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The Inverse Burr Negative Binomial Distribution with Application to Real Data -.:: Natural Sciences Publishing

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The Inverse Burr Negative Binomial Distribution with Application to Real Data -.:: Natural Sciences Publishing J. Stat. Appl. Pro. 5, No. 1, 53-65 (2016) 53 Journal of Statistics Applications & Probability An International Journal http://dx.doi.org/10.18576/jsap/050105 The Inverse Burr Negative Binomial Distribution with Application to Real Data 1, 2 1 1 Abdullahi Yusuf ∗, Badamasi Bashir Mikail , Aliyu Isah Aliyu and Abdurrahaman L. Sulaiman 1 Department of Mathematics, Federal University Dutse PMB 7156 Jigawa State, Nigeria. 2 Department of Mathematics, Bayero University Kano PMB 3011, Kano State, Nigeria. Received: 14 Aug. 2015, Revised: 22 Nov. 2015, Accepted: 25 Nov. 2015 Published online: 1 Mar. 2016 Abstract: We introduce in this paper a four-parameter lifetime model, called the inverse burr negative binomial distribution. We derive some statistical properties of the proposed model that includes moments, quantile functions, median, reliability and entropy. The method of maximum likelihood is used for estimating the model parameters and the observed information matrix is obtained. Simulation study is performed to investigate the performance of the estimation of the model parameters. Two real data sets are used to demonstrate the flexibility of the new proposed model in comparison with other popular lifetime models. Keywords: Negative Binomial Distribution, Inverse Burr Distribution, Maximum likelihood, Observed information matrix. 1 Introduction The negative binomial distribution has been used in compounding distributions to form another flexible model. A lot of disributions have been introduced and applied in survival analysis. [4] pioneered a family of univariate distributions generated by compouding the negative binomial distribution with any continuos model. [3] introduced a lifetime model called the Burr XII negative binomial distribution with application to Lifetime data. The G-negative binomial is defined as follows: For any baseline cumulative distribution function (cdf) G(x), and x R , the G-Negative Binomial (G-NB) family of distributions has probability density function (pdf) f (x) and cumulative∈ density function (cdf) F(x) given by λ k λ 1 fλ , (x)= g(x) 1 k[1 G(x)] − − x > 0 (1) k (1 k) λ 1 { − − } − − − and λ λ (1 k)− 1 k[1 G(x)] − Fλ , (x)= − −{ − − } x > 0 (2) k (1 k) λ 1 − − − The hazard function is given by λ 1 λkg(x) 1 k[1 G(x)] − − h(x)= { − − λ } x > 0 (3) 1 k[1 G(x)] − 1 { − − } − where λ > 0 and k (0,1) The G-NB family∈ has the same parameters of the G distribution plus two additional shape parameters λ > 0 and k (0,1). If X is a random variable having pdf in (1), we write X G NB(λ;k). This generalization is obtained by increasing∈ the number of parameters compared to the G model, this∼ increase− has added more flexibility to the generated distribution. A significant point of the G-NB model is its ability to comparise G distribution as a sub-model when λ = 1 and k 0. However, the inverse burr distribution also known as (BurrIII)has been used in variousfields of sciences. In the actuarial→ literature it is known as the inverse Burr distribution see [9] and as the kappa distribution in the meteorological ∗ Corresponding author e-mail: [email protected] c 2016 NSP Natural Sciences Publishing Cor. 54 A. yusuf et al. : The inverse burr negative binomial... literature see [14] and [13]. It has also been employed in finance, environmental studies, survival analysis and reliability theory see [10], [8] and [6]. Furthermore, [5] proposed an extended BIII distribution in low-flow frequency analysis where its lower tail is of main interest. Recently, [2] introduced the Complementary Burr III Poisson Distribution, [1] proposed the geometric inverse burr distribution . In this paper, we use the Cordeiro G.M and M.Percontini [4] generator to define a model, called the Inverse Burr Negative Binomial (IBNB) Distribution. The main reason for proposing IBNB distribution are: (i) This distribution due to its flexibility became an important model that can be used in a different forms of problems in modeling lifetime data. (ii) It provides a reasonable parametric fit for modeling phenomenon with non-monotone failure rates such as the bathtub-shaped and unimodal failure rates, which are common in reliability and biological studies, unlike the exponetial poison (EP) and generalization of exponential poison (GEP) distributions whose ability’s are only in modeling data with increasing or decreasing failure rates. (iii) The IBNB distribution is a suitable model for fitting skewed data that cannot be properly fitted by existing distributions. The rest of the paper is organized as follows. The immediate section after this introduction is the presence of new model IBNB and some investigation on its properties. Section 3 is the statistical properties of the new model IBNB. In section 4 estimation of the parameters using maximum likelihood method are given. Section 5 is the simulation studies while in section 6 two real data are used to show the flexibility of the proposed model. Finally, the concluding remarks is given in section 7. 2 The IBNB The cumulative distribution function(cdf) and the probability density function(pdf) of the inverse burr distribution are given by α β x αβ 1 α β 1 G(x)= and g(x)= αβx − (1 + x− )− − (4) 1 + xα The inverse burr negative binomial is obtained by substituting cdf and pdf of the inverse burr i.e G(x) and g(x) in (4) in the equations (1) and (2). We therefore have the following results αβ 1 α β 1 α λ 1 kλαβx − (1 + x )− − x β − − fα,β (x; p,α,β)= 1 k[1 ( ) ] x > 0 (5) (1 k) λ 1 − − 1 + xα − − − the corresponding cumulative distribution function is α λ λ x β − (1 k) 1 k[1 ( α ) ] − − − − − 1+x Fα,β (x; p,α,β)= x > 0 (6) (n1 k) λ 1 o − − − the hazard rate function is α s 1 αβ 1 α β 1 x β − − kλαβx − (1 + x )− − 1 k[1 ( α ) ] α β − − 1+x h(x, p, , )= λ x > 0 (7) n xα β o 1 1 k[1 ( α ) ] − − − − − 1+x the survival function is α λ x β − 1 1 k[1 ( α ) ] − − − − 1+x sα,β (x; p,α,β)= x > 0 (8) n (1 k) λ 1 o − − − 2.1 PDF and Hazard Rate Function A lot of failure rate fuction have complex expressions because of the integral in the denominator and therefore the determination of the shapes is not explicit. [11] introduced a method to determine the shape of h(x) with at most one turning point. His method uses the density function instead of the failure rate. A turning point of a function is a point at which the function has a local maximum or a local minimum. f ′ (x) η(x)= (9) − f (x) c 2016 NSP Natural Sciences Publishing Cor. J. Stat. Appl. Pro. 5, No. 1, 53-65 (2016) / www.naturalspublishing.com/Journals.asp 55 Theorem 1([11]) Let η(x) be defined as in (9) (a)if η(x) II(strictly increasing), then h(x) is of type I (b)if η(x) ∈ DD(decreasing), then h(x) is of type D ∈ (c)if η(x) BBTTT(bathtube shape), if there exists x0 such that h′ (x)= 0, then h(x) is of type BBTT . Otherwise is of type I ∈ 2.1.1 Probability Density Function(PDF) The probability density function reported in (5) has the following properties Theorem 2The probability density function of IBNB distribution is decreasing for 0 < α < 1 and unimodal also a constant otherwise. Proof.Taking the log of (5) thereafter differentiating the result, we obtain kαβ(λ + 1)x (α+1)(1 + x α) (β +1) (log( f (x)))′ = n(x) − − − − (1 k[1 (1 + x α) β ]) − − − − α β (α+1) α ( +1)x− ( +1) where n(x)= α x . 1+x− − For 0 < α < 1 the function n(x) is negative. Thus, f ′ (x) < 0 for all x > 0. This shows that f is decreasing for 0 < α < 1. Consider α > 1 implies n will have one exact root x0 and n(x) > 0 for x < x0 and n(x) < 0 for x > x0. Hence, f is a unimodal function with mode at x = x0. However, for α > 1 and some values of β, λ and k, the pdf can be a constant. ⊓⊔ α=0.5, β=0.5, λ=0.5, k=0.5, α=0.5, β=0.5, λ=1, k=1, α=0.3, β=2, λ=7, k=0.5, α=3, β=3, λ=3, k=0.5, α=5, β=0.5, λ=7, k=0.5, b=0.5 α=5, β=0.5, λ=5, k=0.5, α=6, β=5, λ=2, k=0.5, α=5, β=2, λ=2, k=0.5, α=5, β=5, λ=0.3, k=0.3, α=0.3, β=0.3, λ=0.5, k=0.5, f(x) f(x) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 0 1 2 3 4 0 1 2 3 4 x x Fig. 1: Probability Density Function of the Proposed Model 2.1.2 Hazard Rate Function The hazard rate function reported in (7) has the following possible shape for x > 0,λ > 0,k > 0,α > 0,β > 0 in the following theorem. c 2016 NSP Natural Sciences Publishing Cor. 56 A. yusuf et al. : The inverse burr negative binomial... Theorem 3The hazard rate function of the IBNB distribution is decresing for 0 < α 1, increasing and a bathtube shape otherwise.
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