Intelligent Automation & Soft Computing DOI:10.32604/iasc.2021.017890 | |

Article |

Inference of Truncated Lomax Inverse Lomax Distribution with Applications

1Department of Mathematics, Faculty of Science, Jazan University, Jazan, Saudi Arabia

2Faculty of Graduate Studies for Statistical Research, Cairo University, Giza, 12613, Egypt

3The Higher Institute of Commercial Sciences, Al Mahalla Al Kubra, 31951, Egypt

4Deanship of Information Technology, King Abdulaziz University, Jeddah, 21589, Kingdom of Saudi Arabia

5Department of Mathematics, Faculty of Science, Jazan University, Jazan, Saudi Arabia

6Faculty of Business Administration, Sinai University, Cairo, Egypt

*Corresponding Author: M. Elgarhy. Email: m_elgarhy85@sva.edu.eg

Received: 12 February 2021; Accepted: 16 March 2021

Abstract: This paper introduces a modified form of the inverse Lomax distribution which offers more flexibility for modeling lifetime data. The new three-parameter model is provided as a member of the truncated Lomax-G procedure. The new modified distribution is called the truncated Lomax inverse Lomax distribution. The density of the new model can be represented as a linear combination of the inverse Lomax distribution. Expansions for quantile function, moment generating function, probability weighted moments, ordinary moments, incomplete moments, inverse moments, conditional moments, and Rényi entropy measure are investigated. The new distribution is capable of monotonically increasing, decreasing, reversed J-shaped and upside-down shaped hazard rates. Maximum likelihood estimators of the population parameters are derived. Also, the approximate confidence interval of parameters is constructed. A simulation study framework is established to assess the accuracy of estimates through some measures. Simulation outcomes show that there is a great agreement between theoretical and empirical studies. The applicability of the truncated Lomax inverse Lomax model is illustrated through two real lifetime data sets and its goodness-of-fit is compared with that of the recent models. In fact, it provides a better fit to these data than the other competitive models.

Keywords: Truncated Lomax-G family; inverse Lomax distribution; maximum likelihood method; moments

The inverse Lomax (IL) distribution is very flexible in analyzing situations of different types of hazard rate function. It is a member of the inverted family of distributions, that is, the IL distribution is the reciprocal of the well-known Lomax distribution. Kleiber et al. [1] reported that the IL distribution can be used in economics and actuarial sciences. Singh et al. [2] the reliability estimators of the IL distribution were investigated under type-II censoring. Yadav et al. [3] discussed parameter estimators of the IL model in the case of hybrid censored samples. Reyad et al. [4] regarded the Bayesian estimation of a two-component mixture of IL distribution based on a type-I censoring scheme. Bantan et al. [5] provided entropy estimators based on multiple censored scheme. The cumulative distribution function (cdf) of the IL distribution is given by:

where, b and

Recent studies about the generalization and extensions of the IL distribution have been proposed by several authors. Hassan et al. [6] provided the inverse power Lomax distribution. Hassan et al. [7] introduced the Weibull IL distribution. ZeinEldin et al. [8] provided the alpha power transformed IL distribution. Maxwell et al. [9] introduced the Marshall-Olkin IL distribution. ZeinEldin et al. [10] introduced odd Fréchet IL distribution. Hassan et al. [11] proposed Topp-Leone IL distribution.

A classical strategy to generate families of probability distributions consists of adding parameter(s) to baseline distributions. These families have the ability to improve the desirable properties of the probability distributions as well as to extract more information from the several data applied in many areas like engineering, economics, biological studies and environmental sciences. Another useful generator that works with the truncated random variable. The notable studies about the truncated-G families in this regard, are the truncated Fréchet-G [12], truncated inverted Kumaraswamy-G [13], truncated Lomax-G [14], truncated power Lomax-G [15] and truncated Cauchy power-G [16].

Hassan et al. [14] suggested the newly truncated Lomax–G (TL-G) family with the following cdf,

where

The main purpose of this work is to suggest a more flexible and enhance model called the truncated Lomax IL (TLIL) distribution. The key motivations of the TLIL distribution in the practice are (i) to enhance the flexibility of the IL distribution by using TL-G, (ii) to introduce the modified form of the IL distribution whose density can be expressed as a linear combination of the IL distributions, (iii) to develop a new model with increasing, decreasing, reversed J-shaped and upside-down shaped hazard functions and (iv) to provide better fits than the competing models. Further, we study various statistical properties and estimate the model parameters by using the maximum likelihood (ML) method. Finally, simulation studies as well as applications to real data are given.

This work can be arranged as follows. In Section 2, we give the model description of the TLIL distribution and provide a linear representation of its density and distribution functions. Section 3 provides some structural properties of the TLIL distribution. Estimation of parameters along with simulation study is considered in Section 4. Applications are given in Section 5 followed by comments and conclusions.

In this section, we define and provide the density and distribution function of the TLIL distribution. This model is yielded by taking the base-line Eq. (3) to be the cdf of the IL model. A random variable Y is said to have the TLIL, if its cdf is represented as:

where,

Expression of hazard rate function (hrf) is given by:

Graphic features of the pdf and hrf plots of Y ∼ TLIL

Fig. 1 gives density and hazard functions plots for specified values of parameters. It is observed that the pdf of the TLIL can be right-skewed, uni-modal and reversed J-shaped. The hazard function can be increasing, decreasing, reversed J-shaped and upside-down bathtub.

2.1 Quantile Function and Median

The quantile function of Y, denoted by Q(u), is defined by inverting Eq. (5) as follows

where u~ U(0,1). Further, the TLIL distribution can be simply simulated from Eq. (8). The median of Y is obtained by putting u = 0.5 in Eq. (8).

Here, we provide the representation of cdf Eq. (5) and pdf Eq. (6) of the TLIL distribution. As mentioned in Hassan et al. [14], the pdf of the TL-G family is expressed as follows:

where

where,

Furthermore, Hassan et al. [14] provided a useful expression of cdf

where

3.1 Probability-Weighted Moments

The probability-weighted moments (PWMs) are less sensitive to outliers. They are utilized to study characteristics of the probability distributions. They are sometimes used when maximum likelihood estimates are unavailable or difficult to compute. The PWMs, denoted by

where, k and h are positive integers. The class of the PWMs of the TLIL is obtained by substituting Eq. (10) and Eq. (12) in Eq. (13) as follows

where;

3.2 Moments and Related Measures

Here, we provide an infinite sum representation for the n-th moment about the origin, inverse moments and incomplete moments of the TLIL model, since it has a pivotal role in the study of the distribution and real data applications. The n-th moment for the TLIL is obtained as follows:

The first four moments are obtained by setting n =1, 2, 3 and 4 in Eq. (15). The n-th central moment (

Based on Eq. (16), we can obtain the skewness and kurtosis measures using the well-known relationships. Further, the moment generating function of the TLIL distribution for |t| < 1, is given by

The m–th inverse moment, for the TLIL distribution is derived by using pdf Eq. (10) as follows:

The harmonic mean of the TLIL distribution can be obtained by using the first inverse moment.

Additionally, the n-th incomplete moment of Y can be obtained from Eq. (10) as follows:

where, B(.,.,t) is the incomplete beta function.

The conditional moments and the mean residual lifetime function are of interest for lifetime models to be obtained. The r-th conditional moment is defined as:

The r-th moment of the residual life of the TLIL distribution is obtained by substituting Eq. (10) in Eq. (20) as follows

where B(.,.,t) is the incomplete beta function. In particular, the mean residual life of the TLIL model is obtained by substituting r =1 in Eq. (21).

Moreover, the reversed residual life (RRL) is defined as the conditional random variable 𝑡−Y|Y≤𝑡 which denotes the time elapsed from the failure of a component given that its life is less than or equal to 𝑡. The r-th moment of the RRL is defined by:

The r-th moment of the RRL of TLIL distribution is obtained by substituting Eq. (10) in Eq. (22) as follows

The mean of RRL (mean waiting time) represents the waiting time elapsed since the failure of an item on condition that this failure had occurred. For, r = 1 in Eq. (23) we obtain the mean of RRL of the TLIL distribution.

The entropy affords a great tool to evaluate the amount of information (or uncertainty) exists in a random observation relating to its parent distribution. A small value of entropy provides the smaller uncertainty in the data. Hassan et al. [14] mentioned that the Rényi entropy of the TL-G distributions is defined as:

where

So, after some manipulation, the Rényi entropy of the TLIL distribution is given by

where, B(.,.) is the beta function.

4 Estimation and Numerical Study

In this section, the ML estimators of the model parameters are derived in the case of complete samples. Also, approximate confidence intervals (CIs) are obtained. Furthermore, a numerical study is conducted.

Suppose that x1, x2,…,xn be the observed values from TLIL distribution whose lifetimes have the pdf Eq. (6). The log-likelihood based on complete sample, is given by

where,

For interval estimation of the parameters, it is known that under regularity conditions, the asymptotic distribution of ML estimators of elements of unknown parameters for

where,

where

In this sub-section, we perform a simulation study to evaluate the behavior of the ML estimates (MLEs). The attitude of the different estimates is checked in terms of their mean square errors (MSEs), standard errors (SEs) and average lengths (ALs) of the CIs. 10000 random samples of sizes 30, 50, 75 and 100 are generated from TLIL distribution. Four sets of parameter values are chosen as;

Based on the above tables, we conclude the following

• It is clear that MSEs, SEs and ALs decrease as sample size increases for all estimates.

• The MSEs for

• The MSEs for

• As the value of

• As the value of

• The SEs and ALs for

In this section, we provide the applicability of TLIL distribution by using two real data sets. These data have been used by several authors to show the superiority of other competing models. We also provide a formative evaluation of the goodness of-fit of the models and make comparisons with other distributions. The measures of goodness of fit including; the Akaike information criterion (AIC), Consistent AIC (CAIC), Hannan-Quinn information criterion (HQIC), Anderson-Darling(A∗) and Cramér- von Mises (W∗) are calculated to compare the fitted models. Generally, the best fit to the data that is correspond to the lowest values of these statistics.

The first data set was proposed by Murthy et al. [18], which represents the failure times for a particular windshield device. For these data, we shall compare the TLIL model with the following models: the Zubair Lomax (ZL) [19], gamma-Lomax (GL) [20], exponentiated transmuted generalized Rayleigh (ETGR) [21], transmuted Weibull Lomax (TWL) [22], Kumaraswamy Lomax (KwL) [23], Kumaraswamy Weibull (KwW) [24], McDonald Weibull (McW) [25], beta Weibull (BW) [26] and transmuted Marshall-Olkin Fréchet (TMOFr) [27]. The estimated parameters of these models and the corresponding SEs for windshield data are provided in Tab. 3. The statistics AIC, CAIC, HQIC, A* and W* are mentioned in Tab. 4. The estimated pdf, cdf, sf and PP plots for Aircraft Windshield data of the fitted models are displayed in Fig. 2.

It is clear from Tab. 3, Tab. 4 and Fig. 2 that the TLIL provides a better fit to the data and therefore could be chosen as the best model.

This data set describes there mission times (in months) of a random sample of 128 bladder cancer patients studied by Lee et al. [28]. For these data, we compare the fit of the TLIL with some other distributions like; the ZL, transmuted modified Weibull (TMW) distribution [29], transmuted additive Weibull (TAW) distribution [30], generalized inverse gamma distribution distribion [31], beta exponentiated Burr XII (BEBXII) distribution [32], beta Fréchet (BFr) distribution [33] and ETGR distribution The estimated parameters along with their SEs are provided in Tab. 5. The statistics of the fitted models are presented in Tab. 6. We note from Tab. 6 that the TLIL gives the lowest values of AIC, CAIC, HQIC, A∗ and W∗ as compared to the other competitive models. Therefore, TLIL distribution provides the best fit for the cancer patient data. More information can be found in Fig. 3.

As seen from Fig. 3 that the TLIL provides a better fit to cancer data and so could be preferred than the other competitive model.

We provide a new three-parameter lifetime distribution depends on the recent truncated Lomax-G family. The new truncated Lomax inverse Lomax model offers flexibility for modeling lifetime data. Expressions of density and distribution functions are obtained as a linear combination of the inverse Lomax distribution. Several mathematical properties of the new model are derived like; probability weighted moments, quantile function, moment generating function, ordinary and incomplete moments, inverse moments, conditional moments, and Rényi entropy. The maximum likelihood method of estimation is employed to obtain the point and approximate confidence interval of population parameters. We assess the accuracy of estimates viz simulation study. Applications to two real data are utilized to illustrate the importance and usefulness of the new model compared to some models.

Acknowledgement: We would like to thank all four reviewers and the academic editor for their interesting comments on the article, greatly improving it in this regard.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present review.

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