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The Transmuted Inverted Nadarajah-Haghighi Distribution with an Application to Lifetime Data

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The Transmuted Inverted Nadarajah-Haghighi Distribution with an Application to Lifetime Data Pak.j.stat.oper.res. Vol.17 No.2 2021 pp 451-466 DOI: http://dx.doi.org/10.18187/pjsor.v17i2.3734 The Transmuted Inverted Nadarajah-Haghighi Distribution With an Application to Lifetime Data Aliyeh Toumaj1, S.M.T.K. MirMostafaee2∗, G.G. Hamedani3 ∗Corresponding author 1. Department of Statistics, University of Mazandaran, Babolsar, Iran, aliye−[email protected] 2. Department of Statistics, University of Mazandaran, Babolsar, Iran, [email protected] 3. Marquette University, Milwaukee, WI 53201-1881, USA, [email protected] Abstract In this paper, we propose a new lifetime distribution. We discuss several mathematical properties of the new distribu- tion. Certain characterizations of the new distribution are provided. We study the maximum likelihood estimation and asymptotic interval estimation of the unknown parameters. A simulation study, as well as an application of the new distribution to failure data, are also presented. We end the paper with a number of remarks. Key Words: Maximum Likelihood Estimation; Goodness-of-Fit Criteria; Quadratic Rank Transmutation Map; Reversed Hazard Rate Function; Simulation. Mathematical Subject Classification: 60E05, 62E15, 62F10. 1. Introduction Lifetime distributions play important roles in modeling and analysis of many life phenomena. Consequently, many researchers have tried to introduce new competitive lifetime distributions in order to model real data that arise in many areas such as medicine, engineering and economics where the old distributions cannot be employed. Generalizing a lifetime distribution often consists of adding one or more parameters to the baseline distribution. The new parameters could make the generalized distribution more flexible in the sense that it can model the data sets more suitably. There exist many methods of generalizing distributions, see for example Lee et al. (2013) for a discussion regarding a variety of methods of generating statistical distributions developed in recent decades. Among the generalizing methods, an approach, called the quadratic rank transmutation map, has been proposed by Shaw and Buckley (2007) and Shaw and Buckley (2009). Let g(x) and G(x) be the probability density function (pdf) and cumulative distribution function (cdf) of the baseline distribution, respectively, of a random variable X, then the cdf and pdf of the transmuted family of distributions are given by 2 F (x) = (1 + λ)G(x) − λG(x) ; jλj ≤ 1;x 2 R; (1) and f(x) = (1 + λ)g(x) − 2 λg(x)G(x);x 2 R; (2) respectively. The Transmuted Inverted Nadarajah-Haghighi Distribution With an Application to Lifetime Data 451 Pak.j.stat.oper.res. Vol.17 No.2 2021 pp 451-466 DOI: http://dx.doi.org/10.18187/pjsor.v17i2.3734 Many authors use the transmutation procedure to achieve new generalized distributions, see for example the trans- muted generalized inverse Weibull distribution (Khan and King, 2014, and Merovci et al., 2013), the transmuted exponentiated exponential distribution (Merovci, 2013a), the transmuted Lindley distribution (Merovci, 2013b), the transmuted Dagum distribution (Elbatal and Aryal, 2015), the transmuted Burr type III distribution (Abdul-Moniem, 2015) and the transmuted Birnbaum-Saunders distribution (Bourguignon et al., 2017). Nadarajah and Haghighi(2011) introduced an extension of the exponential distribution, called the Nadarajah-Haghighi distribution, with the following pdf α−1 1−(1+βx)α fNH (x) = αβ(1 + βx) e ; x > 0; α; β > 0: Here, we emphasize that the Nadarajah-Haghighi distribution is a special case of the distribution introduced by Dimi- trakopoulou et al.(2007). Tahir et al.(2018) used the transformation Y = 1=X, where X has a Nadarajah-Haghighi distribution, to propose a new inverted model called the inverted Nadarajah-Haghighi (INH) distribution. The pdf of the INH distribution is given by αβ β α−1 β α g(y) = 1 + exp 1 − 1 + ; y > 0; α; β > 0: y2 y y The corresponding cdf of the INH is β α G(y) = exp 1 − 1 + ; y ≥ 0: y In this paper, we use the transmutation map to generalize the INH distribution. To this end, we take g(x) and G(x) in (2) to be the pdf and cdf of the INH distribution, respectively, to obtain the pdf of the transmuted inverted Nadarajah- Haghighi (TINH) distribution as follows αβ β α−1 β α β α f(x) = 1 + exp 1 − 1 + 1 + λ − 2 λ exp 1 − 1 + ; x2 x x x α > 0; β > 0; jλj ≤ 1; x > 0: (3) We write X ∼ TINH(α; β; λ) if the pdf of X is (3). The corresponding cdf of the TINH distribution is given by β α β α F (x) = exp 1 − 1 + 1 + λ − λ exp 1 − 1 + ; x ≥ 0: (4) x x The special cases of the TINH distribution are listed below: • For λ = 0, we obtain the INH distribution (Tahir et al., 2018). • For α = 1, we obtain the transmuted inverted (inverse) exponential distribution, discussed by Oguntunde and Adejumo(2015). • For λ = 0 and α = 1, we obtain the inverted (inverse) exponential distribution. In addition, the hazard rate function (hrf) of the new distribution turns out to be α−1 α α αβ β h β i n h β io x2 1 + x exp 1 − 1 + x 1 + λ − 2 λ exp 1 − 1 + x h(x) = α α ; x > 0: (5) h β i n h β io 1 − exp 1 − 1 + x 1 + λ − λ exp 1 − 1 + x Figures1 and2 show the shapes of (3) and (5) for selected values of the parameters, respectively. From Figure1, we see that the pdf of the new model is unimodal. Figure2 reveals that the hrf shapes are upside-down bathtub (increasing-decreasing). Therefore the new distribution shows a somehow flexibility and it may be used to model the lifetime data sets that cannot be modeled by non-inverted distributions satisfactorily. Here, we emphasize that hrf might not be absolutely decreasing. Tahir et al.(2018) claimed that the pdf and hrf of the INH distribution can be The Transmuted Inverted Nadarajah-Haghighi Distribution With an Application to Lifetime Data 452 Pak.j.stat.oper.res. Vol.17 No.2 2021 pp 451-466 DOI: http://dx.doi.org/10.18187/pjsor.v17i2.3734 decreasing and they plotted pdf and hrf when the scale and shape parameters of the INH distribution set equal to one and 0.4, respectively, to show their claim. However, we plotted the pdf and hrf of the INH distribution (recall that INH is a special case of TINH when λ = 0) and observed that they both are increasing-decreasing, see Figure3. α=0.5 ; β=1 α=3 ; β=1 λ=− 1 λ=− 1 λ=− 0.5 λ=− 0.5 λ=0 λ=0 λ=0.5 λ=0.5 λ=1 λ=1 f(x) f(x) 0.0 0.5 1.0 1.5 2.0 2.5 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 5 10 15 x x Figure 1: Plots of the pdfs of the TINH distribution for selected parameter values. α=0.5 ; β=1 α=3 ; β=1 λ=− 1 λ=− 1 λ=− 0.5 λ=− 0.5 λ=0 λ=0 λ=0.5 λ=0.5 λ=1 λ=1 h(x) h(x) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 5 10 15 x x Figure 2: Plots of the hrfs of the TINH distribution for selected parameter values. The rest of the article is organized as follows: Several mathematical properties of the new model are presented in Section 2. Certain characterizations of the new distribution are provided in Section 3. Section 4 is devoted to maximum likelihood estimation of the parameters, a discussion regarding the asymptotic behavior of the maximum likelihood estimators and a simulation study. An application of the new model is provided in Section 5. Finally, a number of remarks are given in Section 6. 2. Mathematical Properties In this section, we focus on some mathematical properties of the new distribution, such as the quantile function, skewness, kurtosis, moments, reversed hazard rate function, the reliability parameter and order statistics. The Transmuted Inverted Nadarajah-Haghighi Distribution With an Application to Lifetime Data 453 Pak.j.stat.oper.res. Vol.17 No.2 2021 pp 451-466 DOI: http://dx.doi.org/10.18187/pjsor.v17i2.3734 Pdf Hrf f(x) h(x) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0 5 10 15 x x Figure 3: Plots of the pdf (left) and hrf (right) of the TINH distribution when β = 1; α = 0:4 and λ = 0. 2.1. Quantile function The quantile function is one of the important characteristics of a distribution. One application of the quantile function arises when one wants to generate samples from the selected distribution. For the TINH distribution, the quantile function x = Q(p), can be derived via inverting the cdf (4) 8 " 1 #−1 p α > (1+λ)− (1+λ)2−4λp > β 1 − ln − 1 ; if λ 6= 0; <> 2λ Q(p) = (6) > > h 1 i−1 :> β (1 − ln p) α − 1 ; if λ = 0: The median of X ∼ TINH(α; β; λ) is simply obtained by setting p = 0:5 in (6). In order to generate a random sample of size n from TINH(α; β; λ), first generate a random sample, say U1; ··· ;Un from the standard uniform distribution, then the required random sample of size n, X1; ··· ;Xn, can be derived by using the relation Xi = Q(Ui) for i = 1; ··· ; n.
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