Statistical properties of the anomalous scaling exponent estimator based on time-averaged mean-square displacement

The most common way of estimating the anomalous scaling exponent from single-particle trajectories consists of a linear fit of the dependence of the time-averaged mean-square displacement on the lag time at the log-log scale. We investigate the statistical properties of this estimator in the case of fractional Brownian motion (FBM). We determine the mean value, the variance, and the distribution of the estimator. Our theoretical results are confirmed by Monte Carlo simulations. In the limit of long trajectories, the estimator is shown to be asymptotically unbiased, consistent, and with vanishing variance. These properties ensure an accurate estimation of the scaling exponent even from a single (long enough) trajectory. As a consequence, we prove that the usual way to estimate the diffusion exponent of FBM is correct from the statistical point of view. Moreover, the knowledge of the estimator distribution is the first step toward new statistical tests of FBM and toward a more reliable interpretation of the experimental histograms of scaling exponents in microbiology.

Figure 1

Probability density function of TAMSD estimator $\widehat{\beta }$ for different trajectory lengths $N$ ($N=20$, 100, 500, 1000), obtained via Monte Carlo simulations with 1000 independent runs, with $n=10$ and ${\tau }_i=i$ ($i=1,...,10$). The true values of $\beta$ are (a) 0.6, (b) 1.0, and (c) 1.4.

Figure 2

Empirical quantiles of the TAMSD estimator $\widehat{\beta }$ as functions of the trajectory length $N$, obtained via Monte Carlo simulations with 1000 independent runs, with $n=15$ and ${\tau }_i=i$ ($i=1,...,15$). The true values of $\beta$ are (a) 0.6, (b) 1.0, and (c) 1.4.

Figure 3

The (a) bias and (b) variance of the estimator $\widehat{\beta }$ as functions of the trajectory length $N$, obtained via Monte Carlo simulations with 1000 independent runs, with $n=10$ and ${\tau }_i=i$ ($i=1,...,10$). The results are presented for three values of $\beta$: 0.6, 1.0, and 1.4.

Figure 4

The probability for the estimator $\widehat{\beta }$ to be within the interval $[\beta -0.1,\beta +0.1]$ as a function of the trajectory length $N$ and of the parameter $n$, obtained via Monte Carlo simulations with 1000 independent runs. (a), (c), (e) FBM trajectories and (b), (d), (f) FBM perturbed by white noise $\mathcal{N}(0,0.01)$. Top, middle, and bottom rows present the cases $\beta =0.6, \beta =1.0$, and $\beta =1.4$, respectively.