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The Beta-Burr Type V Distribution: Its Properties and Application to Real Life Data

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The Beta-Burr Type V Distribution: Its Properties and Application to Real Life Data International Journal of Advanced Statistics and Probability, 5 (2) (2017) 83-86 International Journal of Advanced Statistics and Probability Website: www.sciencepubco.com/index.php/IJASP doi: 10.14419/ijasp.v5i2.7862 Research paper The beta-burr type v distribution: its properties and application to real life data Husaini Garba Dikko *, Aliyu Yakubu, Alfa Aliyu Saidu Department of Statistics, Ahmadu Bello University Zaria-Nigeria *Corresponding author E-mail: [email protected] Abstract A new distribution called the beta-Burr type V distribution that extends the Burr type V distribution was defined, investigated and estab- lished. The properties examined provide a comprehensive mathematical treatment of the distribution. Additionally, various structural properties of the new distribution verified include probability density function verification, asymptotic behavior, Hazard Rate Function and the cumulative distribution. Subsequently, we used the maximum likelihood estimation procedure to estimate the parameters of the new distribution. Application of real data set indicates that this new distribution would serve as a good alternative distribution function to model real- life data in many areas. Keywords: Burr Type V; Beta Burr V; Hazard Rate; Maximum Likelihood Estimation. tan x 2 tan x 1 1. Introduction e sec x 1 e x 22 fx Burr's type V is one of the twelve distributions introduced in 1942 0 elsewhere by [5], which can be used to fit basically any given set of unimod- al data [7]. So many researchers have established beta –G distribu- tion; these include among others: Nadarajah and Gupta [20] de- (1) fined the beta-Frechet distribution; Famoye, et al. [5] defined the beta-Weibull; Nadaraja and Kotz [19] defined the beta- The cumulative distribution function of Burr type V distribution is: exponential distribution; Kong et al. [15] proposed the beta- gamma distribution. Fischer and Vaughan [12] introduced the tan x beta-hyperbolic secant distribution; beta-Gumbel (BGU) distribu- F x 1 (2) tion was introduced by Nadarajah and Kotz [18]; the beta-Pareto distribution was defined and studied by Akinsete et al. [4]; the Where α and β are the location parameters of the distribution. beta-Rayleigh distribution was proposed by Akinsete and Lowe The rest of the paper is organized as follows: in section two, we [3]; beta-Burr XII by Paranaiba et al. [21]; and very recently, briefly introduce the beta-burr type V (BBV) distribution, its Merovci et al. [17] developed the beta-Burr type X distribution properties is studied in section 3 while estimates of the parameters leaving the rest types of Burr distributions with little or no interest of the distribution using the maximum likelihood estimation tech- from researchers. This is the case with Burr type V distribution. It nique is obtained in section 4, in section 5 we look at the compari- is clear therefore, that this distribution is yet to catch the eyes of son of our new distribution with other known distributions and we researchers. Burr Type V distribution can be used to model real finally conclude in section 6. lifetime data. Ever since the introduction of the Burr distribution, it has some- what been neglected as an option in the analysis of lifetime data. 2. Beta- burr type V (BBV) Even though there are researchers that indicate that this distribu- tion possesses sufficient flexibility to make it a possible model for Beta distribution takes on several problems in reliability analysis various types of data; for example, it was used to examine the since it has been widely acclaim to be very powerful and applica- strengths of 1.5 cm glass fibers [17], Paranaiba et al. [21] studied ble probability distribution. In recent years, development focuses cancer recurrence by using the real data set. on new techniques for building meaningful distributions, which Burr's type V distribution extension with two parameters is include the use of the logit of beta (the link function of the beta through the beta-G generator pioneered by Eugene et al. [8]. We generalized distribution which was introduced by Jones [14] and investigate and discern the properties of the proposed distribution elsewhere [10]). The logit of beta introduced by Jones has proba- from G(x) which was used by Eugene et al [8]. It is known as the bility density function (pdf). beta generalized class of distribution, and it has two shape pa- rameters in the generator, and it is given in equation 3. f(x) g(x) = [F(x)]a−1[1 − F(x)]b−1 (3) The probability density function of Burr type V distribution is: B(a,b) Copyright © 2017HusainiGarbaDikko et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 84 International Journal of Advanced Statistics and Probability Where 3.0 a=0.6, b=0.9, alpha =0.5, beta=0.2 F(x)is the cumulative distribution function of the baseline distri- a=3.0, b=2.5, alpha =0.5, beta=0.8 a=3.0, b=2.5, alpha =1.5, beta=0.8 bution and 2.5 a=3.0, b=2.5, alpha =1.5, beta=2.0 a=0.5, b=0.5, alpha =1.5, beta=2.0 f(x)is the probability density function of the baseline distribution a=1.5, b=1.5, alpha =1.5, beta=1.5 Many researchers have used this technique to come up with many 2.0 compound distributions which include: [1],[3],[4],[6],[8],[9],[12],[15],[16],[17],[18],[19],[20],[21],[23] 1.5 and many others. The aim of this paper is to propose a new model BBTV of Pdf called beta- Burr type V distribution. Using the logit defined in (3), 1.0 the pdf of the proposed distribution is given by equation (4): 0.5 − tan x 2 αβe sec x 0.0 f(x) = [1 + βe− tan x]−αa−1[1 − (1 + β(a,b) -50 0 50 βe− tan x)−α]b−1 (4) X Fig 1: Graph of Beta-Burr Type V Distribution for Various Parameter 3. Properties Values. 3.2. Asymptotic behaviour We will examine the statistical properties of the distribution we have put forth in the previous section. It will be scrutinized by In order to investigate the asymptotic behavior of the proposed verifying the asymptotic behaviour, true probability function, π model (Beta –Burr type V), we find limit as x → and limit as cumulative density and hazard rate function. 2 π x → − of the BBV distribution. 2 3.1. Investigation of proper PDF αβe− tan xsec2x ( ) ( − tan x)−αa−1 [ To verify that the proposed density function is a proper pdf, we limπ f x = limπ 1 + βe 1 x→− x→ − β (a, b) integrate (4) with respect to x and see whether the integral is equal 2 2 − tan x −α b−1 to unity or otherwise. If it is equal to unity, the density function is − (1 + βe ) ] a proper pdf otherwise it is not. This is equal to zero, since π π 2 2 αβ − tan x 2 − tan x −αa−1 ∫ π f(x)dx = ∫ π e sec x[1 + βe ] [1 − − 푡푎푛 푥 2 − 푡푎푛 푥 −훼−1 − − β(a,b) 푙푚휋 훼훽푒 푠푒푐 푥(1 + 훽푒 ) = 0 2 2 푥→ − − tan x −α b−1 2 (1 + βe ) ] dx (5) Similarly, 푙푚휋 푓(푥) = 0 To integrate (5), 푥→ 2 − tan x 2 dp 2 Letp = βe , then dp = −psec x dxand dx = 2 Since 푙푚휋 푠푒푐 푥 = 0 −psec x 푥→ 2 Hence, 3.3. Cumulative density function (CDF) of BBV distri- π ∞ bution 2 α ∫ f(x)dx = ∫ psec2x [1 + p]−αa−1 [1 π − 0 β(a, b) In this section, the cumulative density function of the BBV distri- 2 dp bution will be obtained − (1 + p)−α]b−1 psec2 x 푥 훼훽 − 푡푎푛 푦 2 −푡푎푛푦 –훼푎−1 퐹(푥) = ∫ 휋 푒 푠푒푐 푦[1 + 훽푒 ] [1 − − 훽(푎,푏) ∞ 2 α −푡푎푛푦 −훼 푏−1 = ∫ (1 + p)−αa−1 [1 − (1 + p)−α]b−1 dp (1 + 훽푒 ) ] 푑푦 (6) 0 β(a, b) To integrate (6), 푙푒푡 푝 = 훽푒푡푎푛 푦, then 1 −α − let R = (1 + p) ThenR α = 1 + p 2 푑푝 푑푝 = −푝푠푒푐 푦푑푦And푑푦 = −푝푠푒푐2푦 This implies Therefore, −α−1 dR dR = −α(1 + p) dpAnddp = 1 1+ −αR α ∞ 훼 퐹(푥) = ∫ 푝푠푒푐2푦[1 + 푝]−훼푎−1[1 Therefore, 훽푒− 푡푎푛 푥 훽(푎, 푏) −훼 푏−1 푑푝 π − (1 + 푝) ] 1 2 2 1 β(a, b) 푝푠푒푐 푦 ∫ f(x)dx = ∫ Ra−1(1 − R)b−1 dR = = 1 π − β(a, b) 0 β(a, b) ∞ 2 훼 −훼푎−1 −훼 푏−1 = ∫ [1 + 푝] [1 − (1 + 푝) ] 푑푝 − 푡푎푛 푥 훽(푎, 푏) Hence, f(x) is a proper pdf? 훽푒 The graph of the BBV distribution for various parameter values (a, Let푅 = (1 + 푝)−훼then, b, α and β) is given in fig. 1 훼+1 푑푅 푑푅 = −훼푅 훼 푑푝 And 푑푝 = 훼+1 −훼푅 훼 Hence, International Journal of Advanced Statistics and Probability 85 −훼 ∑푛 (1+훽푒− 푡푎푛 푥) 훼푛훽푛푒− 푖=1 푡푎푛 푥푖 훼 훼푎+1 푑푅 푛 2 −푡푎푛푥푖 −훼푎−1 푏−1 = 푛 ∏ 푠푒푐 푥 (1 + 훽푒 ) [1 − 퐹(푥) = ∫ 푅 훼 [1 − 푅] ( ) 푖=1 푖 훽(푎, 푏) 훼+1 (훽 푎,푏 ) 0 훼 − 푡푎푛 푥푖 −훼 푏−1 훼푅 (1 + 훽푒 ) ] (11) − 푡푎푛 푥 −훼 (1+훽푒 ) 훼푎+1−훼−1 훼 푏−1 푑푅 Taking log of (11), we have the log-likelihood given by: 퐹(푥) = ∫ 푅 훼 (1 − 푅) 0 훽(푎, 푏) 훼 푛 푙(푥; 푎, 푏, 훼, 훽) = 푛 푙표푔 훼 + 푛 푙표푔 훽 − ∑푖=1 푡푎푛푥푖 − 푛 푙표푔(⌈푎) − − 푡푎푛 푥 −훼 푛 2 1 (1+훽푒 ) 푛 푙표푔(⌈푏) + 푛 푙표푔(⌈푎 + 푏) + ∑푖=1 푙표푔(푠푒푐 푥푖) − (훼푎 + 퐹(푥) = ∫ 푅푎−1(1 − 푅)푏−1푑푅 1) ∑푛 푙표푔(1 + 훽푒푡푎푛푥푖) + (푏 − 1) ∑푛 푙표푔[1 − (1 + 훽(푎, 푏) 푖=1 푖=1 0 훽푒− 푡푎푛 푥푖)−훼] (12) Therefore, Differentiating (12) partially with respect to 푎, 푏, 훼푎푛푑훽 gives 훽[{1+훽푒− 푡푎푛 푥}−훼,푎,푏] (13), (14), (15) and (16): 퐹(푥) = (7) 훽(푎,푏) 휕푙(푥:푎,푏,훼,훽) = 푛휓(푎) + 푛휓(푎 + 푏) − 훼 (13) − 푡푎푛 푥 −훼 휕푎 Where훽[{1 + 훽푒 } , 푎, 푏] is an incomplete beta function? 휕푙(푥: 푎, 푏, 훼, 훽) 3.4.
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