Bootstrap Tail Thickness Estimation for Symmetric Alpha-Stable Random Variables

A new theorem shows that a bootstrap algorithm can estimate the impulsivity or tail thickness of symmetric a-stable (SαS) signals. SαS bell curves include the Gaussian bell curve as a special but non-impulsive case. Signals grow more impulsive as the bell curve’s tail thickness increases or as the tail-thickness parameter α falls from 2 toward 0. The thin-tailed Gaussian bell curve has α = 2. The algorithm computes a statistic from SαS samples and then matches the test statistic against a continuum of precomputed values to the find the estimated tail thickness α̂. The theorem and a corollary show that the statistic is invertible because it is a continuous bijection. So the boostrapped α̂ is a consistent estimator of α in general. Simulations show that α̂ is robust for signals with α ∈ [0.2, 2] and that the estimator error decreases as the number of samples increases. 1. Robust Estimation of Symmetric α-Stable Tail Thickness. We show that a bootstrap algorithm can estimate the impulsiveness of a sequence of symmetric α-stable (SαS) random samples [1], [2], [3], [4], [5], [6]. The algorithm estimates the tailthickness parameter α by interpolating a sample statistic between precomputed values. Figure 6 and Table 1 show that the algorithm applies to SαS random variables with α ∈ [0.2, 2]. A theorem shows that each α corresponds to a unique value of a sample statistic τ (Xα). A corollary to the theorem shows that the map is a bijection. The algorithm estimates α by calculating τ (Xα) and then inverting the bijection (Figure 5). The estimator α̂ applies to all finite sequences of independent and identically distributed (i.i.d.) SαS random variables. Symmetric α-stable random variables have thick power-law tails and generalize the Gaussian probability density function (pdf) [7], [8], [9], [10], [11], [12], [13], [14]. The tail-thickness parameter α lies in (0, 2] and controls the impulsivity of samples drawn from the random variable Xα. Figure 1 shows the inverse relation between α and the thickness of the bell-curve tails. Figure 2 shows how α controls impulsiveness of samples from the distribution. The Gaussian pdf takes α = 2. The thickertailed Cauchy pdf takes α = 1. The moments of an α-stable random variable are finite only up to order k for k < α. Only the Gaussian random variable has finite second and higher-order moments. The usual central limit theorem states that a standardized sum of finite-variance random variables converges in distribution to the standard normal distribution Z ∼ N (0, 1) [15], [16]. The generalized central limit theorem states a similar result for infinite-variance α-stable random variables [17], [18]: a standardized sum of α-stable random variables converges in distribution to an α-stable random variable with the same α. It also shows this holds only for α-stable random variables. Brandon Franzke and Bart Kosko are with the Signal and Image Processing Institute, Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089, USA (email: [email protected]) Real noise tends to be impulsive. Natural sources of impulsive signals include condensed and soft matter physics [19], [20], [21], geophysics [22], meteorology [23], biology [24], economics [25], [26], [27], fractional kinetics [28], [29], and communications [14], [30], [31], [32], [33]. Many random models assume that the dispersion of a random variable is equal to its finite variance or its squared-error from the population mean. Impulsive signals violate this finite variance assumption in general. So these model may wrongly dismiss important “rare” events as outliers. Selection of a bell-curve signal model requires empirical tests to estimate the actual tail thickness. Finding the optimal SαS bellcurve for a given symmetric signal pattern is an open research problem. The α-Stable Estimation Algorithm in Section 3 estimates α from a sequence of observed SαS random samples. The algorithm computes the estimator α̂ in two stages: (1) it constructs a bijective τ-map between a test statistic τ (Xα) and α and (2) it computes τ (Xα) for observed random samples and then uses the τ−1-map to find α̂. The α-Stable Estimation Map Theorem in Section 2 ensures that the τ-map is unique. A corollary shows that the τ-map is a bijection and thus has an inverse α̂ = τ−1 (τ (Xα)). The τ-map is also continuous. Thus α̂ converges in probability to α and so is consistent because α̂ is a consistent estimator of α since τ̂ (Xα) is a consistent estimator of τ (Xα) [15]. Section 4 presents simulations to show that α̂ is a good estimator for α ∈ [0.2, 2]. 2. The α-Stable Estimation Map Theorem The α-Stable Estimation Algorithm computes an estimator α̂ of the tail-thickness parameter α. It computes a sample statistic τ (Xα) from a sequence of observed SαS samples and then estimates α through the τ−1-map that maps from τ (Xα) to α. The α-Stable Estimation Map Theorem guarantees that on average the τ-map generates distinct τ (Xα) for two independent sequences of SαS independent and identically distributed (i.i.d.) random variables Xα1 and Xα2 if α1 , α2. The α-Stable Estimation Algorithm uses a corollary to ensure the inverse exists. The corollary thus allows the algorithm to estimate α through the τ−1-map. The algorithm computes a test statistic for Xα that resembles a vector p-norm. The test statistic is finite because the pth-sample moment of a finite sequence of such realizations is finite for any finite p > 0. Suppose Xα is a sequence of N i.i.d. SαS random variables Xα with pdf fα (x). Suppose the random variable has α ∈ (0, 2] and has unit dispersion: γ = 1. Suppose further that N is finite. Then the sample maximum H (xα) = max 1≤k≤N |xα [k]| < ∞ (1) almost surely since limx→∞ fα (x) = 0. Define gp by the length-normalized sample p-norm for finite Fig. 1. Symmetric α-stable probability density functions. The figure shows SαS probability density functions for α = 2.0 (Gaussian), 1.8, 1.5, and 1.0 (Cauchy). The thickness of the bell curve tails increases as α decreases. Thicker tails correspond to more impulsive samples. The Gaussian bell curve is the only SαS distribution with finite moments of order k ≥ 2. Choosing α for a given application is an empirical question. −5 0 5 α=2