate of d. Liu and Shang open with simulation studies built to make that failure plainly visible, running local Whittle and detrended fluctuation analysis on data where the right answer is fixed in advance.

The correction

Their proposed fix has three moving parts, and the order matters. First, swap in a better starting estimate: the local polynomial Whittle with noise estimator, introduced by Frederiksen and colleagues in 2012, which the authors recommend specifically because it holds up better when short-range autoregressive structure is strong. Second, use that preliminary number to prefilter the data, stripping out most of the estimated long memory so that what remains is largely the short-range behaviour. Third, apply a sieve bootstrap to the filtered series. A sieve bootstrap approximates a series with an autoregressive model and then manufactures many synthetic series from it, each one a plausible alternative history with the same short-range character.

Those synthetic series are the point. Because their true memory parameter is known, the researchers can see how far the estimator drifts on them, and subtract that drift from the estimate made on the real data. The abstract describes the result as a potential improvement in bias rather than a solved problem, and that hedge is the authors' own.

One benefit comes almost free. Having generated hundreds of resampled estimates, you can look at how widely they scatter, which yields confidence intervals for d. The authors call this a byproduct, but it addresses a real gap: point estimates of long memory often arrive without any honest statement of how uncertain they are.

Several things this preprint does not do are worth stating clearly. The evidence on offer is simulation, not analysis of real recorded data, and the abstract reports no numbers at all, so the size of the improvement lives inside the 46 pages, 12 figures and 6 tables that readers would need the full paper to see. It has not been peer reviewed. And the title promises functional time series, meaning data where each observation is a whole curve rather than a single value, though the abstract itself stays with the general fractionally integrated setting and does not spell out how the functional case is handled.