  We study a well-known estimator of the fractal index of a stochastic process.
Our framework is very general and encompasses many models of interest; we show
how to extend the theory of the estimator to a large class of non-Gaussian
processes. Particular focus is on clarity and ease of implementation of the
estimator and the associated asymptotic results, making it easy for
practitioners to apply the methods. We additionally show how measurement noise
in the observations will bias the estimator, potentially resulting in the
practitioner erroneously finding evidence of fractal characteristics in a time
series. We propose a new estimator which is robust to such noise and construct
a formal hypothesis test for the presence of noise in the observations.
Finally, the methods are illustrated on two empirical data sets; one of
turbulent velocity flows and one of financial prices.
