The Alpha Power Gompertz Distribution: Characterization, Properties, and Applications

Sankhy¯a: The Indian Journal of Statistics 2021, Volume 83-A, Part 1, pp. 449-475 c 2020, Indian Statistical Institute The Alpha Power Gompertz Distribution: Characterization, Properties, and Applications Joseph Thomas Eghwerido Federal University of Petroleum Resources, Effurun, Nigeria Lawrence Chukwudumebi Nzei University of Benin, Benin City, Nigeria Friday Ikechukwu Agu University of Calabar, Calabar, Nigeria Abstract A new three-parameter model called the Alpha power Gompertz is derived, studied and proposed for modeling lifetime Poisson processes. The advantage of the new model is that, it has left skew, decreasing, unimodal density with a bathtub shaped hazard rate function. The statistical structural properties of the proposed model such as probability weighted moments, moments, order statistics, entropies, hazard rate, survival, quantile, odd, reversed hazard, moment generating and cumulative functions are investigated. The new proposed model is expressed as a linear mixture of Gompertz densities. The parameters of the proposed model were obtained using maximum likelihood method. The behaviour of the new density is examined through simulation. The proposed model was applied to two real-life data sets to demonstrate its flexibility. The new density proposes provides a better fit when compared with other existing models and can serve as an alternative model in the literature. AMS (2000) subject classification. 62E10; 62E15. Keywords and phrases. Alpha power distribution, Gompertz distribution, Gompertz failure rate, Moment generating function of Gompertz, Moment of Gompertz 1 Introduction Modeling lifetime data has received a tremendous attention in recent times. However, modeling lifetime data rely on their distribution. Thus, developing more flexible distributions to model life time data is therefore required to convey the true characteristic of the data. 450 J. T. Eghwerido et al. The Gompertz distribution is a continuous distribution named after (Gom- pertz, 1824). The Gompertz distribution is an important distribution used to describe the lifespans in gerontology, biology, survival analysis and failure rate in computer codes, customer lifetime value and self-avoiding walk. How- ever, it is imperative to note that the Gompertz distribution has a monotone increasing hazard rate function. Thus, the Gompertz distribution is used to model phenomena with increasing failure rate. Unfortunately, in practice one needs to consider scenarios with non-monotonic increasing functions (Sarhan and Kundu, 2011). Thus, to account for this disadvantage, Keller et al. (1982) introduced the exponential distribution. However, the exponential distribution could only account for constant failure rate. Thus, distribution with a linear increasing failure rate such as the Gompertz distribution is introduced to account for the monotone increasing failure rate. This study presents a new class of distribution called the Alpha Power Gompertz (APG) distribution with a bathtub shaped hazard rate function The motivation of this study is to introduce a new class of statistical distribution with a bathtub shaped failure rate that can provide a good fit for real life data than existing well-known distributions. More so, it is moti- vated based on the results obtained from existing literature like Gompertz, transmuted Gompertz distribution (Khan et al., 2016a), transmuted generalized Gompertz distribution (Khan et al., 2016b), alpha power inverse Weibull (Basheer, 2019), alpha power inverted exponential (Unal et al., 2018), Weibull Fréchet (Afify et al., 2016) distributions, we propose the APG distribution using the alpha power transformation characterizations. Several classes of the Gompertz distributions have been proposed in literature. El-Gohary et al. (2013) proposed the generalized Gompertz distribution, Oguntunde et al. (2019) proposed Gompertz Fréchet. (Oguntunde et al., 2018) proposed Gompertz inverse exponential. Eghwerido et al. (2019b) proposed the Gompertz alpha power inverted exponential distribution. Ex- ponentiated Gompertz generated family of distribution was proposed in Cordeiro et al. (2016). Alizadeh et al. (2017) proposed the Gompertz-G family of distributions. Obubu et al. (2019) proposed the Gompertz length biased exponential distribution. Mahdavi and Kundu (2017)proposedanew method for generating distributions. Nassar et al. (2017) proposed the alpha power Weibull distribution. Unal et al. (2018) proposed the alpha power inverted exponential distribution, Nadarajah and Okorie (2017)proposedthe moment of the alpha power transformed power transformed generalized exponential distribution. Hassan et al. (2019) proposed the alpha power transformed power Lindley distribution. Lenart (2014) proposed the method for obtaining the moments of the Gompertz distribution and Agu and Onwukwe The Alpha Power Gompertz Distribution 451 (2019) proposed a modified Laplace distribution. Abdul-Moniem and Seham (2015) proposed the transmuted Gompertz distribution. In this article, we aim at proposing a three parameter distribution called the alpha power Gompertz distribution (APG) for lifetime data. The statistical properties of the proposed density will be established. We also provide the maximum likelihood estimates (MLEs) of the parameters. The moments of the APG distribution are also obtained in a closed form. The probability density function (pdf) of the Gompertz distribution is given as λ f(x)=λexp βx − exp(βx) − 1 x, β, λ > 0. (1.1) β The corresponding cumulative distribution function (cdf) of the Gompertz distribution is given as λ F (x)=1− exp − exp(βx) − 1 x, β, λ > 0, (1.2) β where β is the scale parameter and λ is the shape parameter. Let g(x)andG(x) be the density and cumulative functions of the baseline model. Then, Mahdavi and Kundu (2017) proposed a transformation called the alpha power with the pdf given as ⎧ ⎪ log α g(x)αG(x), if α + −{1} ⎨⎪ (α−1) fAP T (x)=⎪ (1.3) ⎩⎪ g(x), otherwise α =1, where + is the set of positive real number. However, the corresponding cdf is given as ⎧ G(x) ⎪ α −1 , if α + −{1} ⎨⎪ (α−1) FAP T (x)= (1.4) ⎪ ⎩⎪ G(x), otherwise α =1. In this article, the APG distribution with its statistical properties will be given intensive mathematical treatment. However, the flexibility of the propose model is examined by the application of the proposed model to glass fibre data obtained by workers at the UK National Physical Laboratory and a carbon data application. 452 J. T. Eghwerido et al. This article is organized as follows. In Section 2, we defined the APG distribution and provided some plots for its pdf and hazard rate function (hrf). Section 3 discussed the mixture representation of the density. In Sec- tion 4, we discussed some statistical properties of the APG distribution. In Section 5, we obtained the order statistics. The estimates of the parame- tersandsimulationarediscussedinSection6. In Section 7,wepresented applications to validate the proposed distribution. The results obtained are compared with existing distributions in this section. Section 8 is the conclu- sion. 2 APG Distribution In this section, we shall propose a three parameter class of the Gompertz distribution. Let X be a random variable of the APG distribution, then the pdf of the APG distribution is given as ⎧ ⎪ ⎪ 1−exp − λ exp(βx)−1 ⎪ β ⎪ log α λexp βx − λ exp(βx)−1 α , if α + −{1} ⎨⎪ (α−1) β f (x)= AP G ⎪ ⎪ ⎪ ⎪ ⎩ λexp βx − λ exp(βx) − 1 , otherwise α =1. β (2.1) However, the corresponding cdf is given as ⎧ ⎪ 1−exp − λ exp(βx)−1 ⎪ β − ⎪ α 1 , if α + −{1} ⎨⎪ (α−1) FAP G(x)=⎪ (2.2) ⎪ ⎪ ⎩⎪ − − λ − 1 exp β exp(βx) 1 , otherwise α =1. where α is additional extra shape parameter that controls the skewness and kurtosis of the newly proposed distribution. Figure 1 is the plot of the APG density for selected values of parameters α, λ and β. The plot shows that the APG distribution can be skewed to the left, decreasing and unimodal. The reliability function of the APG distribution is given as ⎧ ⎪ 1−exp − λ exp(βx)−1 −1 ⎪ α 1 − α β , if α + −{1} ⎪ α−1 ⎨⎪ − SAP G(x)=1 FAP G(x)=⎪ (2.3) ⎪ ⎪ ⎩ exp − λ exp(βx) − 1 , otherwise α =1. β The Alpha Power Gompertz Distribution 453 α=3.0, β=3.7, λ=1.6, α=7.0, β=1.4, λ=2.5, α=6.5, β=2.3, λ=4.2, α=1.5, β=1.5, λ=2.5, α=0.2, β=1.5, λ=1.1, α=0.5, β=3.0, λ=2.0, α=2.0, β=2.0, λ=2.0, f(x) 01234 0.0 0.2 0.4 0.6 0.8 1.0 1.2 x Figure 1: The plots of the APG pdf for some parameter values The failure rate function of the APG distribution is given as ⎧ −exp − λ exp(βx)−1 ⎪ β ⎪ λexp βx − λ exp(βx)−1 α log α, if α + −{1} ⎪ β − − λ − ⎪ exp exp(βx) 1 ⎪ 1−α β ⎨⎪ fAP G(x) hAP G(x)= = S (x) ⎪ AP G ⎪ λ ⎪ λexp βx− exp(βx)−1 ⎪ β ⎪ , otherwise α =1. ⎩⎪ exp − λ exp(βx)−1 β (2.4) Figure 2 shows the plot of the APG hazard rate function for selected values of parameters α, λ and β. The plots show that the APG distribution is bathtub shaped. The cumulative hazard rate function of the APG distribution is given as HAP G(x)=− ln SAP G(x) (2.5) 3 Mixture Representation In this section, we shall derive the algebraic expression for the APG distribution. Let the quantity 1−exp − λ exp(βx)−1 α β 454 J. T. Eghwerido et al. α=5.0, β=0.7, λ=2.0, α=2.0, β=1.7, λ=1.1, α=3.5, β=0.3, λ=1.8, α=4.0, β=1.5, λ=2.0, α=0.2, β=1.5, λ=1.1, α=3.5, β=3.0, λ=2.0, α=2.0, β=1.0, λ=2.0, h(x) 02468 0.0 0.2 0.4 0.6 0.8 1.0 1.2 x Figure 2: The plots of the APG hrf for some parameter values in Eq.

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