Sankhy¯a: The Indian Journal of 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 proper- ties of the proposed model such as weighted moments, moments, order statistics, entropies, hazard rate, survival, quantile, odd, reversed haz- ard, 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 com- pared 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 , 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 . However, the exponen- tial distribution could only account for constant failure rate. Thus, distribu- tion 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 gene- ralized Gompertz distribution (Khan et al., 2016b), alpha power inverse Weibull (Basheer, 2019), alpha power inverted exponential (Unal et al., 2018), Weibull Fr´echet (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 lit- erature. El-Gohary et al. (2013) proposed the generalized Gompertz distri- bution, Oguntunde et al. (2019) proposed Gompertz Fr´echet. (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 . Unal et al. (2018) proposed the alpha power in- verted exponential distribution, Nadarajah and Okorie (2017)proposedthe moment of the alpha power transformed power transformed generalized ex- ponential distribution. Hassan et al. (2019) proposed the alpha power trans- formed 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 . 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 statis- tical 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 and λ is the . 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 pro- pose 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. 2.1 be B. Then, it can be expanded as

∞ i log α λ B = 1 − exp − exp(βx) − 1 i. (3.1) i! β i=0

Also, let the quantity

λ 1 − exp − exp(βx) − 1 i β in Eq. 3.1 be K. Then, the binomial expansion of K is given as

i i! λj K = −1 jexp − exp βx − 1 . (3.2) i − j !j! β j=0

Thus, in power series, the APG distribution can be expressed as

⎧ ⎪ i+1 ⎪ ∞ i log α 1 − j − λ − + −{ } ⎪ i=0 j=0 λ − 1 exp exp(βx) 1 j +1 +βx , if α 1 ⎨⎪ α 1 i−j !j! β fAP G(x)=⎪ ⎪ ⎪ ⎩⎪ − λ − λexp βx β exp(βx) 1 , otherwise α =1. (3.3) The Alpha Power Gompertz Distribution 455

The corresponding cdf in power series is given as

⎧ i ⎪ log α ⎪ 1 ∞ i − j − λ − − + −{ } ⎪ − − 1 exp exp(βx) 1 1 , if α 1 ⎨⎪ α 1 i=0 j=0 (i j)!j! β FAP G(x)= ⎪ ⎪ ⎩⎪ − − λ − 1 exp β exp(βx) 1 , otherwise α =1. (3.4) However, the mixture representation given in Section 3 is used in obtaining simplified properties of the APG distribution. The mixture representation also helps in expressing the new distribution in terms of Gompertz distribu- tion. The reversed hazard function is given as

⎧ ⎪ i+1 j ⎪ λ log α 1 −1 exp − λ exp(βx)−1 j+1 +βx ⎪ β ⎪ i−j !j! ⎪ ∞ i + −{ } ⎪ i=0 j=0 i , if α 1 ⎪ log α j ⎪ − − λ − − ⎪ − 1 exp exp(βx) 1 1 ⎨⎪ (i j)!j! β r(x)= ⎪ ⎪ ⎪ ⎪ λexp βx− λ exp(βx)−1 ⎪ β ⎪ , otherwise α =1. ⎪ ⎩ 1−exp − λ exp(βx)−1 β (3.5) The Odds function of the APG distribution is given as

⎧ i ⎪ ∞ i log α j ⎪ 1 −1 exp − λ exp(βx)−1 −1 ⎪ α−1 i=0 j=0 (i−j)!j! β + −{ } ⎪ i , if α 1 ⎪ log α ⎨ − 1 ∞ i − j − λ − − FAP G(x) 1 − − 1 exp exp(βx) 1 1 O(x)= = α 1 i=0 j=0 (i j)!j! β ⎪ SAP G(x) ⎪ ⎪ ⎪ ⎩⎪ λ − − exp β exp(βx) 1 1, otherwise α =1. (3.6)

4 Statistical Properties of the APG Distribution This section investigates some statistical properties of the APG distri- bution. These include quantile function and random number generation, skewness and kurtosis, moments, generating function, probability weighted moment, entropies, moments of the residual and reversed residual lifes and order statistics. 456 J. T. Eghwerido et al.

4.1. Quantile Function and Random Number Generation Let X be a random variable such that X ∼ AP G(α, λ, β). Then, the quantile function of X ⎧for u ∈ (0, 1) is given as ⎪ − β −1 ⎪ β 1 log 1 − log 1 − log α log u(α − 1) + 1 , 0

4.2. Skewness and Kurtosis The Bowley’s formula for coefficient of skewness is given as x − 2x + x Sk = 0.75 0.5 0.25 . x0.75 − x0.25 However, the Moor’s formula for coefficient of kurtosis is given as x − x − x + x Ks = 0.875 0.625 0.375 0.125 . x0.75 − x0.25 th th The skewness (Sk), kurtosis (Ks), median (M), 25 percent (Q1)and75 percent (Q3) of the APG distribution is evaluated. Data are generated using −1 − β − −1 − xu = β log 1 λ log 1 log α log u(α 1) + 1 , 0

Table 1: (continued) λ =2.5 Sk Ks M Q1 Q3 0.2202 0.4798 0.1909 0.0815 0.3620 0.2054 0.4433 0.1860 0.0805 0.3459 0.1487 0.3155 0.1655 0.0762 0.2862 0.0999 0.2154 0.1457 0.0713 0.2368 0.0548 0.1269 0.1249 0.0652 0.1915 0.1569 0.3397 0.2532 0.1161 0.4414 0.1407 0.3042 0.2449 0.1142 0.4183 0.0822 0.1838 0.2118 0.1058 0.3367 0.0359 0.0939 0.1816 0.0968 0.2727 −0.0039 0.0179 0.1518 0.0864 0.2166 0.1402 0.3054 0.2706 0.1271 0.4610 0.1239 0.2703 0.2611 0.1248 0.4361 0.0657 0.1524 0.2242 0.1150 0.3487 0.0207 0.0656 0.1910 0.1045 0.2811 −0.0173 −0.0070 0.1586 0.0926 0.2224 0.0934 0.2104 0.3258 0.1664 0.5181 0.0772 0.1770 0.3125 0.1627 0.4875 0.0221 0.0680 0.2622 0.1467 0.3830 −0.0180 −0.0090 0.2193 0.1306 0.3049 −0.0502 −0.0711 0.1791 0.1132 0.2386 0.0671 0.1552 0.3658 0.1994 0.5558 0.0515 0.1233 0.3493 0.1941 0.5212 −0.0003 0.0215 0.2886 0.1721 0.4051 −0.0363 −0.0482 0.2385 0.1509 0.3199 −0.0644 −0.1029 0.1927 0.1288 0.2489

4.3. The Moments of the APG Distribution The moment of the APG distribution is given in Laplace transform as

⎧ ⎪ ∞ i ∞ − λ − − + −{ } ⎪ i=0 j=0 μij 0 λexp β exp(βx) 1 j +1 + βx exp sx dx, if α 1 ⎨⎪ L(s)=⎪ ⎪ ⎪ ⎩ ∞ − λ − − 0 λexp βx β exp(βx) 1 exp sx dx, otherwise α =1. (4.5) where (−1)j log α (i+1) μij = . j!(i − j)! α − 1 460 J. T. Eghwerido et al.

By substituting q = exp βx ,theEq.4.5 can be expressed as ⎧ ∞ ∞ − s ⎪ i μ λ exp λ (j +1) exp − λ (j +1)q q β dq, if α + −{1} ⎨⎪ i=0 j=0 ij β β 1 β L(s)= ⎪ ⎩⎪ ∞ − s λ λ − λ β β exp β 1 exp β q q dq, otherwise α =1. (4.6) Thus, by Abramowitz and Stegun (1965) 5.1.4, ∞ exp(−zt) En(z)= n dt, n > 0,Re(z) > 0. (4.7) 1 t

λ s where z = β (j +1)andn = β . Thus, ⎧ ∞ i λ λ λ + ⎪ μij exp (j +1) E s ( (j +1)) λ, β > 0, if α −{1} ⎨⎪ i=0 j=0 β β β β L(s)= (4.8) ⎪ ⎩ λ λ λ exp E s ( ), otherwise α =1. β β β β

The rth moment of the APG distributed random variable X is given as ⎧ ∞ ∞ − − ⎪ i μ exp λ (j +1) r x 1 exp − λ (j +1)x ln(x) r 1 dx, if α + −{1} ⎨⎪ i=0 j=0 ij β 1 βr β r E X = ⎪ ⎪ ⎩ λ ∞ r −1 − λ r−1 exp β 1 β x exp β x ln(x) dx, otherwise α =1. (4.9) However, by Milgram (1985), the integral representation and the generalized integro-exponential function in Eq. 4.9 can be expressed as ∞ m 1 m −s − Es (z)= ln x x exp zx dx. (4.10) Γ(m +1) 1 m More so, the quantity Es can be defined as m −1 ∂m Em(z)= E (z). (4.11) s m! ∂sm s Clearly, the rth moment of the APG distributed random variable is given as ⎧ ⎪ ∞ i μ r! exp λ (j +1) Er−1 λ (j +1) , if α + −{1} ⎨⎪ i=0 j=0 ij βn β 1 β E Xr = ⎪ ⎩ r! λ r−1 λ βn exp β E1 β , otherwise α =1, (4.12) r−1 λ where the quantity E1 β (j +1) is given as ∞ − λ p r n p − λ 1 β (j +1) −1 r! λ − d Er 1 (j +1) = + ln (j +1) r p lim Γ 1 − υ , 1 − r − υ→0 p β p=1 p p! r! p=0 r p !p! β dυ (4.13) The Alpha Power Gompertz Distribution 461 for p ∈ 0, 1, 2, 3, 4 ,wehave dp lim Γ 1 − υ =0; p =0, υ→0 dυp dp lim Γ 1 − υ = γ!; p =1, υ→0 dυp dp π2 lim Γ 1 − υ = γ2 + ; p =2, υ→0 dυp 6 dp π2 lim Γ 1 − υ = γ3 + γ +2ε(3); p =3, υ→0 dυp 2 dp 3 lim Γ 1 − υ = γ4 + γ2π2 +8γε(3) + π4; p =4 υ→0 dυp 20 ∞ −s where γ =0.57722 is the Euler- Mascheroni constant and ε(s)= r=0 r is the Riemann zeta function. 4.4. Generating Function The probability generating function of the random variable X that is APG distributed is given as ∞ ∞ ∞ p x x p log t M(t)=E t = t fAP G(x)dx = x fAP G(x)dx for |t| > 1,x>0. −∞ p! p=0 1 (4.14) This implies ∞ ∞ i ∞ λ M(t)=E tx = ω xpλexp βx− exp(βx)−1 j+1 dx ijp β i=0 j=0 p=0 1 (4.15) where (log t)p (log α)i+1 1 ω = (−1)j. ijp p! α − 1 j!(i − j)! Integrating by parts and simplifying, gives

⎧ − ⎪ ∞ i ∞ ω p! exp( λ (j +1))Ep 1 λ (j +1) , if α + −{1} ⎨⎪ i=0 j=0 p=0 ijp βp β 1 β M(t)= ⎪ ⎩ (log t)p p! λ p−1 λ p! βp exp( β )E1 β , otherwise α =1, (4.16) However, the moment generating function (mgf) of the random variable X is given as

tx ∞ tx MX (t)=E e = −∞ e fAP G(x)dx ∞ i ∞ − λ − = λ i=0 j=0 μij 0 exp βx β exp(βx) 1 j +1 + tx dx. (4.17) 462 J. T. Eghwerido et al.

However, using the same substitution of Eq. 4.6, the mgf MX (t)canbe expressed as ∞ i ∞ λ λ λ s M (t)= μ exp (j +1) exp − (j +1)q q β dq. (4.18) X ij β β β i=0 j=0 0 More so, by Abramowitz and Stegun (1965) 5.1.5, ∞ n ψn(z)= exp(−zt)t dt, n > 0,Re(z) > 0. (4.19) 1 λ s where z = β (j +1)andn = β Thus, we have the mgf of the APG distribution expressed as ⎧ ∞ i λ λ λ + ⎪ μij exp (j +1) ψ s ( (j +1)) λ, β > 0, if α −{1} ⎨⎪ i=0 j=0 β β β β

MX (t)= ⎪ ⎩ λ λ λ exp ψ s ( ), otherwise α =1. β β β β (4.20)

4.5. Probability Weighted Moments Probability weighted moment (PWM) is used to obtain the quantiles and parameters of a distribution that may not be obtained explicitly. The (s, r)th PWM of the random variable X is expressed as ∞ r s r s ρ(s,r) = E[X F (X) ]= x FAP G(x) fAP G(x)dx. 0 The function F (x)sf(x) can be expressed as ∞ i s λ F (x)sf(x)= η λexp− (exp(βx) − 1)(j − jm +1)+βx ijm β i=0 j=0 m=0 where 1 (log α)2i+m+1 η =(−1)2j+s−m . ijm (α − 1)s+1 ((i − j)!j!)m+1 Now, ∞ i s ∞ r − λ − − ρ(s,r) = ηijm 0 x λexp β (exp(βx) 1)(j jm +1)+βx i=0 j=0 m=0 ∞ i s (4.21) r! λ − r−1 λ − = ηijm βr exp β (j jm +1) E1 β (j jm +1) i=0 j=0 m=0

⎧ ∞ − Thus, ⎪ i s η r! exp λ (j − jm +1) Er 1 λ (j − jm +1) , if α + −{1} ⎨⎪ i=0 j=0 m=0 ijm βr β 1 β ρ(s,r) = ⎪ ⎪ ⎩ r! λ r−1 λ βr exp β E1 β , otherwise α =1. (4.22) The Alpha Power Gompertz Distribution 463

4.6. Entropies The R´enyi entropy of the random variable X that is APG distributed measures the variation of the uncertainty. Thus, it is given as ∞ 1 δ Rδ(X)= log fAP G(x)dx δ > 0,δ =0. (4.23) 1 − δ −∞ Thus, this can be expressed as

∞ i ∞ δ λδ Rδ(X)= log mij λ exp δβx − j +1 exp(βx) − 1 dx −∞ β i=0 j=0 (4.24) where δ(i+1) (−1)jδ log α mij = . 1 − δ j!(i − j)! δ α − 1 δ In Laplace form, we have ∞ i ∞ λδ L (X)= log m λδ exp δβx− j +1 exp(βx) − 1 exp −sx dx. (4.25) δ ij β i=0 j=0 0 On integrating gives ∞ i λδ δλ λ Lδ(X)= log mij exp (j +1) E s ( (j +1)) λ, β > 0. (4.26) β β β β i=0 j=0 However, on simplifying ⎧ ⎪ ∞ i λδ δλ λ + ⎪ log m + (j +1) + log E s ( (j +1)) , if α −{1} ⎪ i=0 j=0 ij β β β ⎨⎪ β

Lδ(X)= ⎪ ⎪ ⎪ λδ δλ λ ⎩ log + + log E s ( )), otherwise α =1. β β β β (4.27) The Shannon entropy of the random variable X with APG distributed is given as

∞ i λ Se(X)=E − log fAP G(x) = j +1 E exp(βx) − 1 − βE x − log μij λ . (4.28) β i=0 j=0 On simplifying, we have ⎧ ∞ i 2 ⎪ λ λ s λ − λ ⎪ i=0 j=0 exp( (j +1))ψ ( (j +1)) (j +1) ⎪ β β β β β ⎨⎪ − λ − + −{ } exp β (j +1) log μij λ , if α 1 Se(X)= ⎪ ⎪ ⎪ ⎩ λ 2 λ λ λ λ exp( )ψ s ( ) − − exp − log μij λ, otherwise α =1. β β β β β β (4.29) 464 J. T. Eghwerido et al.

4.7. Moment of the Residual The nth moment of the residual life, say n dn(t)=E X − t |X>t for n =1, 2,... uniquely determines FAP G(x) (see Navarro et al. 1998). However, the nth moment of the residual life is given as ∞ 1 − n dn(t)= − X t dFAP G(x). (4.30) 1 FAP G(t) t This implies,

∞ i ∞ 1 n λ dn(t)= μij (x − t) λexp βx − (exp(βx) − 1)(j +1) dx 1 − F (t) β AP G i=0 j=0 t

∞ i n ∞ 1 n−k n λ = μij t λx exp βx − (exp(βx) − 1)(j +1) dx 1 − FAP G(t) β i=0 j=0 k=0 t

⎧ ⎪ 1 ∞ i n n−k n! λ n−1 λ ⎪ − μij t n exp E1 (j +1) ⎪ 1 FAP G(t) i=0 j=0 k=0 β β β ⎪ ⎨⎪ | arg t| <π, if α + −{1} dn(t)= ⎪ ⎪ ⎪ ⎩⎪ − 1 n tn−k n! exp λ En 1 λ , | arg t| <π otherwise α =1. 1−F (t) k=0 βn β 1 β (4.31)

5 Order Statistics

Let X1,X2, ··· ,Xn be a random sample of size n from the fAP G(x) distribution and X(1),X(2), ··· ,X(n)be the corresponding order statistics, Then, probability density function of the kth order statistics Xi:n,sayfi:n(x) is given as − − n! k 1 n k fk(x)= FAP G(x) fAP G(x) 1 − FAP G(x) −∞

On simplifying (5.1) after substitution, we have ⎧ − λ (eβx−1) ⎪ βx− λ eβx−1 β n−k ⎪ λ log αn! e β 1 − αe ⎪ − − ⎪ k 1 ! n k ! ⎪ − λ (eβx−1) − λ (eβx−1) ⎪ β k−1 β ⎪ × α1−e − 1 α1−e +n+k, if α + −{1} ⎨⎪ f (x)= k ⎪ ⎪ k−1 ⎪ n! − − λ − ⎪ 1 exp exp(βx) 1 ⎪ k−1 ! n−k ! β ⎪ k−1 ⎩⎪ × − λ − − λ − λexp βx β exp(βx) 1 exp β exp(βx) 1 , otherwise α =1. (5.2) The Alpha Power Gompertz Distribution 465

6 Estimation of Parameters Several approaches have been employed for parameter estimation in lit- erature. In this article, the maximum likelihood method is adopted to obtain the parameters of the APG. Let x =(x1,...,xn) be a random sample from the APG model with unknown parameter vector θ =(α, β, λ)T . Then log-likelihood function of the APG can be expressed as n − − − λ − = n log log α +logλ n log α 1 + βxi β (exp(βx) 1) n i=0 − − λ − + 1 exp( β (exp(βx) 1)) log α. i=0 (6.1) However, the partial derivative of the with respect to each parameter and equating to zero is given as

n n 1 n λ = − + 1 − exp(− (exp(βx ))) , (6.2) ∂α α log α α − 1 α β i i=1

1 n 1 n 1 λ = − exp(βx ) + exp(βx )−1 exp − (exp(βx )−1) log α (6.3) ∂λ λ β i β i β i i=1 i=0

n λ − − λxi ∂β = xi + β exp(βxi) 1 β exp(βxi) i=1 n − λ − λ − λ − β2 (exp(βxi) 1) + β xiexp(βxi) exp β (exp(βxi) 1) . i=1 (6.4) The estimates of the unknown parameters can be obtained by equating the vector to zero. The solution to the vector is obtained analytically using Newton-Raphson algorithm. Software like MATLAB, R, MAPLE, and so on could be used to obtain the estimates. 6.1. Simulations study A simulation is carried out to test the flexibility and efficiency of the APG distribution. Table 1 shows the simulation for different values of parameters for the APG distribution. The simulation is performed as follows: • −1 − β − −1 − Data are generated using xu = β log 1 λ log 1 log α log u(α 1) + 1 . 0

In this simulation study, we investigated the mean estimates (MEs), , biases and root means squared errors (RMSEs) of the maximum likelihood estimate MLEs. The bias is calculated by (for S = α, λ, β)

1 1000 Biasˆ = Sˆ − S . S 1000 i i=1 Also, the MSE is obtained as

1 1000 2 MSEˆ = Sˆ − S , S 1000 i i=1 Table 2 shows the results of the Monte Carlo study. The results in Table 2 show that the biases, RMSEs of the MLEs of the parameters decreases as the sample size increases and approach zero. This corresponds to the first-order asymptotic theory. The mean estimates of the parameters approach the true parameter values as the sample size increases. The variance decreases in all the cases as the sample size increases.

7 Real Life Data Application In this section, we illustrate empirically the flexibility of the APG model by means of two real life data sets. We compare the fits of the APG, Ku- maraswamy Gompertz (KUG), Weibull Gompertz (WG), alpha power in- verted exponential (APIE) (Unal et al., 2018), Gompertz (G), Topp Leone Gompertz (TLG), transmuted Gompertz (TG) (Khan et al., 2016a), trans- muted generalized Gompertz (TGG) (Khan et al., 2016b), Weibull Fr´echet (WFr) (Afify et al., 2016), alpha power inverse Weibull (APIW) (Basheer, 2019), Gompertz Weibull (GW) and alpha power Weibull (APW) (Nassar et al., 2017) distributions. The first data set consists of 63 observations of the strength of 1.5cm glass fibres obtained by workers at the UK National Physical Laboratory (Smith and Naylor, 1987) as used in Efe-Eyefia et al. (2019), Eghwerido et al. (2019a), Zelibe et al. (2019), Bourguignon et al. (2014) and Afify et al. (2016). The second set of data is the observations of carbon data used in Nichols and Franco (2006) and Afify et al. (2016). In order to compare the distribution under study, the following criteria were used: Akaike Information Criteria (AIC), Consistent Akaike Informa- tion Criteria (CAIC), Bayesian Information Criteria (BIC), and Hannan The Alpha Power Gompertz Distribution 467

Table 2: Simulation results for mean estimates, biases and root mean squared errors ofα, ˆ λˆ and βˆ for the APG distribution n Parameter ME Variance Bias RMSE 10 α =2.5 4.0602 41.5433 1.5602 6.6315 λ =5.7 4.5689 8.4072 -1.1311 3.1123 β =0.6 2.8479 8.7239 2.2479 3.7117

50 α =2.5 3.4171 25.2779 0.9171 5.1107 λ =5.7 5.1991 5.3214 −0.5009 2.3606 β =0.6 1.2323 1.7696 0.6323 1.4729

100 α =2.5 3.0825 8.7287 0.5825 3.0113 λ =5.7 5.4241 3.3999 −0.2759 1.8644 β =0.6 0.9246 0.9465 0.3246 1.0256

150 α =2.5 2.9065 5.4617 0.4065 2.3721 λ =5.7 5.4742 2.6973 −0.2258 1.6578 β =0.6 0.8374 0.6491 0.2374 0.8399

250 α =2.5 2.7059 3.1795 0.2059 1.7950 λ =5.7 5.4827 1.9200 −0.2173 1.4026 β =0.6 0.7920 0.4789 0.1920 0.7182

350 α =2.5 2.5392 1.7921 0.0392 1.3393 λ =5.7 5.4754 1.4094 −0.2246 1.2082 β =0.6 0.7643 0.3415 0.1643 0.6070 and Quinn Information Criteria (HQIC). The test statistics are given as fol- − ˆ − ˆ − ˆ 2kn lows: AIC = 2 +2k, BIC = 2 + k log(n), CAIC = 2 + n−k−1 , HQIC = −2ˆ+2k log(log(n)), where n isthesamplesize,k is the number of model parameters and ˆ is minus twice the maximized log-likelihood. In general, model with the smallest or least AIC value is regarded as the best model to fit the data. Tables 3 and 4 provide the values of the test statistics for the models fitted. The model parameters MLEs and their corresponding standard errors (in parenthesis) are also included in the tables. The results in these tables are obtained using the R PROGRAM. In Tables 3 and 4, the APG model gives the lowest values for the AIC, CAIC, BIC and HQIC statistics among all fitted models for this data. Thus, 468 J. T. Eghwerido et al. 32.58 32.99 39.01 35.11 0.1073 0.6288 74) . 33) 36) 57) 59) 02) 66) 80) 07) 02) 34.30 34.71 40.73 36.82 0.1660 0.9319 01) 54) 29) 38.80 39.49 47.37 42.16 0.2472 1.3566 03) 34.89 35.09 39.18 36.58 0.1306 0.7544 47) 53) 49) 08) 42) 20) 31) 36.83 37.52 45.41 40.20 0.1809 1.0123 05) 37.48 38.17 46.05 40.85 0.1800 1.0098 20) 34.85 35.54 43.42 38.22 0.1516 0.8444 ...... 82(34 . 75(0 01(0 83(0 67(0 10(0 95(0 39(0 09(0 02(0 12(0 58(0 24(0 03(0 46(0 82(0 92(0 56(0 85(0 03(0 23(1 21(0 02(0 ...... =25 =0 =1 =0 =2 =0 =-0 =0 =0 =-0 =3 =1 =0 =0 =2 =2 =2 =1 =0 =0 =3 =0 =0 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ β b α α λ α β β a b λ a b α α β β α β α β λ a ˆ ˆ ˆ Table 3: Performance rating of the APG distribution with glass fibres dataset APG Distribution Parameter MLEs AIC CAIC BIC HQIC W A WG TLG KUG TGG WFr TG The Alpha Power Gompertz Distribution 469 Table 3: (continued) 28) 14) 49) . . . 03) 81) 24) 28) 141.38 141.58 145.6 143.06 2.0184 3.4230 10) 38.18 38.6 44.61 40.71 0.1748 0.9626 04) 38.38 39.07 46.95 41.75 0.2330 1.2832 16) 82.58 82.99 89.01 85.11 0.9853 5.2956 08) 196.33 196.53 200.61 198.01 0.7815 4.2596 82) 30) 78) 51) 51) ...... 45(79 10(48 86(20 . . . 55(8 22(0 51(0 15(0 01(0 78(0 50(0 31(0 73(0 80(0 48(4 80(0 62(0 ...... =83 =6 =0 =61 =0 =0 =0 =0 =0 =0 =4 =3 =1 =0 =16 =5 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ a α λ b α β λ α β a b α β λ α β ˆ ˆ Bold font differentiate the proposed model with existing related models Distribution Parameter MLEs AIC CAIC BIC HQIC W A APIE APW GW APIW G 470 J. T. Eghwerido et al. 288.78 289.02 296.60 291.94 0.0582 0.4087 42) . 53) 00) . . 74) 32) 07) 36) 11) 290.25 290.68 300.68 294.47 0.0650 0.3850 01) 31) 07) 34) 28) 290.77 291.19 301.19 294.99 0.0690 0.4170 01) 291.43 291.68 299.25 294.60 0.0757 0.4079 87) 12) 15) 37) 46) 23) 27) 290.56 290.99 300.98 294.78 0.0648 0.3834 94) 72) 290.53 290.95 300.95 294.75 0.0666 0.3901 ...... 59(136 . 25(0 36(1 69(0 26(0 01(0 40(3 26(2 20(0 98(0 62(0 01(0 83(0 34(0 40(0 68(0 09(0 14(0 49(14 26(59 20(5 52(2 85(3 ...... =3 =0 =0 =0 = 150 =0 =0 =0 =2 =0 =0 =2 =-0 =0 =0 =0 =0 =3 =0 =2 =3 =2 =-0 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ α β β α β λ a a b α α β λ β a a b α α β a b α ˆ ˆ ˆ Table 4: Performance rating of the APG distribution with carbon data dataset Distribution Parameter AIC CAIC BIC HQIC W A APG WFr TGG ˆ APW WG GW The Alpha Power Gompertz Distribution 471 Table 4: (continued) 56) 56) . . 02) 17) 01) 11) 296.12 296.37 303.94 299.28 0.0721 0.4266 08) 529.66 529.68 534.87 533.00 0.1768 1.3637 01) 300.15 300.40 307.97 303.32 0.1262 1.0363 20) 328.48 328.73 336.30 331.65 0.5362 2.9868 01) 10) 417.84 417.97 423.05 419.95 0.3778 2.0679 08) 08) 54) 14) 02) 301.95 302.37 312.37 306.17 0.0742 0.5620 ...... 82(75 67(85 . . 08(0 73(0 97(0 51(0 79(0 04(0 11(0 54(0 30(0 90(0 47(0 96(0 28(0 16(0 ...... =0 =0 =0 = 109 = 105 =0 =0 =0 =1 =0 =0 =0 =0 =1 =2 =0 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ β λ α β α β λ a b α β β α a α λ ˆ Bold font differentiate the proposed model with existing related models DistributionTLG Parameter AIC CAIC BIC HQIC W A G TG KUG APIW APIE 472 J. T. Eghwerido et al.

APG APG WFr WFr KUG KUG WG WG TLG TLG GW GW TG TG f(x) f(x) 0.0 0.5 1.0 1.5 0.0 0.1 0.2 0.3 0.4 0.5

0.5 1.0 1.5 2.0 2.5 0 1 2 3 4 5 6

x x

Figure 3: The plots of estimated APG density

it could be chosen as the best model among them. Figure 3 shows the plots of the estimated densities of the models under consideration. Figure 4 is the plots of the estimated cdfs. These plots show that the APG distribution produces a better fit than others models for both data. However, Figure 5 shows the quantile-quantile plot of the data sets for APG distribution that further validates the APG pdf plot. Fn(x) Fn(x)

ecdf ecdf APG APG WFr WFr KUG KUG WG WG TLG TLG GW GW TG TG 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0 0123456

x x

Figure 4: The plots of estimated cdf of the APG model The Alpha Power Gompertz Distribution 473

APIE KUG WG APIE KUG WG Exp Exp Expected probability Expected probability Expected probability Expected probability 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0

Observed probability Observed probability Observed probability Observed probability Observed probability Observed probability

TLG APG TG TLG APG TG Expected probability Expected probability Expected probability Expected probability Expected probability Expected probability 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0

Observed probability Observed probability Observed probability Observed probability Observed probability Observed probability

Figure 5: The Q-Q plots for the dataset

8 Conclusion Inthis article, a three-parameter model called APG distribution has been successfully derived. This density extends the Gompertz distribution and makes the Gompertz density more flexible. The basic statistical properties of the APG distribution such as the order statistics, hazard function, cu- mulative hazard function, quantile, reversed hazard function, median, odds function have been successfully established. The APG distribution was ex- plicitly expressed as a linear function of the Gompertz distribution. The PWMs and entropies of the proposed distribution were also derived. A sim- ulation study of the proposed model was also illustrated. The simulation shows that the shape of the proposed distribution could be bathtub, uni- modal or decreasing (depending on the value of the parameters). The new distribution was applied to a real life data. It shows that the APG distri- bution performed better than, KUG, WG, TG, TGG, TLG, WFr, APIW, APW, KL, GW, APIE and G models. Author Contributions. The study proposes a new class of the family of the Gompertz distribution. The Statistical properties of the APG distribution are established. Compliance with Ethical Standards. Conflict of interests. The authors declare no potential conflict of inter- ests. Financial disclosure. No financial grant for this research. 474 J. T. Eghwerido et al.

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Joseph Thomas Eghwerido Department of Mathematics, Federal University of Petroleum Resources, Effurun, Delta State, Nigeria E-mail: [email protected]

Lawrence Chukwudumebi Nzei Department of Mathematics, University of Benin, Benin City, Nigeria

Friday Ikechukwu Agu Department of Statistics, University of Calabar, Calabar, Nigeria

Paper received: 11 July 2019.