This repository contains R programs for the article, “Taylor’s law of fluctuation scaling for semivariances and higher moments of heavy-tailed data.” This article has been submitted for publication.
Prior to using R programs on this repository, please download the main R program TL_Library.R. R packages, stabledist and xtable, are used to generate random sample from one-sided stable law F(c,α) with c>0 and 0<α<1, and organize simulation results, respectively.
Here we generate random samples from F(1,α) and evaluate the performance of the proposed tail-index estimators of α from the Bias (x 10^3; average of estimate minus true α) and MSE (mean squared [estiamte minus true α]).
For the Bias and MSE of the estimators, B1, B2, B3, HI.N, HI.M, HI.Opt, and MHB3 with 0<α<1, c=1, and sample size n=104 based on 104 independent repetitions, one can use the R codes
Table 1-2 to reproduce Tables 1 and 2.
Our calculation took approximately 10 minutes for n=105 on a computer with a 3.0GHz processor and 64GB of memory.
1.2. Bias and MSE of the estimators with 0<α<1 and c=0.5,1,2 (Tables 1-6 in Supporting Information).
For the Bias and MSE of the estimators, B1, B2, B3, HI.N, HI.M, HI.Opt, and MHB3 with 0<α<1, one can use the rcodes Table 1-2 in SI for c=1,
Table 3-4 in SI for c=0.5, and
(Table 5-6 in SI) for c=2, based on 104
independent repetitions.
One can change the sample size n in the R codes to obtain the Bias and MSE for sample size n in {102, 103, 104, 105}. Our calculation took approximately 30 minutes for n=105 on a computer with a 3.0GHz processor and 64GB of memory.
Part 2. Convergence of the number of observations exceeding the sample mean and the generalized Taylor's law.
We used the R codes
Figure 1 for the histgrams and QQ-plot of U-α/Γ(1-α) and Nn+/nα when α=0.25
and
Figure 2 for α=0.5.
Our calculation took approximately 10 minutes for n=105 on a computer with a 3.0GHz processor and 64GB of memory.
We use Table 7-8 in SI to obtain the Bias and MSE to demonstrate the Taylor's law for lower semivariance when 0<α<1 based on 104 independent repetitions. Our calculation took approximately 10 minutes for n=105 on a computer with a 3.0GHz processor and 64GB of memory.
2.3. Effect of sample size to the convergence of the moment ratios (Theorems 3 and 6) and Nn+/nα in Theorems 9 (Tables 9-11 in Supporting Information).
We use the following R codes,
Table 9 in SI(m=1000),
Table 10 in SI(m=200), and
Table 11 in SI(m=2000),
to perform the two-sample Kolmogorov-Smirnov test and evaluate the speed of the convergence in Theorems 3, 6, and 9, where m is a parameter to approximate the random vectors (Uh1, Uh2) inspired by LePage et al. (1981) and Cohen et al. (2020)
See more details in the Supporting Information.
For each m, our calculation took approximately 6 hours for n=105 on a computer with a 3.0GHz processor and 64GB of memory.
2.4. Effect of smaple size to the convergence of the ratio of sample mean and lower semivariance in Theorem 1, the Taylor's Laws for semivariances (Tables 12 in Supporting Information).
Since the ratio of squared sample mean and lower semivariance converges almost surely to 1, we use the R codes
Table 12 in SI
to evaluate the pace of convergence by the proportions of the difference between the forgoing ratio and 1 within tolerances ε from 10-3 to 10-6 with large sample size n=108. Our calculation took approximately 2 hours for n=105 on a computer with a 3.0GHz processor and 64GB of memory.
- Lepage, Raoul and Woodroofe, Michael and Zinn, Joel (1981). Convergence to a stable distribution via order statitics. The Annals of Probability, 624-632
- Cohen, Joel E. and Davis, Richard A. and Samorodnitsky, Genneady (2020). Heavy-tailed distributions, correlations, Kurtosis and Taylor's Law of fluctuation scaling. Proceedings of the Royal Society A 476, 2020010.