Topp-Leone moment exponential distribution: properties and applications

This article is published under the Creative Commons CC-BY-ND License (http://creativecommons.org/licenses/by-nd/4.0/). This license permits use, distribution and reproduction, commercial and non-commercial, provided that the original work is properly cited and is not changed in anyway. Abstract: In this article, a new three parameter lifetime model is proposed as a generalisation of the moment exponential distribution. The proposed model is named as Topp-Leone moment exponential distribution. The induction of two additional shape parameters will enhance the capability of the proposed model to handle the complex scenarios in modelling. Several properties of the proposed model are discussed. The model parameters are estimated using method of maximum likelihood. Real life applications of the proposed model have been carried out by using datasets from the fi elds of botany, archaeology and ecology.


INTRODUCTION
The selection of an appropriate model to analyse the behaviour of real data is an attractive and complicated task. In certain situations, the existing classical models are not suitable to express the utility of real data in several applied areas. These complexities clearly demand useful models to handle these scenarios. A comprehensive discussion is available in the literature (Azzalini, 1985;Freimer et al., 1988;Marshall & Olkin, 1997;Eugene et al., 2002;Azzalini & Capitanio, 2003;Alzaatreh et al., 2013) about the generalisation of classical distributions.
In lifetime data analysis, exponential (Ex) distribution is strongly recommended due to its interesting 'lack of memory' property, which makes it more fl exible for modifi cation. The probability density function (PDF) of this distribution is decreasing with a constant hazard rate function (HRF). This limitation opens doors for various generalisations of the exponential distribution. Various extensions of the Ex distribution have been proposed in literature, see for example, the exponentiated Ex (Gupta & Kundu, 2001), extended exponentiated Ex (Abu-Youssef et al., 2015), gamma exponentiated Ex (Ristic & Balakrishnan, 2012), beta Ex (Nadarajah & Kotz, 2006), odd exponentiated half logistic Ex (Afi fy et al., 2018) and generalised odd log-logistic Ex (Afi fy et al., 2019), among others. Dara and Ahmed (2012)  In practice, the most suitable form of hazard rate is the bathtub form. Topp and Leone (1955)  where is a baseline CDF with parameter vector is the corresponding baseline PDF, and are positive shape parameters.
In this article, we have provided a new generalisation of the MEx distribution using the TL-G family (Rezaei et al., 2017). The proposed model is referred to as Topp-Leone moment exponential (TLMEx) distribution. The induction of two shape parameters escalates the adaptability of the TLMEx model. The closed form of its hazard rate makes it more fl exible to model life time datasets.

THE TLMEX DISTRIBUTION
In this section, the CDF and PDF of the TLMEx distribution are defi ned. The CDF of the TLMEx follows using equations (1) and (3) as and the corresponding PDF is ... (06) where β is a scale parameter and α and γ are shape parameters. The plot of density function for various combinitions of parameters is given in Figure 1.

Mo ments
Moments are the fundamental property for any distribution function. The qth raw moment of a distribution is defi ned as Using the PDF equation (7) and assuming one can write The mean of the TLMEx distribution is easily obtained by setting q =1, and is The moment generating function (MGF) of X is defi ned by Using and equation (7), the MGF of the TLMEx distribution reduces to The moment generating function provides all the moments of the distribution.

Re sidual life and reversed residual life
The nth moment of the residual life is defi ned by After some simplifi cations, the above expression reduces to Using equation (7) to solve the and after some simplifi cations, we have We can fi nd the mean residual life (MRL) from the above equation by replacing . The MRL is the The PDF of the TLMEx distribution can be expressed in a mixture form as … (7) where is the constant term given by Based on equation (7), many statistical properties of the TLMEx model can be easily studied.

Mathematical characteristics
In this section, some mathematical properties of the TLMEx distribution are discussed. These include moments, moment generating function, reliability analysis, Lorenz curve and a certain characterisation.
September 2020 Journal of the National Science Foundation of Sri Lanka 48 (3) expected additional life length for a unit which is alive at age t.

The moment of reserved residual life is given by
We can write Using equation (7), the moment of the reversed residual life of the TLMEx distribution is

Lo renz curve
The Lorenz curve (LC) is considered to explain the graph of ratio for any positive random variable in favour of with the property , and The LC is defi ned as The LC is used to model the income data. Some notable applications of LC can be found in Lindley (1958). Using equation (7), we have The LC for the TLMEx model becomes

Ch aracterization
In this section, we provide a characterisation of the concomitant of the record statistics for X. On integrating the last expression with respect to w, we obtain where Thus

Es timation
This section is devoted to maximum likelihood estimation of parameters of the TLMEx distribution. For this suppose be a random sample from this distribution with a parameter vector . The log-likelihood function of is immediately written as where .
The components of the score vector are and where The MLEs of the can be obtained by equating the above nonlinear system of equations to zero and solving them simultaneously. This can be done by using nonlinear optimisation methods such as the quasi Newton algorithm to maximize numerically. The entries of the Fisher information matrix are given in Appendix A.

SIMULATION RESULTS
In this section, a simulation study is conducted to assess the performance of the MLEs of the TLMEx parameters.  The simulation study is conducted by drawing random samples of diff erent sizes from this distribution. Since the CDF is not easily invertible, a numerical method of solving nonlinear equation is used, where u is a uniform random number, for various choices of the parameters as discussed by Lange (2010). Maximum likelihood estimators of unknown parameters α, β and γ for diff erent sample sizes have been computed. The procedure was repeated for 5,000 times and then the average estimates and standard errors (SEs) were computed. Table 1 provides the simulation results including the average estimates of the parameters and their SEs.
The results show that the MLEs is consistent as the estimated values are very close to the true values. Further, the standard errors of the estimates decrease when the sample size increases.

Applications
Real data applications of the TLMEx distribution are discussed in this section. We have used three datasets from the fi eld of ecology, botany, and archaeology. These datasets are listed in Appendix A. The TLMEx distribution is compared with some existing distributions which are listed in Table 2. The unknown parameters of each distribution are estimated using the maximum likelihood method. We use -2 (where is the maximized log-likelihood) and AIC (Akaike information criterion) as goodness-of-fi t measures.
The fi rst dataset refers to the level of mercury in 34 albacore caught in the Eastern Mediterranean obtained from Mol et al. (2012). The second dataset represents the petal length (cm) for a random sample of 35 iris virginica from (Anderson, E., Bull. Amer. Iris Soc). The third dataset represents the length (cm) of a random sample     of 61 projectile points found at the Wind Mountain Archaeological by Woosley and McIntyre (1996).
The estimated parameters of the fi tted models, -2 and AIC are given in Tables 3-5 for the three datasets, respectively. The values in these tables reveal that the TLMEx distribution provides better fi t as compared with the other models. The fi tted density, empirical CDF and total time on test (TTT) plots of the TLMEx distribution are displayed in Figures 3 to 5 for the three datasets, respectively. These fi gures show that the TLMEx distribution fi ts to all datasets properly. The TTT plot shows that all datasets have monotonic increasing of HRFs. From Tables 3 to 5 we can see that the TLMEx distribution provides best fi t to all three datasets as the AIC value for this distribution is the smallest for all three datasets. The same is refl ected from Figures 3 to 5.

CONCL USIONS
In this paper we have introduced a new three-parameter model called Topp-Leone moment exponential (TLMEx) distribution, which generalises the moment exponential distribution proposed by Dara and Ahmad (2012). The new TLMEx model is found to be more fl exible and adaptable for modelling life time data in ecology, reliability, and environmental sciences. Some mathematical properties of the proposed model are studied. A suitable characterisation of the TLMEx distribution is presented. The unknown parameters of the TLMEx model are estimated via the maximum likelihood method and evaluated through a simulation study. Applicability of the new distribution is illustrated by means of three real datasets from the fi eld of ecology, Botany, and Archaeology. The proposed model is a reasonably better fi t to the three datasets used as compared with the competing models.