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The Generalized Odd Nakagami-G Family of Distributions: Properties and Applications

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The Generalized Odd Nakagami-G Family of Distributions: Properties and Applications NATURENGS, MTU Journal of Engineering and Natural Sciences 1:2 (2020) 1-16 NATURENGS MTU Journal of Engineering and Natural Sciences https://dergipark.org.tr/tr/pub/naturengs DOI: 10.46572/nat.2020.12 The Generalized Odd Nakagami-G Family of Distributions: Properties and Applications Ibrahim ABDULLAHI1*, Job OBALOWU2 1Department of Mathematics and Statistics, Faculty of Science, Yobe State University, 620, Yobe, Nigeria. 2Department of Statistics, Faculty of Physical Science, University of Ilorin, 240003, Kwara, Nigeria. (Received: 15.08.2020; Accepted: 01.10.2020) ABSTRACT: In this study, the Generalized Odd Nakagami-G distribution has been empirically investigated using Frechet distribution as the baseline distribution. Some mathematical statistics properties viz: quantile function, moments, probability-weighted moment, entropies, and order statistics were derived among others. The method of maximum likelihood estimation is suitable for the derivation of estimators of the distribution parameters. And the Generalized odd Nakagami Frechet (GONak-Fr) distribution gives the best fit vis-a-vis its competitors via application to real life data sets. Keywords: GONak-G, GONak-Fr, Probability-weighted moment, Entropies, Order statistics. 1. INTRODUCTION There have been recent developments focus on generalized odd classes of continuous distributions by adding at least one shape parameters to the baseline distribution, has led mathematical statistician researcher to make of the new flexible distributions and studying the properties of these distributions and using these distributions to model data in many applied areas which include engineering, biological studies, environmental sciences and economics. Numerous methods for generating new families of distributions have been proposed [1] by many researchers. The beta-generalized family of distribution was developed, Kumaraswamy generated a family of distributions [2], Beta-Nakagami distribution [3], Weibull generalized family of distributions [4], Additive Weibull generated distributions [5], Kummer beta generalized family of distributions [6], the Exponentiated-G family [7], the Gamma-G (type I) [8], the Gamma-G family (type II) [9], the McDonald-G [10], the Log-Gamma-G [11], A new beta generated Kumaraswamy Marshall-Olkin-G family of distributions with applications [12], Beta Marshall- Olkin-G family [13] and Logistic-G family [14]. In this paper, the so-called generalized odd Nakagami generator and the corresponding family of distributions are called generalized odd Nakagami (GONak-G) family of distributions. To the best of the researcher's knowledge, it is new in the literature. *Corresponding Author: [email protected] ORCID number of authors: 1 0000-0002-7280-3035, 2 0000-0001-5232-1509 Abdullahi and Obalowu, NATURENGS, MTU Journal of Engineering and Natural Sciences 1:2 (2020) 1-16 2. CONSTRUCTIONS OF THE GENERALIZED ODD NAKAGAMI DISTRIBUTION The Generalized Odd Nakagami-G distribution (GONak-G) would be made τ Gx( ;η ) λt2 τ λ − − η 2λ F( x;,λβτη ,, ) = 1;Gx( ) t21λ− eβ dt (1) ∫0 Γ(λβ) λ 2 τ λ Gx( ;η ) Fx( .,λβη ,) = γ λ , (2) * τ β 1;− Gx( η ) ( ) where 22 ττ λλGx( ;;ηη) 1 Gx( ) γλ, =,λ * ττ β(1;−−Gx( ηη) ) Γ(λβ) (1;Gx( ) ) where λβτ,,> 0 are three additional parameters, η is the parameter for baseline G(x) and γσ( , x) =∫ veσ −−1 v ∂ v is the incomplete gamma function. By differentiation, the probability density function (pdf) of the GONak-G distribution will be obtained as follows: 2 τ λ Gx( ;η ) λ 21λτ − − β τ 2λτg( xGx) ( ) 1;−Gx( η ) fx( ;,λβτη ,, ) = e (3) τ 21λ+ Γ−(λβ) λ (1 Gx( ) ) The survival function hazard rate function (hrf) is given by 2 τ λ Gx( ;η ) (4) Sx( ;,λβτη ,,) = 1 − γ λ , * τ β (1;− Gx( η ) ) 2 τ λ Gx( ;η ) 21λτ − − λ τ β 2λτg( xGx) ( ) 1;−Gx( η ) 21λ+ e λ τ Γ−(λβ) (1 Gx( ) ) (5) λβτη = hx( ;, ,, ) 2 τ λ Gx( ;η ) 1,−γλ * τ β (1;− Gx( η ) ) 3. THE GENERALIZED ODD NAKAGAMI- FRECHET (GONak-Fr) In this section, we study generalized odd Nakagami Frechet (GONakFr) distributions. Consider the Frechet distribution with cdf and pdf given by: 2 Abdullahi and Obalowu, NATURENGS, MTU Journal of Engineering and Natural Sciences 1:2 (2020) 1-16 ρ v − (6) Gx( ;,ρν) = ex and v ν ρ (7) ρν − gx( ;,ρν) = ex xν +1 Then the pdf of the GONak-Fr distribution is given by − v 2 v ρ ρ λ τ −2λτ −−x λν e 1 x β 2λ τρ ν e fx( ;,λβτη ,, ) = 21λ+ e (8) ρ v −τ xeνλ+1Γ−(λβ) 1 x 3.1 Investigation of the Proposed GONak-Fr Distribution To show that the proposed GONak-Fr distribution is a pdf, we proceed as follows: − v 2 v ρ ρ τ λ x −2λτ −−e 1 ∞ λν x β 2λ τρ ν e ex∂ (9) v 21λ+ ∫ ρ 0 −τ xeνλ+1Γ−(λβ) 1 x −2 ρ v λ τ Let ue=x −1 β 3 ρ v τ β exx −1 ∂=xuv ∂ v ρ ρ τ 2λτν e x x Eq. (9) becomes λ−1 λβλ−1 ∞ u eu−u∂ λ−1 ∫ (10) Γ(λβ) 0 λ then integral in Eq. (10) becomes ∞ 11λ−−1 u ∫ue∂= u Γ(λ) =1 ΓΓ(λλ) 0 ( ) Hence Generalized Odd Nakagami-Frechet Distribution is pdf. 3 Abdullahi and Obalowu, NATURENGS, MTU Journal of Engineering and Natural Sciences 1:2 (2020) 1-16 Various forms of shapes are observed, showing the great flexibility of GONak-Fr distribution. Figure 1. GONak-Fr density function Figure 2. GONak-Fr distribution function Figure 3. GONak-Fr survival function 4 Abdullahi and Obalowu, NATURENGS, MTU Journal of Engineering and Natural Sciences 1:2 (2020) 1-16 3.2 Shape of the Crucial Functions The shapes of the pdf of the GONak-G family can be defined analytically. The critical points of the GONak-G are given by Eq. (3) are the roots of the resulting equation: 21τ − gx( ;η)′ gx( ;η) τηgx( ;;2;;) Gx( ηλτηη) gx( ) Gx( ) +(2λτ − 1) +−( 21λ ) − =0 gx( ;;ηη) Gx( ) τ τ 3 (11) (1;− Gx( η ) ) βη(1;− Gx( ) ) 3.3 Linear Representation In this subsection, we provide a very useful linear representation for the GONak-G density function. We obtain an expansion for density function defined in Eq. (3) as follows: using Taylor expansion the GONak-G pdf becomes 21λτ − i i τ 2i 21λτλ gxGx( ) ( ) ∞ (− ) λ Gx( ) λβτη = fx( ;, ,, ) 21λτ+ ∑ (12) λ τ β Γ−(λβ) (1 Gx( ) ) i=0 i! 1− Gx( ) Using generalized binomial series we obtain −λ + + ∞ ττ21( i) 2(λ ++ij) j (1−=Gx( ) ) ∑Gx( ) (13) j=0 j therefore, inserting Eq. (13) in Eq. (12) the GONak-G pd f will take the following form λ+i ∞ i 2τλ(−1) 2(λ ++ij) 21τλτ(ij++−) f( x;,λβτη ,, ) = ∑ g( xGx) ( ) (14) Γ(λβ) ij,0= i! j this can be written as ∞ 2τ 21τλτ(ij++−) f( x;,λβτη ,, ) = ∑ πij, g( xGx) ( ) (15) Γ(λ) ij,0= i λ+i 2τλ(−1) 2(λ ++ij) π ij, = (16) Γ(λβ) i! j then, 5 Abdullahi and Obalowu, NATURENGS, MTU Journal of Engineering and Natural Sciences 1:2 (2020) 1-16 ∞ λβτη= ω fx( ;, ,, ) ∑ ij, 2τλτ(ij++) ( x) (17) ij,0= where π ω = ij, , ij, 2τλτ(ij++) 21τλτ(ij++−) =τλτ ++ η η 2τλτ(ij++) ( x) 2( i) j gx( ;;) Gx( ) Eq. (17) is the infinite linear representation of the GONak-G pdf in terms of the exponentiated generated distribution. Similarly, the cdf of the GONak-G family can also be expressed as a linear combination of exponentiated generated cdf given by ∞ λβτη= π Fx( ;, ,, ) ∑ ij, L2τλτ(ij++) ( x) (18) ij,0= where, 2τλτ(ij++) L2τλτ(ij++) ( x) = Gx( ;η ) is the cdf of the exponentiated generated family with the power parameter 2.τλτ(ij++) 4. MATHEMATICAL AND STATISTICAL PROPERTIES In this section, we provide some mathematical properties of the GONak-G distribution. 4.1. Moments th The r moment of GONak-G family can be obtained from pdf in Eq. (17) as follows ∞ ∞ r µω′ = ∂ r ij, ∑ ∫ x2τλτ(ij++) ( xx) (19) ij,0= 0 that is, ′ µωr= ij, Irri,2τλτ( ++) j , =1, 2, 3, (20) where ∞ ∞ r = ∂ Irij,2τλτ( ++) ∑ ∫ x2τλτ(ij++) ( xx) ij,0= 0 6 Abdullahi and Obalowu, NATURENGS, MTU Journal of Engineering and Natural Sciences 1:2 (2020) 1-16 Finally, the mean and variance of the GONak-G family is given by EX( ) = ωij, I1,2τλτ(ij++) , (21) and 2 Var X= ωω I − I (22) ( ) ij,,2,2τλτ(ij++) ij 1,2τλτ(ij++) , 4.2. Moment Generating Function The moment generating function of GONak-G family can be obtained from Eq. (20) as follows: ∞∞r rω t µ′ tIij, ri,2τλτ( ++) j =r = (23) Mt( X ) ( ) ∑∑ rr=00rr!!= 4.3. Probability Weighted Moments [15] stated that for a random variable X, the probability weighted moments (pwm) is given by: ∞ rr ξ = EXFxss= xFx fx∂ x sr, ( ) ∫ ( ) ( ) (24) −∞ The PWM of GONak-G is obtained by substituting (2) and (17) into (24) 2 r τ ∞ η ∞ s λ Gx( ; ) ξ= x γλ, ω ( xx)∂ (25) sr,,∫ * β τ ∑ i j 2τλτ(ij++) 0 1;− Gx( η ) ij,0= ( ) using an expansion of incomplete gamma ratio function leads to λ + 2 rk τ λ Gx( ;η ) 2 r τ τ β − η (1;Gx( ) ) ∞ (26) λ Gx( ;η ) γλ, = c * β τ r ∑ rk, (1;− Gx( η ) ) Γ(λ) k =0 using generalized binomial one can obtain 7 Abdullahi and Obalowu, NATURENGS, MTU Journal of Engineering and Natural Sciences 1:2 (2020) 1-16 −+λ ∞ ττ2( rk) 21(λrk+) +− n n (1;−=Gx( ηη) ) ∑Gx( ;) (27) n=0 n substituting Eq. (27) into Eq. (26) leads to λrk+ ∞ λ 21(λrk+) +− n τλ2( rk +) + n = c Gx( ;η ) r λrk+ ∑ rk, (28) Γ(λβ) kn,0= n using generalized binomial one can obtain ∞ l τλ2( rk +) + n lq+ τλ21( rk++−) n l q ηη= − Gx( ;1) ∑∑( ) Gx( ;) (29) lq=00 = l q substituting Eq.
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