The generalized Cauchy family of distributions with applications
- Ayman Alzaatreh^{1}Email author,
- Carl Lee^{2},
- Felix Famoye^{2} and
- Indranil Ghosh^{3}
DOI: 10.1186/s40488-016-0050-3
© The Author(s). 2016
Received: 23 February 2016
Accepted: 13 July 2016
Published: 3 August 2016
Abstract
A family of generalized Cauchy distributions, T-Cauchy{Y} family, is proposed using the T-R{Y} framework. The family of distributions is generated using the quantile functions of uniform, exponential, log-logistic, logistic, extreme value, and Fréchet distributions. Several general properties of the T-Cauchy{Y} family are studied in detail including moments, mean deviations and Shannon’s entropy. Some members of the T-Cauchy{Y} family are developed and one member, gamma-Cauchy{exponential} distribution, is studied in detail. The distributions in the T-Cauchy{Y} family are very flexible due to their various shapes. The distributions can be symmetric, skewed to the right or skewed to the left.
Keywords
T-R{Y} framework Quantile function Moments Shannon’s entropy1. Introduction
The Cauchy distribution, named after Augustin Cauchy, is a simple family of distributions for which the expected value does not exist. Also, the family is closed under the formation of sums of independent random variables, and hence is an infinitely divisible family of distributions. The Cauchy distribution was used by Stigler (1989) to obtain an explicit expression for P(Z _{1} ≤ 0, Z _{2} ≤ 0) where (Z _{1}, Z _{2})^{ T } follows the standard bivariate normal distribution. The Cauchy distribution has been used in many applications such as mechanical and electrical theory, physical anthropology, measurement problems, risk and financial analysis. It was also used to model the points of impact of a fixed straight line of particles emitted from a point source (Johnson et al. 1994). In Physics, it is called a Lorenzian distribution, where it is the distribution of the energy of an unstable state in quantum mechanics.
Eugene et al. (2002) introduced the beta-generated family of distributions using the beta as the baseline distribution. Based on the beta-generated family, Alshawarbeh et al. (2013) proposed the beta-Cauchy distribution. The beta-generated family was extended by Alzaatreh et al. (2013) to the T-R(W) family. The cumulative distribution function (CDF) of the T-R(W) distribution is \( G(x)={\displaystyle {\int}_a^{W\left(F(x)\right)}r(t)dt,} \) where r(t) is the probability density function (PDF) of a random variable T with support (a, b) for − ∞ ≤ a < b ≤ ∞. The link function W : [0, 1] → ℝ is monotonic and absolutely continuous with W(0) → a and W(1) → b.
Aljarrah et al. (2014) considered the function W(.) to be the quantile function of a random variable Y and defined the T-R{Y} family. In the T-R{Y} framework, the random variable T is a ‘transformer’ that is used to ‘transform’ the random variable R into a new family of generalized distributions of R. Many families of generalized distributions have appeared in the literature. Alzaatreh et al. (2014, 2015) studied the T-gamma and the T-normal families. Almheidat et al. (2015) studied the T-Weibull family. In this paper, a family of generalized Cauchy distribution is proposed and studied.
This article focuses on the generalization of the Cauchy distribution and studies some new distributions and their applications. The article gives a brief review of the T-R{Y} framework and defines several new generalized Cauchy sub-families. It contains some general properties of the T-Cauchy{Y} distributions. A member of the T-Cauchy{Y} family, the gamma-Cauchy{exponential} distribution, is studied. The study includes moments, estimation and applications. Some concluding remarks were provided.
2. The T-Cauchy{Y} family of distributions
Alzaatreh et al. (2013, 2014, 2015) studied, respectively, the T-R{exponential}, T-normal{Y} and T-gamma{Y} families of distributions. Aljarrah et al. (2014) studied some general properties of the T-R{Y} family. Next, we define the T-Cauchy{Y} family.
Quantile functions for different Y distributions
Y | Q _{ Y } (p) |
---|---|
(a) Uniform | p |
(b) Exponential | −log(1−p) |
(c) Log-logistic | p / (1−p) |
(d) Logistic | log[p /(1−p)] |
(e) Extreme value | log[−log(1−p)] |
(f) Fréchet | −1/log(1−p) |
2.1 T-Cauchy{uniform} family of distributions
2.2 T-Cauchy{exponential} family of distributions
Note that the CDF and the PDF in (8) and (9) can be written as F _{ X }(x) = F _{ T }(H _{ C }(x)) and f _{ X }(x) = h _{ C }(x)f _{ T }(H _{ C }(x)) where h _{ C }(x) and H _{ C }(x) are the hazard and cumulative hazard functions for the Cauchy distribution, respectively. Therefore, the T-Cauchy{exponential} family of distributions arises from the ‘hazard function of the Cauchy distribution’.
2.3 T-Cauchy{log-logistic} family of distributions
2.4 T-Cauchy{logistic} family of distributions
Note that (13) can be written as \( {f}_X(x)=\frac{h_C(x)}{F_C(x)}\times {f}_T\left( \log \left(\frac{F_C(x)}{1-{F}_C(x)}\right)\right) \), which is a family of generalized Cauchy distributions arising from the ‘logit function’ of the Cauchy distribution.
2.5 T-Cauchy{extreme value} family of distributions
The CDF in (14) and the PDF in (15) can be written as F _{ x }(x) = F _{ T }(log H _{ C }(x)) and f _{ X }(x) = {h _{ C }(x)/H _{ C }(x)}f _{ T }(log H _{ C }(x)) respectively.
2.6 T-Cauchy{Fréchet} family of distributions
3. Some properties of the T-Cauchy{Y} family of distributions
In this section, we discuss some general properties of the T-Cauchy family of distributions. The proofs are omitted for straightforward results.
- (i)
The random variable X = − θ cot(πF _{ Y }(T)) follows the T-Cauchy{Y} distribution.
- (ii)
The quantile function for T-Cauchy{Y} family is Q _{ X }(p) = − θ cot(πF _{ Y }(Q _{ T }(p))).
The Shannon’s entropy (Shannon 1948) of a random variable X is a measure of variation of uncertainty and it is defined as η _{ X } = − E{log(f(X))}. The following proposition provides an expression for the Shannon’s entropy for the T-Cauchy{Y} family.
Next proposition gives a general expression for the r-th moment for the T-Cauchy{Y} family.
As an example of the applicability of the results in Lemma 1 and Propositions 1 and 2, we use these results and apply them on the T-Cauchy{exponential}. One can get similar results by choosing any of the T-Cauchy{Y} families.
- (i)
The random variable X = θ cot(πe ^{− T }) follows a distribution in the T-Cauchy{exponential} family.
- (ii)
The quantile functions for T-Cauchy{exponential} family is \( {Q}_X(p)=\theta \cot \left(\pi {e}^{-{Q}_T(p)}\right). \)
Proof: The result follows from Proposition 1 and the fact that E(log f _{ Y }(T)) = μ _{ T }.□
Proof: The result follows from Proposition 2 and the fact that cot(πF _{ Y }(u)) = − cot(πe ^{− u }).□
Proof: For Cauchy distribution, one can show that \( {f}_C^{\prime }(x)=-2\pi {\theta}^{-1}x{f}_C^2(x) \) and \( {h}_C^{\prime }(x)=-2\pi {\theta}^{-1}x{h}_C(x)+{h}_C^2(x). \) On finding \( {f}_X^{\prime }(x) \) by using Eq. (9) and setting the derivative to zero, it is easy to get the result in (24). □
4. Gamma-Cauchy{exponential} distribution
- (i)
GC(1, β, θ) is the exponentiated Cauchy distribution proposed by Sarabia and Castillo (2005). In particular GC(1,1,1) is the standard Cauchy distribution.
- (ii)
GC(1, n ^{− 1}, θ), n ∈ ℕ is the distribution of the minimum of n independent Cauchy random variables.
- (iii)
GC(n + 1, 1, θ), n ∈ ℕ is the distribution of the nth upper record in a sequence of independent Cauchy random variables.
- (i)
If a random variable Y follows a gamma distribution with parameters \( \alpha \) and β, then X = θ cot(πe ^{− Y }) follows the GC(α, β, θ) distribution.
- (ii)
The quantile function of GC(α, β, θ) is \( Q(p)=\theta \cot \left(\pi {e}^{-\beta {\gamma}^{-1}\left[\alpha, p\varGamma \left(\alpha \right)\right]}\right),\kern0.5em 0<p<1. \)
- (iii)
The Shannon’s entropy for the GC(α, β, θ) distribution is given by
\( {\eta}_X=\alpha \left(1+\beta \right)+ \log \left({\pi}^{-1}\theta \beta \varGamma \left(\alpha \right)\right)+\left(1-\alpha \right)\psi \left(\alpha \right)-2{\displaystyle \sum_{j=1}^{\infty }{V}_j{\left(1+2j\beta \right)}^{-\alpha },} \) where ψ(.) is the digamma function and V _{ j } is defined in Eq. (21).
Proof: It is not difficult to show that the mode of f(x) in (25) is the solution of k(x/θ) = 0, where k(x) is defined above. Therefore, the mode of f(x) is at m = θx where k(x) = 0. To show the unimodality of f(x), consider A(x) = log(π ^{− 1} cot^{− 1}(x)) and B(x) = 2x cot^{− 1}(x). Clearly A(x) is a strictly decreasing function (since it is equal to log(1 − F _{ C }(x))). Furthermore, A(x) < 0 for all x ∈ ℝ. Now, B′(x) = 2[−x/(1 + x ^{2}) + cot^{− 1}(x)]. Therefore, B′(x) > 0 for all x ≤ 0. If x > 0, we have B′(x) < B′(0) = π/2 since B″(x) < 0. Since \( \underset{x\to \infty }{ \lim }{B}^{\prime }(x)=0. \) we get B′(x) > 0 for all x > 0. Therefore, B(x) is strictly increasing for all x ∈ ℝ. Now, let us prove the claim that η(x) = A(x)B(x) is a decreasing function on ℝ.
Proof of the claim: Let 0 ≤ x ≤ y, then 0 ≤ − A(x) ≤ − A(y) and 0 ≤ B(x) ≤ B(y). This implies that η(x) ≥ η(y). Now let x < 0, then η′(x) = − 2x/(x ^{2} + 1) − 2(x ^{2} + 1)^{− 1} x log(π ^{− 1} cot^{− 1}(x)) + 2 cot^{− 1}(x)log(π ^{− 1} cot^{− 1}(x)). Since the middle term in η′(x) is negative, consider \( \psi (x)=\frac{x}{x^2+1}-{ \cot}^{-1}(x) \log \left({ \cot}^{-1}(x)/\pi \right). \) On differentiation, \( {\psi}^{\prime }(x)=\frac{1}{x^2+1}\left\{\frac{2}{x^2+1}+ \log \left({ \cot}^{-1}(x)/\pi \right)\right\}. \) It is easy to show that the term \( \zeta (x)=\frac{2}{x^2+1}+ \log \left({ \cot}^{-1}(x)/\pi \right) \) is strictly increasing on x ≤ 0 with ζ(0) > 0 and ζ(−∞) → 0. Thus, ζ(x) > 0 for all x < 0. This implies that ψ(x) is strictly increasing on x < 0 with ψ(0) > 0 and ψ(−∞) → 0. That is, ψ(x) > 0 for all x < 0. Therefore η′(x) ≤ 0 for all x < 0. Hence, η(x) = A(x)B(x) is a decreasing function in ℝ. This completes the proof of the claim. The fact that η(−∞) → 2 and η(∞) → − ∞ implies that η(x) = 0 has a unique solution. Now, B(x) − 1 + 1/β is only a shift by − 1 + 1/β and therefore remains a strictly increasing function. One can show that the term A(x)[B(x) − 1 + 1/β] remains a decreasing function for all x ∈ ℝ and hence k(x) remains a decreasing function in ℝ with k(−∞) → α + 1 > 0 and k(∞) → − ∞. This ends the proof.□
In the following subsection, we provide some results related to the moments of GC(α, β, θ) distribution.
4.1 Moments of gamma-Cauchy{exponential} distribution
The next proposition establishes the condition for the existence of r-th moment of the GC(α, β, θ) distribution.
Proposition 5: The r-th moment of the GC(α, β, θ) distribution exists if and only if α > r and β ^{− 1} > r.
Let \( \zeta (x)=\frac{x^r}{1+{x}^2}{\left(1/2+{\pi}^{-1}ta{n}^{-1}(x)\right)}^{\alpha -1}{\left(1/2-{\pi}^{-1}ta{n}^{-1}(x)\right)}^{1/\beta -1}. \) As x → ∞, ζ(x) ~ x ^{− (1/β − r + 1)}. So I _{1} exists if and only if 1/β > r. Similarly one can show that I _{2} exists if and only if α > r.□
Next, we consider recursive relation for the r-th moment of the GC(α, β, θ) distribution.
- (i)
\( {\mu}_{2n}^{\prime}\left(\alpha, \beta \right)=\frac{1}{\pi \beta {\left(1-\beta \right)}^{\alpha -1}}{\displaystyle \sum_{j=1}^n\frac{{\left(-1\right)}^{j-1}}{2n-2j+1}}\left\{{\mu}_{2n-2j+1}^{\prime}\left(\alpha, \frac{\beta }{1-\beta}\right)-{\mu}_{2n-2j+1}^{\prime}\left(\alpha -1,\frac{\beta }{1-\beta}\right)\right\}+{\left(-1\right)}^n. \)
- (ii)
\( {\mu}_{2n+1}^{\prime}\left(\alpha, \beta \right)=\frac{1}{\pi \beta {\left(1-\beta \right)}^{\alpha -1}}{\displaystyle \sum_{j=1}^n{\displaystyle \sum_{i=0}^j\frac{{\left(-1\right)}^{n-j}}{2j}}}\left(\begin{array}{c}\hfill n\hfill \\ {}\hfill j\hfill \end{array}\right)\left(\begin{array}{c}\hfill j\hfill \\ {}\hfill i\hfill \end{array}\right)\left\{{\mu}_{2i}^{\prime}\left(\alpha, \frac{\beta }{1-\beta}\right)-{\mu}_{2i}^{\prime}\left(\alpha -1,\kern0.5em \frac{\beta }{1-\beta}\right)\right\}+{\left(-1\right)}^n{\mu}^{\prime}\left(\alpha, \beta \right). \)
Mean, variance, skewness and kurtosis calculations for GC(α, β,1)
α | β | Mean | Variance | Skewness | Kurtosis |
---|---|---|---|---|---|
4 | 0.01 | -10.7296 | 56.4986 | -5.2367 | * |
0.1 | -0.8567 | 0.7824 | -3.8549 | * | |
0.2 | 0.0774 | 0.6096 | 0.2118 | * | |
0.3 | 0.8738 | 2.7446 | 18.0382 | * | |
5 | 0.01 | -8.0673 | 21.2643 | -3.4321 | 44.2181 |
0.1 | -0.5021 | 0.3953 | -1.7868 | 17.6345 | |
0.2 | 0.4480 | 0.6855 | 2.4384 | 56.5459 | |
0.3 | 1.5732 | 6.8542 | 22.1028 | * | |
8 | 0.01 | -4.6280 | 3.5337 | -1.9178 | 5.9542 |
0.1 | 0.1248 | 0.2153 | 0.7560 | 3.2951 | |
0.2 | 1.6343 | 2.7528 | 6.3891 | 395.2955 | |
0.3 | 5.3887 | 124.9174 | 13.2288 | * | |
10 | 0.01 | -3.5986 | 1.6310 | -1.5609 | 15.6352 |
0.1 | 0.4431 | 0.2448 | 0.9704 | 8.1938 | |
0.2 | 2.7883 | 8.3416 | 8.3283 | 802.9949 | |
0.3 | 11.1914 | 840.3657 | 5.1306 | * |
4.2 Estimation of parameters by maximum likelihood method
Therefore, the MLE \( \widehat{\alpha}, \) \( \widehat{\beta} \) and θ are obtained by setting the Eqs. (35), (36) and (37) to zero and solving them simultaneously. Note that the number of equations can be reduced to two by using Eq. (36) to get \( \beta ={\displaystyle \sum_{i=1}^n\frac{ \log \left({z}_i\right)}{n\alpha }}. \) The initial value for the parameter θ can be taken as θ _{0} = 1. From Remark (i) in Gamma-Cauchy{exponential} distribution, the random variable Y _{ i } = − log[0.5 − π ^{− 1} tan^{− 1}(X _{ i }/θ _{0})], i = 1, 2, …, n follows a gamma distribution with parameters α and β. Therefore, by equating the sample mean and sample variance of Y _{ i } with the corresponding population mean and variance, the initial estimates for α and β are, respectively, \( {\alpha}_0={\overline{y}}^2/{s}_y^2 \) and \( {\beta}_0={s}_y^2/\overline{y} \) where \( \overline{y} \) and \( {s}_y^2 \) are the sample mean and sample variance for y _{ i }, i = 1, …, n.
4.3 Simulation study
Estimates and standard deviations for the parameters using MLE method
Sample size | Actual values | ML estimates | Standard deviation | ||||||
---|---|---|---|---|---|---|---|---|---|
n | α | β | \( \theta \) | \( \widehat{\alpha} \) | \( \widehat{\beta} \) | \( \widehat{\theta} \) | \( \widehat{\alpha} \) | \( \widehat{\beta} \) | \( \widehat{\theta} \) |
50 | 1 | 1 | 1 | 1.1605 | 0.9260 | 1.1455 | 0.0518 | 0.0460 | 0.0538 |
0.5 | 1 | 2 | 0.5377 | 0.9552 | 2.1399 | 0.0180 | 0.0426 | 0.0978 | |
0.5 | 3 | 2 | 0.5351 | 2.8849 | 2.1952 | 0.0154 | 0.1207 | 0.1065 | |
0.9 | 0.5 | 1 | 1.0229 | 0.4960 | 1.0579 | 0.0481 | 0.0222 | 0.0442 | |
2 | 0.5 | 1 | 2.9571 | 0.4448 | 1.2255 | 0.4271 | 0.0282 | 0.0917 | |
5 | 0.5 | 1 | 7.9697 | 0.4487 | 0.8862 | 0.9537 | 0.0183 | 0.0737 | |
100 | 1 | 1 | 1 | 1.0717 | 0.9617 | 1.0499 | 0.0216 | 0.0230 | 0.0248 |
0.5 | 1 | 2 | 0.5182 | 1.0090 | 2.0843 | 0.0084 | 0.0229 | 0.0457 | |
0.5 | 3 | 2 | 0.5165 | 2.9594 | 2.0505 | 0.0065 | 0.0591 | 0.0441 | |
0.9 | 0.5 | 1 | 0.9375 | 0.4932 | 1.0309 | 0.0181 | 0.0105 | 0.0203 | |
2 | 0.5 | 1 | 2.2258 | 0.4753 | 1.0866 | 0.0646 | 0.0131 | 0.0281 | |
5 | 0.5 | 1 | 6.3436 | 0.4737 | 0.9243 | 0.4015 | 0.0089 | 0.0367 | |
300 | 1 | 1 | 1 | 1.0244 | 0.9983 | 1.0270 | 0.0094 | 0.0114 | 0.0111 |
0.5 | 1 | 2 | 0.5076 | 0.9908 | 2.0320 | 0.0041 | 0.0102 | 0.0229 | |
0.5 | 3 | 2 | 0.5113 | 2.9636 | 2.0718 | 0.0039 | 0.0315 | 0.0241 | |
0.9 | 0.5 | 1 | 0.9242 | 0.4953 | 1.0159 | 0.0086 | 0.0053 | 0.0102 | |
2 | 0.5 | 1 | 2.1286 | 0.4844 | 1.0369 | 0.0270 | 0.0071 | 0.0123 | |
5 | 0.5 | 1 | 5.2815 | 0.4987 | 0.9562 | 0.1368 | 0.0040 | 0.0149 |
4.4 Applications
10, 33, 44, 56, 59, 72, 74, 77, 92, 93, 96, 100, 100, 102, 105, 107, 107, 108, 108, 108, 109, 112, 121, 122, 122,124,130, 134, 136, 139, 144,146, 153, 159, 160, 163, 163,168, 171, 172, 176,113, 115, 116, 120, 183,195, 196, 197, 202, 213, 215, 216, 222, 230,231, 240, 245, 251, 253, 254, 255, 278, 293, 327, 342, 347, 361,402, 432, 458, 555 |
The data is skewed-to-the right with skewness = 1.3134 and kurtosis = 3.8509.
1120, 1160, 963, 1210, 1160, 1160, 813, 1230, 1370, 1140, 995, 935, 1110, 994, 1020, 960, 1180, 799, 958, 1140, 1100, 1210, 1150, 1250, 1260, 1220, 1030, 1100, 774, 840, 874, 694, 940, 833, 701, 916, 692, 1020, 1050, 969, 831, 726, 456, 824, 702, 1120, 1100, 832, 764, 821, 768, 845, 864, 862, 698, 845, 744, 796, 1040, 759, 781, 865, 845, 944, 984, 897, 822, 1010, 771, 676, 649, 846, 812, 742, 801, 1040, 860, 874, 848, 890, 744, 749, 838, 1050, 918, 986, 797, 923, 975, 815, 1020, 906, 901, 1170, 912, 746, 919, 718, 714, 740 |
The data is approximately symmetric with skewness = 0.3175 and kurtosis = 2.6415.
Parameter estimates for the survival time data
Distribution | Cauchy | Gamma-Pareto | Beta-Cauchy | Gamma-Cauchy{exponential} |
---|---|---|---|---|
Parameter Estimates | ĉ = 139.3079 (9.4281)^{a} \( \widehat{\theta} \) = 48.1262 (7.6793) | \( \widehat{\alpha} \) = 6.030 (0.9770) ĉ = 0.4497 (0.0760) \( \widehat{\theta} \) = 10 | \( \widehat{\alpha} \) = 13.9274 (18.5335) \( \widehat{\beta} \) = 4.5828 (3.6504) ĉ = 117.9055 (37.6269) \( \widehat{\theta} \) = 27.0884 (99.7681) | \( \widehat{\alpha} \) = 16.1591 (2.6666) \( \widehat{\beta} \) = 0.1027 (0.0238) \( \widehat{\theta} \) = 110.1742 (35.4345) |
Log-likelihood | -437.5967 | -465.4670 | -424.4339 | -424.4423 |
AIC | 879.1934 | 934.9340 | 856.8679 | 854.8847 |
K-S | 0.1416 | 0.2606 | 0.0760 | 0.0752 |
K-S p-value | 0.1114 | 0.0001 | 0.8005 | 0.8105 |
Parameter estimates for the annual flow of the Nile River data
Distribution | Cauchy | Gamma-Pareto | Beta-Cauchy | Gamma-Cauchy{exponential} |
---|---|---|---|---|
Parameter Estimates | ĉ = 879.3679 (17.3969)^{a} \( \widehat{\theta} \) = 103.8804 (13.44841) | \( \widehat{\alpha} \) = 5.0437 (0.6902) ĉ = 0.1357 (0.0195) \( \widehat{\theta} \) = 456 | \( \widehat{\alpha} \) = 50.9201 (66.0939) \( \widehat{\beta} \) = 25.1275 (29.7478) ĉ = 712.2062 (445.6252) \( \widehat{\theta} \) = 482.3092 (361.7110) | \( \widehat{\alpha} \) = 322.5715 (6.4901) \( \widehat{\beta} \) = 0.0103 (0.0003) \( \widehat{\theta} \) = 103.5797 (76.1343) |
Log-likelihood | -674.4637 | -696.7975 | -653.4892 | -654.3825 |
AIC | 1352.9270 | 1397.5950 | 1314.9780 | 1314.7650 |
K-S | 0.1311 | 0.1705 | 0.0736 | 0.0637 |
K-S p-value | 0.0642 | 0.0060 | 0.6515 | 0.8120 |
5. Concluding remarks
A family of generalized Cauchy distributions, T-Cauchy{Y} family, is proposed using the T-R{Y} framework. Several properties of the T-Cauchy{Y} family are studied including moments and Shannon’s entropy. Some members of the T-Cauchy{Y} family are presented. A member of the T-Cauchy{Y} family, the gamma-Cauchy{exponential} distribution, is studied in detail. This distribution is interesting as it consists of exponentiated Cauchy distribution and distributions of record values of Cauchy distribution as special cases. Various properties of the gamma-Cauchy{exponential} distribution are studied, including mode, moments and Shannon’s entropy. Unlike the Cauchy distribution, the gamma-Cauchy{exponential} distribution can be right-skewed or left-skewed. Also, the moments of the gamma-Cauchy{exponential} distribution exist under certain restrictions on the parameters. In particular, the r-th moment for the gamma- Cauchy{exponential} distribution exists if and only if α, β ^{− 1} > r and this is not the case for the Cauchy distribution. The flexibility of the gamma-Cauchy{exponential} distribution and the existence of the moments in some cases make this distribution as an alternate to the Cauchy distribution in situations where the Cauchy distribution may not provide an adequate fit.
Declarations
Acknowledgement
The first author gratefully acknowledges the support received from the Social Fund Policy Grant at Nazarbayev University.
Authors’ contributions
The authors, AA, CL, FF and IG with the consultation of each other carried out this work and drafted the manuscript together. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis 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.
Authors’ Affiliations
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