Tsallis–Mittag-Leffler distribution and its applications in gas prices

Journal Pre-proof Tsallis-Mittag-Leffler distribution and its applications in gas prices Hamzeh Agahi, Mohsen Alipour

PII: DOI: Reference:

S0378-4371(19)32049-7 https://doi.org/10.1016/j.physa.2019.123675 PHYSA 123675

To appear in:

Physica A

Received date : 31 May 2019 Revised date : 7 November 2019 Please cite this article as: H. Agahi and M. Alipour, Tsallis-Mittag-Leffler distribution and its applications in gas prices, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123675. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier B.V.

*Highlights (for review)

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• Tsallis-Mittag-Leffler distribution is introduced. • Our results include some well-known q-distributions.

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• A real data set on daily Henry Hub Natural (HHN) gas price is analyzed.

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• An application in time series data based on 100 days moving average is given.

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Journal Pre-proof *Manuscript Click here to view linked References

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Tsallis-Mittag-Leffler distribution and its applications in gas prices a

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Hamzeh Agahia∗, Mohsen Alipoura†

Department of Mathematics, Faculty of Basic Science,

Babol Noshirvani University of Technology, Shariati Ave.,

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Babol, 47148-71167, Iran

Abstract

The theory of Mittag-Leffler process is a fundamental concept in probability and stochastic process. This paper introduces the class of Tsallis-Mittag-Leffler distributions. In special case, our results include a new Tsallis q-Weibull distribution, one-parametric Mittag-Leffler distribution and some other well-known distributions. Finally, to illustrate the applicability of our distribution, a real data set on gas prices in time series data based on 100 days moving average is analyzed.

Introduction

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Keywords: Probability density function; Gas prices; Tsallis-Mittag-Leffler distribution; MittagLeffler distribution; Tsallis q-Weibull distribution.

In 1994, Tsallis [1] defined the deformed logarithm of order q as: lnq (x) =

x1−q − 1 , 1−q

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which converges to ln x when q → 1. The inverse function of lnq x is the deformed exponential: 1

eq (x) = [1 + (1 − q) x] 1−q , q 6= 1

(1.1)

which converges to ex when q → 1. The concept of q-statistics is an efficient tool in statistical mechanics [1, 2, 3, 4, 17]. This concept opens new horizons in statistics as well as statistical distributions [5, 6, 7]. ∗

†

h [email protected] (H. Agahi) Corresponding author. e-mail: [email protected], [email protected] (M. Alipour)

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1.1

History of Mittag-Leffler stochastic process

One-parametric Mittag-Leffler function is defined as [10]

k=0

zk , z ∈ C, Re (α) > 0, Γ (kα + 1)

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Eα [z] =

∞ X

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where Γ is the complete gamma function.

As an increasing L´evy process on the spatial half-line x > 0, Pillai proposed the theory of Mittag-Leffler process in 1990 [9].

Definition 1.1 A random variable X is said to have the one-parametric Mittag-Leffler distribution

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with parameter 0 < α ≤ 1, denoted by X ∼ M L (α), if it’s cumulative distribution function (cdf ) and it’s probability density function (pdf ) are given by

FαM L (x) = 1 − Eα [−xα ] , x ≥ 0, 0 < α 6 1, d fα (x) = − Eα [−xα ] = xα−1 Eα,α [−xα ] , x > 0, 0 < α 6 1, dx

(1.2)

respectively, where Eα,η [·] is the Mittag-Leffler function that is defined as the power series [12, 11, 10] ∞ X

zk , z, η ∈ C, Re (α) > 0. Γ (kα + η)

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Eα,η [z] =

k=0

(1.3)

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The stochastic process {X(t), t > 0} with stationary independent increment with X(0) = 0, X(1) and the Laplace transform

L[X(1)](λ) = (1 + λα )−1 , 0 < α 6 1, λ > 0,

i.e., the Laplace transform of one-parametric Mittag-Leffler random variable, is called the Mittag-

by

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Leffler stochastic process.

Then the Laplace transform of X(t), denoted by L[X(t)](λ), is given

L[X(t)](λ) = (1 + λα )−t , 0 < α 6 1, λ > 0.

(1.4)

The numerical simulations of Mittag-Leffler functions are not very easy [14, 8]. The statistical software R with package “MittagLeffleR” can help us to simulate the Mittag-Leffler functions, see e.g. [15]. Some applications of Mittag-Leffler functions in statistical mechanics can be found in [16, 18]. 2

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1.2

What is our idea?

• How we increase the flexibility of the one-parametric Mittag-Leffler distribution (1.2) by Tsallis

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statistics?

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This paper answers this question and introduces the Tsallis-Mittag-Leffler distribution, including some well-known distributions in Section 2. In Section 3, we present an application of the TsallisMittag-Leffler distribution in gas daily price.

Main results: Tsallis-Mittag-Leffler distributions

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2

Now, we propose the class of Tsallis-Mittag-Leffler distributions, which is the main result in this paper.

Definition 2.1 A random variable X is said to have the Tsallis-Mittag-Leffler distribution with

(2.1)

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parameters α, q, γ, β, denoted by X ∼ T M L (α, q, γ, β), if its pdf is given by   βγxγ−1 1 γ Eα,α ln (1 − βx (1 − q)) , fT M L (x; α, q, γ, β) = βαxγ (q − 1) + α 1−q x > 0, q > 1, β > 0, γ > 0, 0 < α 6 1,

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where Eα,α [·] is defined in (1.3).

Figures 1-4 show the T M L (α, q, γ, β) distribution with different parameters. The following properties of the Tsallis-Mittag-Leffler distribution (2.1) are interesting:

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Property 2.2 If q → 1, then

lim fT M L (x; α, q, γ, β) =

q→1

βγ γ−1 x Eα,α [−βxγ ] , α

which is a new generalization of Mittag-Leffler distribution (1.2), denoted by X ∼ M L (α, β, γ). Property 2.3 If α = 1, then βγxγ−1 eq (−βxγ ) fT M L (x; α = 1, q, γ, β) = βxγ (q − 1) + 1 3

(2.2)

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which is a new version of Tsallis q-Weibull distribution. In this case, we denote X ∼ N T W (q, β, γ) . In particular, if γ = α = 1, then βeq (−βx) , βx (q − 1) + 1

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fT M L (x; α = 1, q, γ = 1, β) =

(2.3)

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which is a new version of Tsallis q-exponential distribution. In this case, we denote X ∼ N T E (q, β) . Property 2.4 If γ = α, q → 1, then

lim fT M L (x; α, q, γ = α, β) = βxα−1 Eα,α [−βxα ] ,

q→1

Property 2.5 If α = 1, q → 1, then

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which is the pdf of Mittag-Leffler distribution (1.2), denoted by X ∼ M L (α, β).

γ

lim fT M L (x; α = 1, q, γ, β) = βγxγ−1 e−βx ,

q→1

which is the pdf of the classical Weibull distribution.

f f f f

1.0

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1.5

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f x ,0.5 ,q ,Γ ,Β

2.0

x ,0.5 ,3,1,0.5 x ,0.5 ,2 ,5 ,0.5 x ,0.5 ,1.5 ,1,3 x ,0.5 ,1.5 ,3,0.5

0.5

0.0

0.5

1.0

1.5

x

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0.0

Figure 1: Plot of T M L (α, q, γ, β) for α = 0.5 with different values of q, γ, β. Theorem 2.6 If X ∼ T M L (α, q, γ, β), then its cumulative distribution function is given by Z t T ML FX (t) = fTML (x; α, q, γ, β) dx 0   1 γ = 1 − Eα ln (1 + βt (q − 1)) , 1−q where q > 1, β > 0, γ > 0 and 0 < α 6 1.

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0.8 Β=2 Γ=2 Α = 0.5

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f TML

0.6

q=2

0.4

q=3

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q=4

0.2

0.0 0.0

0.5

1.0

1.5

2.0

q=5

2.5

3.0

x

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Figure 2: Plot of T M L (α, q, γ, β) for α = 0.5, β = γ = 2 and several values of q > 1. 1.4 1.2 1.0

f TML

0.8 0.6

Β=2 Α = 0.8 q =2 Γ=2 Γ=3 Γ=4

0.4

0.0

0.5

1.0

1.5

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0.0

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Γ=5 0.2

2.0

2.5

3.0

x

Figure 3: Plot of T M L (α, q, γ, β) for α = 0.8, β = q = 2 with some values of γ > 0.

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1 Proof. By changing variable ω α = q−1 ln (1 + βxγ (q − 1)), we can write   Z t xγ−1 βγ 1 T ML γ FX (t) = Eα,α ln (1 − βx (1 − q)) dx γ 1−q 0 βαx (q − 1) + α 1 Z 1 α γ 1 ( q−1 ln(1+βt (q−1))) = αω α−1 Eα,α [−ω α ] dω α 0   1 γ = 1 − Eα ln (1 + βt (q − 1)) .2 1−q

Corollary 2.7 If X ∼ N T W (q, β, γ), then its cumulative distribution function is given by Z t 1 NT W FX (t) = fNTW (x; q, β, γ) dx = 1 − (1 + βtγ (q − 1)) 1−q , 0

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1.2

1.0

Α = 0.6 Γ=2 q =2

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f TML

0.8

Β=1

0.6

Β=2 Β=3

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0.4

0.2

0.0 0.0

0.5

1.0

1.5

2.0

2.5

Β=4

3.0

x

where q > 1, β > 0 and γ > 0.

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Figure 4: Plot of T M L (α, q, γ, β) for α = 0.6, q = γ = 2 and several values of β > 0.

Corollary 2.8 If X ∼ N T E (q, β), then its cumulative distribution function is given by Z t 1 NT E FX (t) = fNTE (x; q, β) dx = 1 − (1 + βt (q − 1)) 1−q , 0

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where q > 1 and β > 0 . Theorem 2.9 If X ∼ N T W (q, β, γ) and n ∈ N, then ∞

xn f N T W

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E (X n ) =

Z

0

where 1 < q <

γ n

+ 1, β > 0 and γ > 0.



   n Γ − Γ γ +1  , (x; q, β, γ) dx = n n 1 β γ (q − 1) γ +1 Γ q−1 +1 1 q−1

n γ

(2.4)

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Proof. By changing variable y γ = (q − 1) βxγ , we have Z ∞ n E (X ) = xn fN T W (x; q, β, γ) dx 0     1 Γ q−1 − nγ Γ nγ + 1   .2 = n n 1 β γ (q − 1) γ +1 Γ q−1 +1 Corollary 2.10 If X ∼ N T E (q, β) and n ∈ N, then E (X n ) =

Z

0

∞

 − n n!  , xn fN T E (x; q, β) dx = 1 β n (q − 1)n Γ q−1 Γ

6



1 q−1

(2.5)

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where 1 < q <

1 n

+ 1 and β > 0.

where 1 < q <

γ n

+ 1, n ∈ N, β > 0 and γ > 0.

Proof. From Theorem 2.9, we have tx

(t) = E e =

Z

0

=

=

Z

∞

etx fN T W (x; q, β, γ) dx

0 +∞ n n ∞X n=0

t x fN T W (x; q, β, γ) dx n!

+∞ n Z X t n=0

=




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MXN T W

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Theorem 2.11 The moment generating function of X ∼ N T W (q, β, γ) is given by     1 +∞ tn Γ q−1 − nγ Γ nγ + 1 X   , MXN T W (t) = n n +1 1 γ (q − 1) γ Γ + 1 n! β n=0 q−1

n!

0

xn fN T W (x; q, β, γ) dx

   n Γ + 1 γ t   .2 n n n! β γ (q − 1) γ +1 Γ 1 + 1 q−1

+∞ n X

Γ



1 q−1

−

n γ

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n=0

∞

Figures 5 and 6 show the N T W (q, β, γ) distribution with some values of q, β and γ.

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1.0

0.8

Β=1 Γ=1 q=2

f NTW

0.6

q=3 q=4

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0.4

q=5

0.2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

x

Figure 5: Plot of N T W (q, β, γ) for β = γ = 1 and several values of q > 1.

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1.0

Β=1 q =2

0.8

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Γ=1

f NTW

0.6

Γ=2 Γ=3

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Γ=4

0.2

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

x

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Figure 6: Plot of N T W (q, β, γ) for β = 1, q = 2 and several values of γ > 0.

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Applications to gas daily prices

In this section, we present an application of our proposed results in gas daily price. We first extract the daily Henry Hub Natural (HHN) gas price per Million British Thermal Units (MBTU) in period of 1997/06/30 to 2017/06/301 [13]. Then we work on the data of the absolute difference of daily

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prices from its last 100 days moving average (see Table 1 and Figure 1).

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Table 1: Summary statistics N Min. Median Mean Max. sd skewness kurtosis 4874 0.0003 0.45915 0.7021214 13.4588 0.8015658 3.099821 23.26366 Throughout this section, we also used the statistical software R version 3.4.1 for estimating the parameters. We apply the Akaike (AIC) Bayesian (BIC) information criteria for comparing the

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various models that are defined as

b AIC := 2k − 2 ln L,

b + k ln n, BIC := −2 ln L

b and n are the number of estimated parameters, the maximum value of the likelihood where k, L

function and the number of data, respectively. Note that the maximum likelihood estimates (MLEs) 1

It is available at https://fred.stlouisfed.org.

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Histogram of W

Figure 7: The histogram of data

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0.4 0.2

Density

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0.8

0.787

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0.147

0.0

0.042

0

1

2

0.017

3

0.005

4

0.002

5

6

W

of the parameters α, q, γ, β based on an iid sample x1 , . . . , xn can be obtained from Xn h

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(γ − 1) ln(xi ) + ln(β) + ln(γ) − ln(α) − ln (βxγi (q − 1) + 1) i 1 + ln Eα,α [− ln (1 + βxγi (q − 1))] , q−1 i=1

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− ln L (α, β, γ, q) = −

numerically. We use the packages “nlminb”, “MittagLeffleR” and “fitdistrplus” from software R version 3.4.1.

Table 2 shows that our proposed distribution is better than other nominated distributions.

Concluding remarks

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We have introduced the Tsallis-Mittag-Leffler distribution. Then some properties of this distribution have been discussed. We have also presented an application of the this distribution on gas prices in time series data based on 100 days moving average.

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Table 2: Fitting models b − log L

Distribution

q

γ

β

α

M L (α, β)

−

−

1.4699812

0.9840884

1.202573

1.080089

1.759101

1.000

3107.649

6223.3

6249.26

−

−

1.42426

−

3150.315

6302.63

6309.12

1.10095

−

1.78509

−

3115.633

6235.27

6248.25

−

0.8518199

1.6889330

−

3127.439

6258.88

6271.86

−

0.952473

0.686087

−

3139.98

6283.97

6296.95

E (β) Exponential T QE (β, q)

GE (γ, β) G-Exponential W E (γ, β) Weibull

Acknowledgment

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TsallisQExponential

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Proposed

BIC

6268.39

6281.37

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T M L (α, β, γ, q)

3132.193

AIC

The author’s are very grateful to the anonymous reviewers for their comments. This work was supported by Babol Noshirvani University of Technology with Grant Program No. BNUT/392100/98

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and BNUT/388063/98.

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References

[1] C. Tsallis, What are the numbers that experiments provide? Quimica Nova. 17 (1994) 468. [2] C. Tsallis, Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years. Braz. J. Phys. (2009) 39 337-356.

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[3] W. Thistleton, J.A. Marsh, K. Nelson, C. Tsallis, Generalized Box–Muller method for generating q-Gaussian random deviates, IEEE Transactions on Information Theory 53 (2007) 4805. [4] L. Borland, The pricing of stock options, in Nonextensive Entropy – Interdisciplinary Applications, eds. M. Gell-Mann and C. Tsallis Oxford University Press, New York, 2004.

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[5] J. Juniper, The Tsallis Distribution and Generalised Entropy: Prospects for Future Research into Decision-Making under Uncertainty, Centre of Full Employment and Equity, The Univer-

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sity of Newcastle, Australia 2007.

[6] F. Zhang, H. K. T. Ng, Yimin Shi On alternative q-Weibull and q-extreme value distributions:

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Properties and applications, Physica A 490 (2018) 1171-1190.

[7] K.K. Jose, S.R. Naik, On the q-Weibull distribution and its applications, Comm. Statist. Theory Methods 38 (2009) 912–926.

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[8] D. Val´erio, J. A. Tenreiro Machado, On the numerical computation of the Mittag-Leffler function, Communications in Nonlinear Science and Numerical Simulation 19(10) (2014) 3419-3424. [9] R.N. Pillai, On Mittag-Leffler functions and related distributions. Ann. Inst. Statist. Math. 42(1) (1990) 157-161.

[10] M.G. Mittag-Leffler, Une generalization de l’integrale de Laplace-Abel. Comp. Rend. Acad.

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Sci. Paris 136 (1903) 537-539. [11] R.P. Agarwal, A propos d’une note de M. Pierre Humbert. C. R. Acad. Sci. Paris 236 (1953)

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[12] R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler functions, related topics and applications, Springer Heidelberg New York Dordrecht London (2014).

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[13] H. Mehri-Dehnavi, H. Agahi, R. Mesiar, Pseudo-exponential distribution and its statistical applications in econophysics, Soft Comput. 23(1) (2019) 357-363. [14] R. Garrappa, Numerical evaluation of two and three parameter Mittag-Leffler functions, SIAM J. Numer. Anal. 53-3 (2015) 1350-1369. [15] G. Gill, P. Straka, MittagLeffleR: Using the Mittag-Leffler distributions in R (2018). doi: 10.6084/m9.figshare.6235898.v1, https://strakaps.github.io/MittagLeffleR/.

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[16] T.A. Sulaiman, M. Yavuz, H. Bulut, H. M. Baskonus, Investigation of the fractional coupled viscous Burgers equation involving Mittag-Leffler kernel, Physica A 527 (2019) 121126.

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[17] F. Zhang, H. K. T. Ng, Y. Shi, Information geometry on the curved q-exponential family with

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application to survival data analysis, Physica A 512 (2018) 788-802.

[18] M. A. F. dos Santos, Mittag-Leffler functions in superstatistics, Chaos, Solitons & Fractals, in

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press, (2019), doi.org/10.1016/j.chaos.2019.109484.

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Journal Pre-proof *Declaration of Interest Statement

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Compliance with ethical standards Conflict of interest The authors declare that they have no conflict of interest.

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Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors.

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