Locally optimal detector design in impulsive noise with unknown distribution
Abstract
This paper designs the locally optimal detector (LOD) in additive white impulsive noise with unknown distribution. Unlike traditional LODs derived from a known or approximated noise probability density function (PDF), the LOD proposed in this paper is achieved by designing the zero-memory non-linearity (ZMNL) function based on real data. After the PDF estimation in a nonparametric way by a kernel method, the ZMNL function is designed as a piecewise differentiable function consisting of a polynomial function and inverse proportional functions. Then, we analyze the detection performance and develop the constant false alarm ratio technique. Simulation results show that the LOD design is near-optimal in α-stable noise and the optimal in real atmospheric data, compared with the maximum likelihood detector of α-stable distribution.
Keywords
Locally optimal detector ZMNL function Non-Gaussian distribution Polynomial fittingAbbreviations
- AZMNL
algebraic-tailed ZMNL
- CFAR
constant false alarm ratio
- CZMNL
Cauchy ZMNL
- FAR
false alarm ratio
- GSNR
generalized SNR
- GZMNL
Gaussian-tailed ZMNL
- KDE
kernel density estimation
- LOD
locally optimal detector
- MFD
matched filter detector
- MLD
maximum likelihood detector
probability density function
- PZMNL
polynomial ZMNL
- SαS
symmetricα
- -stable; SNR
signal-to-noise ratio
- ZMNL
zero-memory nonlinearity
1 Introduction
Generally, signal detection in additive noise can be viewed as a problem of binary or multiple hypothesis testing [1]. Most existing digital systems use linear detectors which are the optimal in additive white Gaussian noise. However, nonlinear processing is required for optimal detection in impulsive noise with heavy PDF tails. This is necessary for many environments, such as vehicular communication, power line communication, low-frequency communication, and underwater communications [2, 3, 4]. In low signal-to-noise ratio (SNR), the LOD can be realized via a simple structure by a nonlinear preprocessor based on ZMNL processing followed by a linear correlator [5, 6]. As is well-known, the ZMNL function can be derived directly if the noise PDF in closed form is known.
However, the ZMNL function needs designing when the analytical PDF is unavailable. For instance, the α-stable noise, which is widely used for modeling the impulse noise, does not provide the PDF in closed form generally [4]. As a result, researchers have developed various ZMNL functions for different α-stable distributions [7, 8, 9]. However, real-data processing results demonstrate that the impulsive noise does not always strictly obey the α-stable distribution, though it generally has a unimodal PDF like the α-stable noise. Therefore, the ZMNLs which are designed for α-stable noise are not surely optimal in real-data processing, as illustrated by the simulations in Section 6.4.
This paper focuses on the LOD design when the noise distribution is unknown, which is not fully discussed before. Actually, this is a reasonable consideration since the noise in real world would probably be varying and disobey the assumed distributions, e.g., the α-stable distribution. To solve this problem, this paper proposes to develop a practical approach for designing the ZMNL based on the real-noise data instead of knowing the PDF or assuming the noise distribution.
In the LOD design, lessons are drawn from the existing ZMNL functions of the symmetric α-stable (S αS) noise which is practically useful as a heavy-tailed distribution. In this paper, the ZMNL is designed as a piecewise function which follows linearity, nonlinearity, and differentiability in different regions. Then, the detection performances are analyzed theoretically. In simulations, the proposed approach is demonstrated in simulated S αS noise and in real atmospheric noise. The detection performances will be compared with several existing detectors.
The remainder of this paper is organized as follows. Section 2 briefly reviews several detectors for known noise PDFs. Section 3 introduces the preparation works for unknown noise PDF. Then, the ZMNL function is designed in Section 4, and detection performances are analyzed in Section 5. Section 6 presents the simulations in α-stable noise and real atmospheric noise. Finally, conclusions are drawn in Section 7.
2 Detectors in white noise with known PDF
where s_{ i }, ξ_{ i }, and x denote the desired signal, the signal amplitude, and the additive white noise, respectively [1]. For the case of multiple sample detection, r, s_{ i }, and x can be considered as M-dimensional row vectors.
where the symbol “ \(\doteq \) ” denotes “being equivalent to.”
where (·)^{ † } denotes transpose. Obviously, the MFD depends on the linear correlation.
is called as the ZMNL function [1].
3 Preparation of LOD design for unknown PDF
This paper proposes to design the LOD and g(x) based on real data when the PDF is unknown. Preparatory work raised by unknown PDF is introduced in this section. The detailed ZMNL design approach will be presented in the next section.
3.1 Guidelines for the LOD design
This paper focuses on impulsive noise which comes from unimodal heavy-tailed distributions and has a PDF similar to the PDFs of existing heavy-tailed distributions such as the S αS noise and the Middleton class A noise [2]. However, on the one hand, due to the remarkable differences among the PDFs, the ZMNL functions of existing noise models are possibly not the optimal for the impulsive noise in various scenarios, giving rise to the necessity of LOD design. On the other hand, since the PDF shapes are similar, the knowledge of S αS noise is referable for our LOD design.
Analysis on the existing ZMNL functions of impulsive noise shows that most of them can be regarded as piecewise functions which consist of the main body in the near-linear region and the tails in the non-linear regions. At the breakpoints, natural ZMNLs deduced from heavy-tailed distributions are generally differentiable, while designed ZMNLs may be continuous or discontinuous.
In this paper, g(x) is designed as a piecewise function, following three guidelines. (i) The main body of g(x) is estimated by polynomial fitting. (ii) The tails of g(x) are modeled as reciprocal functions. (iii) The breakpoints are localized for continuity and differentiability. The detailed reasons of the guidelines will be presented in Section 4 where we develop the LOD design procedures. Before that, the rest of this section will introduce the preparatory work for the LOD design.
3.2 ZMNL sample calculation via the KDE
When the noise distribution is unknown, the PDF can be estimated by nonparametric estimation methods. Herein, this paper proposes to measure the PDF samples by the kernel density estimation (KDE) method, i.e., the Parzen-Rosenblatt window method [10, 11].
where \(\mathcal {K}(\cdot) \) is the kernel function and h>0 is the smoothing parameter called the bandwidth.
simulates f^{′}(x) at x=x_{ l }. Specially, let \(\widetilde {g}[\!x_{1}]=\widetilde {g}[\!x_{L}]=0\).
It is obvious that the value of bandwidth h is critical for the PDF estimation. In fact, since the PDF and ZMNL samples are simulated numerically in discrete points, the increment Δx is also very important. It must be evaluated reasonably so that \(\widetilde {f}[\!x_{l}]\) can describe the shape of f(x) effectively.
As said before, this paper emphasizes on the impulsive noise from heavy-tailed distributions. Although the noise may disobey the α-stable distribution, the α-stable distribution characteristics are also referable in some aspects. In our practice, the α-stable methods are found to be efficient in the evaluation of h and Δx.
The evaluation approach is developed as follows. Firstly, estimate the α-stable distribution parameters based on the noise samples \(\widetilde {x}[\!n]\). An estimation method can refer to paper [12] which is based on the quantile of samples.
By this approach, the PDF estimate \(\widetilde {f}[\!x_{l}]\) can be smooth and well represent the probability density of noise samples.
3.3 ZMNL sample extraction
The ZMNL samples must be classified before using them for LOD design. The reason is as follows. As can be seen from (5), when \(\widetilde {f}(x_{l})\) is very small, the error between \(\widetilde {f}_{\Delta }[\!x_{l}]\) and f^{′}(x_{ l }) can be greatly increased when entering \(\widetilde {g}[\!x_{l}]\), resulting in huge gap between \(\widetilde {g}[\!x_{l}]\) and g(x) (see simulations in Section 6). Thus, \(\widetilde {g}[\!x_{l}]\) samples for small \(\widetilde {f}[\!x_{l}]\) are unreliable.
where \(\overline {x_{i}}\) and \(\overline {\widetilde {g}[\!x_{i}]}\) denote the mean of x_{ i } and \(\widetilde {g}[\!x_{i}]\), respectively, for \(x_{i} \in {\widetilde {\Omega }}\).
Algorithm of ZMNL sample extraction
Step 1: Find the maximum of \(\widetilde {f}[x_{l}]\) as \(\widetilde {f}\left [x_{L_{max}} \right ]\), where L_{ max } denotes its position in {x_{1},x_{2},…,x_{ L }}. |
Step 2: Find the two values of \(\widetilde {f}[\!x_{l}]\) which are nearest to \(\widetilde {f}[x_{L_{max}}]/2\) for l<L_{ max } and l^{′}>L_{ max }, respectively. Denote the corresponding number as L_{ lowh } and L_{ uph }. Initialize L_{ low }=L_{ lowh } and L_{ up }=L_{ uph }. |
Step 3: Use (12) to calculate two linearity values, i.e., ρ_{ low } for \(\widetilde {g}[x_{l}]\) in \(\Omega _{low}=\left [x_{(L_{low}-1)}, x_{L_{up}}\right ]\) and ρ_{ up } for \(\widetilde {g}[x_{l}]\) in \(\Omega _{up}=\left [x_{L_{low}}, x_{(L_{up}+1)}\right ]\). |
Step 4: If max(ρ_{ low },ρ_{ up })≤ρ_{th}, go to Step 5. Otherwise, continue. If ρ_{ low }>ρ_{ up }, let L_{ low }=L_{ low }−1; if ρ_{ low }≤ρ_{ up }, let L_{ up }=L_{ up }+1. Then, go to Step 3. |
Step 5: The near-linear region is defined as \(\Omega =\left [x_{L_{low}}, x_{L_{up}}\right ]\). The samples \(\widetilde {g}[x_{l}]\) in Ω are extracted and denoted as \(\breve {{g}}[x_{i}]\). |
which has I=L_{up}−L_{low}+1 samples.
4 Design of the LOD and ZMNL function
In the LOD design, the ZMNL function is modeled as a piecewise function. The main body is estimated by polynomial fitting. The tails and the breakpoints are determined for continuity and differentiability.
4.1 Polynomial fitting for near-linear region
The proposal of polynomial fitting is based on the basic property of ZMNL functions. As can be seen in (4), the ZMNL function transforms the observations r into the domain related to probability, acting like a weighting function. It may be linear (in Gaussian case) or nonlinear (in non-Gaussian cases). In either case, g(x) is supposed to keep close correlation to x for the majority of x, so that the test statistic \( g(\boldsymbol {r}) \boldsymbol {s}_{i}^{\dag }\) could be optimized. Therefore, the near-linear region always exists in g(x) for heavy-tailed distributions.
This paper employs a polynomial function to describe the nonlinearity property and also keep certain linearity in the near-linear region. This is unlike traditional ZMNL designs which define the nonlinear region very differently from the linear region and produce unavoidable error to the real LOD.
Thus, the main body of \(\widehat {g}(x)\) is achieved.
4.2 Tail design and breakpoint localization
Usually, the tail of g(x) functions as a controller which limits the effects of large-magnitude samples from heavy-tailed distributions. The tail function can be developed in various methods. For instance, Nikais and Shao proposed two kinds of nonlinearity for x∉Ω, including a “hole puncher” to null g(x) and a “clipper” to set g(x) equal to the nearest g(x_{ i }) as constants. The tail of g(x) is also modeled as x^{1−α} or x^{−1} in α-stable noise [4, 7]. Besides, the breakpoints which connect the tails with the main body are also important for the LOD design.
where the breakpoint locations x_{ ne } and x_{ po } and two coefficients B_{ ne } and B_{ po } are the parameters to determine.
Algorithm of ZMNL function design
Step 1: Given P and \(\breve {g}[x_{i}]\), calculate \(\widehat {\boldsymbol {A}}\) by formulas (16)-(19). |
Step 2: Solve the function \(\sum \limits _{p=0}^{P} (p+1)\widehat {A}_{p} x^{p} =0, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(23) \) |
and evaluate x_{ n e } and x_{ p o } by the largest negative root and the smallest positive root respectively. |
Step 3: Compute the two parameters |
\( B_{ne} = \sum \limits _{p=0}^{P} \widehat {A}_{p} x_{ne}^{p+1}, B_{po} = \sum \limits _{p=0}^{P} \widehat {A}_{p} x_{po}^{p+1} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(24) \) |
Step 4: Obtain \(\widehat {g}(x)\) by (21). |
As far as we know, this is the first approach to use polynomial fitting for designing the ZMNL function \(\widehat {g}(x)\). Herein, we call the proposed ZMNL design as polynomial ZMNL (PZMNL).
One merit of the PZMNL is that it can approximate the main body of real LOD very well. As shown in Fig. 1, the LOD g(x) changes smoothly without a breakpoint to clearly classify the near-linear region and the nonlinear region. However, traditional ZMNL functions change dramatically at the breakpoints, from proportional functions in the near-linear region into nonlinear functions in the nonlinear region. They are greatly different from g(x) and thus have an upper bound of being sub-optimal. Unlike them, the PZMNL can fit well around the near-linear region and achieve nearly optimal performances.
Notation: About the polynomial order P, practical experience shows that 10≤P≤20 is suitable for polynomial fitting. Otherwise, if P is too small, the designed polynomial cannot represent the nonlinearity beyond the near-linear region and also reduces the solution to Eq. (23). On the contrary, if P is too large, the designed polynomial is not smooth enough and also generates unfavorable solutions to (23), resulting in a narrow near-linear region. Examples can refer to the simulations in Section 6.
5 Detection performance analysis of the LOD
This section analyzes the detection performances of the detectors based on various ZMNL functions. Supposing h(x) as the ZMNL function before a linear detector, based on the signal model (1), the test statistic is
for hypothesis H_{ i }. If the PDF of T(r) is known, the detection performance can be deduced accordingly. However, the PDF of T(r) is uneasy to obtain generally.
Herein, we consider this topic in low SNR as a usual scenario in real world. Obviously, low SNR means that ξ_{ i } is rather small. As a result, to detect a transmitted signal bit, communication systems need to accumulate massive samples as M is large. Besides, the noise samples are assumed to be independently and identically distributed (i.i.d.). Therefore, by using the central limit theorem, we suppose that the test statistic T(r) obeys the Gaussian distribution. Its PDF is available as long as we know its expectation and covariance.
Before the deduction, the following assumption is made to simplify analysis.
Assumption 1
Then, the asymptotic distribution of T(r) in low SNR is derived, concluded as Theorem 1.
Theorem 1
The proof is referred in Appendix. Note that \(\mathcal {E}_{hg}\) and \(\mathcal {E}_{hh}\) are actually the expectations of h(x)g(x) and h^{2}(x), respectively.
As clearly shown in (29) and (32), the calculation of threshold η depends on the information of PDF f(x). Thus, for the LOD in the noise with unknown PDF, the CFAR technique is actually unavailable.
The following develops the CFAR technique for the noise with unknown PDF. Notice that false alarm is related to hypothesis H_{0} and independent of hypothesis H_{1}. In (31), hypothesis H_{0} requires f(x) to calculate the covariance coefficient \(\mathcal {E}_{hh}\). Though g(x) is also unknown, it is unnecessary in hypothesis H_{0}. In fact, g(x) is only required in hypothesis H_{1} to calculate the expectation coefficient \(\mathcal {E}_{hg}\) in (28) and finally affects the detection probability P_{ D } in (33).
Therefore, this paper proposes the CFAR technique by using the PDF estimate \(\widetilde {f}[x_{l}]\) instead of the unknown real PDF f(x). By employing \(\widetilde {f}[\!x_{l}]\) in (6), we can compute \(\mathcal {E}_{hh}\) in (29) and then the threshold η by formula (32) for the desired FAR. Simulation results show that this approach can achieve a steady FAR which is very close to the desired FAR.
6 Results and discussion
This section presents the results of LOD design in the S αS noise and real atmospheric noise. Discussions about the proposed design of LOD and PZMNL are provided.
6.1 Detector and simulation settings
The general approach to suppress heavy-tailed interference is to pass the received observation through a ZMNL limiter. Two simple examples include a hole puncher and a clipper [4]. Some limiters are designed carefully according to the noise model.
where \(\tau = \sqrt {K(\alpha)\Gamma (1/\alpha)/\Gamma (3/\alpha)}\) uses K(α)=α^{2}. The GZMNL formula can refer to [7, 9] and is not introduced here for the sake of simplification.
As the analytical PDF of S αS noise is generally unavailable, the MLD is achieved by a numerical approach as follows. Firstly, the discrete PDF is computed by the inverse Fourier transform of the characteristic function. Then, assuming the signal amplitude is known, decision rule (2) is realized by linear interpolation. The ideal LOD by (5) is also obtained by linear interpolation of the discrete differential. Simulation results show that the MLD and the LOD perform very close to each other. Thus, the LOD and the MLD are depicted by the MLD curve. Obviously, linear interpolation bears expansive computational cost, but it is necessary for the MLD.
where γ denotes the dispersion of S αS noise. In simulations, γ is fixed as 1, so that the AZMNL in (36) can be used. The GSNR is changed by adjusting the signal amplitude ξ. The sinusoidal signal is transmitted, with uniform energy ss^{ † }=M=1024. The probabilities of detection and false alarm are the results of 10^{5} Monte Carlo simulations.
6.2 ZMNL design in S αS noise
α-Stable distribution is widely used in impulse noise modeling. As its PDF is generally unavailable in closed form, the analytical LOD does not exist for most α-stable distributions. Herein, we simulate the LOD of S αS noise numerically. S αS noise data is simulated for α=1.5, N=10^{4}. The KDE method uses the Gaussian kernel for \(\mathcal {K}(\cdot)\).
Then, \(\widehat A_{p}\) is calculated for various order P. Figure 2 depicts the designed PZMNL functions, as well as the ZMNL of S αS distribution \(\acute {g}(x)\) by numeric simulation. It can be seen that the PZMNL \(\widehat {g}(x)\) for P=10, with the breakpoints at x_{ ne }=− 3.3 and x_{ po }=3.6, is almost the same as \(\acute {g}(x)\). The PZMNLs for P=5 and 20 are less similar to \(\acute {g}(x)\). Moreover, the PZMNL for P=40 is with a narrow near-linear region and quite different from \(\acute {g}(x)\). Hence, setting P around 10 is suitable.
By comparison with Fig. 1, it is clear that the AZMNL, the CZMNL, and the GZMNL functions fit \(\acute {g}(x)\) worse than the PZMNLs for P=5, 10, and 20. Note that the PZMNL generally changes in another simulation which contains different noise samples, and the PZMNLs for P=40 is not always as bad as that of Fig. 2.
6.3 Detection performances in S αS noise
As can be seen Fig. 3b, the CZMNL and the GZMNL curves are similar, worse than the other ZMNL detectors. The MLD is the best, and the AZMNL and the PZMNLs for P=5,10,20 are close to the MLD. Besides, the PZMNL for P=40 is worse, as a result of less fitting to \(\acute {g}(x)\). It is worth noting that all the ZMNL functions achieve significant improvement compared to the MFD.
6.4 Experimental results on real data
Atmospheric noise is known to possess a significant impulsive nature. This paper uses it to demonstrate the proposed PZMNL method. The raw data is recorded by a magnetotelluric sounding system at sampling frequency 512 Hz and then whitened to eliminate the power line interference. The output shows typical characteristics of impulsive noise and called as “real data.” The real data used for illustration is recorded in a sunny day at about 10:20 a.m., June 7, 2017, in Qianjiang, Hubei province, China. Its estimate of α is 1.42, near to α=1.5 of the previously simulated S αS noise. In preprocessing, the power of real data is adjusted so that the estimate of dispersion γ is 1.
6.5 Extended discussion
The above results demonstrate our proposed methods of LOD and PZMNL design are effective based on noise samples, no matter whether their distribution model is known. This is a significant advantage over other methods which are developed based on priori known models. In fact, impulsive noise has several different models, but there is no clear agreement or conclusion about their optimality. Once any detectors make false assumptions about the model of real noise, their performances degrade greatly. However, our methods do not have such a risk.
As our methods are proposed for impulsive noise, they may be used in other applications. Some limitations are worthy noting. Firstly, the PZMNL algorithm is developed for unimodal noise with heavy tails. Otherwise, multimodal noise has more than one near-linear regions and complicates the polynomial fitting. Secondly, Assumption 1 is necessary for the CFAR technique. It is easily satisfied in impulsive noise whose PDF is generally symmetric to zero, but may not for others. Finally, the PZMNL’s order P may vary for different distributions. It can be optimized in preprocessing and updated continuously in real time.
7 Conclusions
This paper proposes a novel approach for the LOD design in impulsive noise from unknown distributions. The ZMNL function is designed based on noise samples without assuming distribution models. As it is mainly developed by polynomial fitting, we call it as PZMNL. Simulations on α-stable noise show that the PZMNL achieves the detection performances similarly to the maximum likelihood detector which knows the noise PDF. Experimental results on real atmospheric noise demonstrate that the PZMNL outperforms other detectors which make false assumptions on the distribution models of real noise.
8 \thelikesection Appendix
8.1 \thelikesubsection Proof of Theorem 1
Notes
Acknowledgements
The authors would like to thank Yangyong Zhang from 722 Research Institute of China Shipbuilding Industry Corporation (CSIC) for assisting in the experiments of collecting the real data of atmospheric noise.
Funding
This work was supported by the National Natural Science Foundation of China (nos. 61701067, 61771085, and 61671095) and the Research Project of Chongqing Educational Commission (nos. KJ1600427 and KJ1600429).
Availability of data and materials
In this paper, the observations of non-Gaussian noise are collected by an environmental noise sounding device. The LOD design approach proposed in this paper can be used as long as the noise samples are collected. For the practical applications, the noise samples are collected by the receivers of communication systems, radar systems, etc. Compared with the conventional detector, it raises no extra requirement on the data or material.
Authors’ contributions
ZL is the first and corresponding author. He proposes the guidelines for the LOD design and the idea of estimating the ZMNL function by polynomial fitting and designing the tails as inverse proportional functions. He analyzes the detection performances of the LOD and develops the CFAR technique. He also simulates the LOD design algorithm and the detection performances. Finally, he writes this paper. PL investigates the α-stable noise and writes the programs of α-stable distribution, including generating noise samples, simulating the discrete PDF, and estimating the parameters. GZ collects the real data of atmosphere noise in low-frequency band, at a sampling frequency 512 Hz, by the magnetotelluric sounding system of 722 Research Institute, China. All authors read and approved the final manuscript.
Competing interests
The authors declared that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
- 1.SM Kay, Fundamentals of statistical signal processing, volume II: detection theory (Prentice Hall PTR, Englewood Cliffs, 1998).Google Scholar
- 2.H Oh, H Nam, Design and performance analysis of non-linearity preprocessors in an impulsive noise environment. IEEE Trans. Veh. Technol. 66(1), 364–376 (2017).Google Scholar
- 3.Y Yabuuchi, et al, in Proc. IEEE ISPLC. Low rate and high reliable modulation schemes for in-vehicle power line communications (IEEE explore digital library, 2011), pp. 249–254.Google Scholar
- 4.CL Nikias, M Shao, Signal processing with alpha stable distribution and applications (John Wiley and Sons, Inc., New York, 1995).Google Scholar
- 5.X Li, et al, Normalisation-based receiver using BCGM approximation for α-stable noise channels. Electron. Lett. 49(15), 965–967 (2013).CrossRefGoogle Scholar
- 6.A Kadri, in European Wireless 2014; 20th European Wireless Conference; Proceedings of VDE. Suboptimal receivers for weak M-ary chirp signals in non-Gaussian noise (IEEE explore digital library, 2014), pp. 1–6.Google Scholar
- 7.X Li, et al, Near-optimal detection with constant false alarm ratio in varying impulsive interference. IET Signal Proc. 7(9), 824–832 (2013).CrossRefGoogle Scholar
- 8.GA Tsihrintzis, CL Nikias, Performance of optimum and suboptimum receivers in the presence of impulsive noise modeled as an alpha-stable process. IEEE Trans. Commun. 43(234), 904–914 (1995).CrossRefGoogle Scholar
- 9.A Swami, BM Sadler, On some detection and estimation problems in heavy-tailed noise. Elsevier J. Signal Process. 82(12), 1829–1846 (2002).CrossRefzbMATHGoogle Scholar
- 10.V Bhatia, B Mulgrew, Non-parametric likelihood based channel estimator for Gaussian mixture noise. Signal Proc. 87(11), 2569–2586 (2007).CrossRefzbMATHGoogle Scholar
- 11.N Markovich, Nonparametric analysis of univariate heavy-tailed data: research and practice (John Wiley and Sons, New York, 2008).zbMATHGoogle Scholar
- 12.JH McCulloch, Simple consistent estimators of stable distribution parameters. Commun. Stat. Simul. Comput. 15(4), 1109–1136 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
Copyright information
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.