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# The Lambert-G Family: Properties, Inference, and Applications

1 Accounting Department, Ashur University, Baghdad, 10011, Iraq

2 Department of Statistics, Mathematics and Insurance, Benha University, Benha, 13511, Egypt

3 Department of Management Information Systems and Production Management, College of Business and Economics, Qassim University, Buraydah, 51452, Saudi Arabia

4 Department of Basic Sciences, Common First Year (CFY), King Saud University, Riyadh, 11362, Saudi Arabia

5 Department of Mathematics and Statistics, Osim Higher Institute of Administrative Science, Osim, 12961, Egypt

* Corresponding Authors: Ahmed Z. Afify. Email: ; Badr Alnssyan. Email:

(This article belongs to the Special Issue: Frontiers in Parametric Survival Models: Incorporating Trigonometric Baseline Distributions, Machine Learning, and Beyond)

*Computer Modeling in Engineering & Sciences* **2024**, *140*(1), 513-536. https://doi.org/10.32604/cmes.2024.046533

**Received** 05 October 2023; **Accepted** 15 January 2024; **Issue published** 16 April 2024

## Abstract

This study proposes a new flexible family of distributions called the Lambert-G family. The Lambert family is very flexible and exhibits desirable properties. Its three-parameter special sub-models provide all significant monotonic and non-monotonic failure rates. A special sub-model of the Lambert family called the Lambert-Lomax (LL) distribution is investigated. General expressions for the LL statistical properties are established. Characterizations of the LL distribution are addressed mathematically based on its hazard function. The estimation of the LL parameters is discussed using six estimation methods. The performance of this estimation method is explored through simulation experiments. The usefulness and flexibility of the LL distribution are demonstrated empirically using two real-life data sets. The LL model better fits the exponentiated Lomax, inverse power Lomax, Lomax-Rayleigh, power Lomax, and Lomax distributions.## Keywords

Many approaches have been suggested to propose new families of distributions or to generalize some of the classical distributions. These families and generalized distributions provide more flexibility in modeling real-life data in different applied fields. The most common feature of the new families and generalized distributions is represented by having one or more extra shape parameters. Hence, the statistical literature contains many families to generate new distributions by adding one or more shape parameters. Some examples include the Kumaraswamy-G by Cordeiro et al. [1], exponentiated T-X by Alzaghal et al. [2], Weibull-G by Bourguignon et al. [3], odd moment exponential-G by Haq et al. [4], Burr XII-G by Cordeiro et al. [5], generalized odd Burr III-G by Haq et al. [6], generalized odd half-logistic-G by Altun et al. [7], Marshall–Olkin alpha power by Nassar et al. [8], new exponential-X by Ahmad et al. [9], new extended heavy-tailed family by Aljohani et al. [10], and modified generalized-G by Shama et al. [11].

One of the most notable approaches to generating new distributions is constructed by using the Lambert-W (LW) function which is also known as the product logarithm function. This approach is discussed by Corless [12]. It is defined (for

The above equation contains only one real-valued solution. Recently, the LW function has been used in the distribution of prime numbers as discussed by Visser [13]. Goerg [14] adopted the LW function to introduce new families of distributions in the context of random variable transformations. Iriarte et al. [15] generated the Lambert-F class as an alternative family for positive data analysis.

This study introduces a new wider class based on the LW function called the Lambert-G (LG) family, in which the baseline distribution is a continuous distribution with positive support. The proposed approach applies a transformation to a baseline cumulative distribution function (cdf), as illustrated in Definition 1. The newly generated cdf of the LG family, with two extra shape parameters, has the quantile function (qf), which is expressed in a closed form in terms of the LW function; hence, the proposed generator is called the LG family.

The LG family has some desirable properties and it can be justified as follows. (i) The three-parameter special sub-models of the LG family are capable of modeling all important hazard rate (hr) shapes including increasing, decreasing, unimodal, J-shape, reversed J-shape, bathtub, and modified-bathtub failure rates; (ii) Moreover, the densities of its sub-models accommodate reversed J shaped, right-skewed, symmetric, left-skewed, decreasing-increasing-decreasing densities; (iii) The LG special sub-models generalize some well-known distributions in the distribution theory literature such as the modified Weibull model [16]; (iv) The LG special models provide better fit than other generalized models under the same baseline distribution as shown in case of the Lambert-Lomax (LL) model.

The paper is organized in the following sections. In Section 2, the LG family is presented. In Section 3, we provide three special sub-models of the LG family. The properties of the LL distribution along with its analytical shapes are explored in Section 4. In Section 5, the parameters of the LL distribution are estimated via six classical estimation methods. Section 6 presents simulation results to address the behavior of different estimators. To show the empirical importance of the LL distribution, two real-life data sets are analyzed in Section 7. Final remarks are given in Section 8.

Definition 1. A random variable X is said to follow the LG family, denoted by

where

The probability density function (pdf) corresponding to Eq. (2) reduces to

where

The hrf of the LG family becomes

According to Eq. (4), the hrf of the LG family has a flexible property because it depends on the value of the extra parameter

Eq. (5) is called the reduced Lambert-G (RLG) family.

The pdf and hrf of the RLG are given, respectively, by

and

3 Special Models of the LG Family

In this section, we provide three specific models of the LG family. These special distributions provide modified flexible forms of some standard distributions namely the exponential, Pareto, and Lomax distributions. The special sub-models of the LG family are called the Lambert-exponential (LE), Lambert-Pareto (LP), and LL distributions. These special models are capable of modeling all important hrf shapes including increasing, decreasing, unimodal, J-shape, reversed J-shape, bathtub, and modified bathtub failure rates. Moreover, the densities of these sub-models can also provide reversed J-shaped, right-skewed, symmetric, left-skewed, decreasing-increasing-decreasing densities.

The LE cdf follows from Eq. (2) by setting

where

The corresponding pdf and hrf of the LE distribution take the forms

and

The hrf shapes of the LE distribution depend only on the value of

Consider the sf of the Pareto distribution

where

and

The shape of the LP hrf depends on the values of

By taking the sf of the Lomax distribution,

where

The corresponding pdf and hrf are

and

The Lomax distribution follows as a special case of the LL distribution with

4 Properties of the LL Distribution

In this section, we provide some basic statistical properties of the LL distribution.

4.1 Behavior of the Density and Hazard Rate Functions

The pdf limits of the LL distribution as

Theorem 4.1. The graph of the pdf of the LL distribution is log-concave if

Proof. Setting

Differentiating twice concerning

Note that

The hrf limits of the LL distribution as

Theorem 4.2. The hrf of the LL distribution is

• Increasing for

• Decreasing for

• Unimodal for

• Bathtub for

• Decreasing-increasing-decreasing for

Proof. From Eq. (16), we have

The derivative of

where

where

Clearly, both the hrf of the LL distribution and

Case 1: For

Case 2: If

Case 3: If

Case 4: If

To discuss other cases, we have to get two critical values of

and

Case 5: For

Case 6: The function

Case 7: For

Theorem 4.3. The

Proof. The

Let

By using the binomial and series expansions, the

Let

Let

Finally, we have

which completes the proof.

As well as the measures of skewness, kurtosis, and asymmetry of the LL distribution are obtained by the following relations:

Table 1 provides some important LL measures for various parametric values. The numerical values show that the LL distribution can be right skewed for different values of

The qf of the LL distribution follows, by inverting its cdf (15), as

where

and

The order statistic for the LL distribution will be discussed in this section. It will also be useful to derive the pdf of the

We have

and

Substituting (21) and (22) in (20), one can write

Hence, the largest order statistic density follows as

The smallest order statistic pdf reduces to

5 Estimation of the LL Parameters

In this section, different techniques are used to estimate the LL parameters.

The LL parameters are estimated by the maximum likelihood (ML). Consider a random sample from the LL distribution denoted by

where

The ML estimators (MLE) of

and

5.2 Least-Squares and Weighted Least-Squares

The least squares (LS) and the weighted LS (WLS) methods are introduced by Swain et al. [17]. The LS estimators (LSE) of the LL parameters are calculated by minimizing the following function concerning

where

and

where

and

The WLS estimators (WLSE) of the LL parameters follow by minimizing the function

where

Also, these estimators are determined by solving the following non-linear system:

where

The Cramér–von Mises (CM) method was introduced by Choi et al. [18] depending on the CM statistics (Boos [19]). Then, the CM estimators (CME) of the LL parameters minimize the following function:

with respect to

where

5.4 Anderson–Darling and Right-Tail Anderson–Darling

Depending on the Anderson–Darling (AD) statistic, the AD method was proposed by Anderson et al. [20] and [21]. Hence, the AD estimators (ADE) of the LL parameters are calculated by minimizing the function

Therefore, the ADE is also obtained by solving the following non-linear system:

where

Luceno [22] applied some motivations to the AD statistic called the right-tail AD (RAD) statistic. Hence, to obtain the RAD estimators (RADE) of the LL parameters, we minimize the following function:

Additionally, the RADE is obtained by solving the non-linear system

where

All above mentioned non-linear systems of equations have no exact solutions, so the optim and nlminb functions in R software can be adopted for this purpose.

This section presents numerical simulation results to explore the efficiency and performance of different estimators for the LL parameters. The following algorithm is adopted to evaluate different estimators:

1. Set different initial values of sample size

2. Generate several random samples of size

3. The outcomes in the previous step are used to calculate the parameter estimates,

4. The three above steps are repeated 6,000 times.

5. Based on

and

From the LL distribution 6,000 samples are generated for

The AB and RMSE of the MLE, CME, LSE, ADE, WLSE, and RADE are presented in Tables 2–4. Moreover, the partial and overall ranks of the estimators are calculated in Table 5. From the values in these tables, one can note that

1. All estimates show the property of consistency, i.e., the AB and RMSE decrease as sample size increases for all parametric combinations.

2. According to AB and RMSE, the ordering of performance of estimators (from best to worst) for all parameters is the MLE, ADE, WLSE, RADE, LSE and CME.

In this section, we analyze two real-life data sets to demonstrate the performance of the LL distribution in practice. Two real-life data sets are fitted to compare the proposed LL model with other five known competitors, namely:

1. The Exponentiated Lomax (EL) distribution [23] with pdf

2. The Poisson–Lomax (PoL) distribution [24] with pdf

3. The Lomax-Rayleigh (LR) distribution [25] with pdf

4. The power Lomax (PL) distribution [26] with pdf

5. The Lomax (L) distribution [27] with pdf

The first data represent 63 service times (thousand hours) of aircraft windshield (unit in thousand hours) as reported in Murthy et al. [28]. The data are as follows:

The second data represents 63 strengths of 1.5 cm glass fibers which are measured by the National Physical Laboratory, in England as reported in Smith et al. [29]. The data are as follows:

Table 6 provides a brief summary for both data sets.

The parameters of the fitted distributions are estimated using the ML method and some discrimination measures are calculated to explore the efficiency of the competing distributions. These measures include the Akaike information criterion (AIC), Bayesian IC (BIC), corrected AIC (CAIC), Hannan–Quinn IC (HQIC), and –ℓ, where ℓ is the maximized log-likelihood. Additionally, goodness-of-fit statistics such as Anderson–Darling (An), Cramér–von Mises (Cr), and Kolmogorov–Smirnov (K-S) with its corresponding p-value (K-S p-value) are also calculated. More details about the goodness-of-fit statistics can be explored by Shama et al. [30].

Tables 7 and 8 present the MLEs of the parameters of the fitted distributions along with their discrimination measures for both data sets, respectively. Based on Tables 7 and 8, we conclude that the LL distribution is the best model as compared to other Lomax extensions. Tables 9 and 10 provide the values of goodness-of-fit measures of the LL model and other models for the two data sets. These results also indicate that the LL model provides a better fit to aircraft windshield and glass fibers data as compared to other Lomax models.

The probability-probability (P-P) plots of the fitted distributions for both data sets are provided in Figs. 4 and 5. The fitted density, cdf, sf, and P-P plots of the LL distribution for both data sets are displayed in Figs. 6 and 7. The plots support the results in Tables 7–10 and illustrate that the LL distribution provides a better approximation between the theoretical and empirical curves. Furthermore, Figs. 8 and 9 present the histograms of aircraft windshield and glass fibers data along with the fitted densities of the LL model and other studied distributions. All plots provide evidence that the LL distribution is the most well-adjusted model for aircraft windshield and glass fibers data.

This study introduces a new flexible family called the LG family. Its special sub-models can represent various shapes of aging failure criteria, including monotonic and non-monotonic failure rates. The densities of the sub-models of the LG family can be reversed-J shaped, right-skewed, symmetric, left-skewed, decreasing-increasing-decreasing densities. One of its special models, namely the LL, is studied in detail. The failure rate shapes of the LL distribution are derived and proved mathematically. In addition, various statistical properties of the LL distribution are investigated. Six estimation methods are employed to estimate the LL parameters, and their performance is explored via simulation results. The numerical experiments illustrate the accuracy of the maximum likelihood; hence, they are recommended for estimating the LL parameters. Two real-life datasets are analyzed, indicating that the LL distribution can provide a better fit for modeling actual data compared to some competing Lomax models.

The perspectives of this study can include the development of a bivariate LL distribution and the construction of a discrete version of the LL model.

Acknowledgement: The authors would like to thank the editor and the referees for valuable comments which greatly improved the paper.

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

Author Contributions: The authors confirm contribution to the paper as follows: study conception and design: J.N.A.A., A.Z.A., B.A., M.S.S.; data collection: A.Z.A., M.S.S.; analysis and interpretation of results: J.N.A.A., A.Z.A., B.A., M.S.S.; draft manuscript preparation: J.N.A.A. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: This work is mainly a methodological development and has been applied on secondary data which are provided in the manuscript.

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

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**APA Style**

*Computer Modeling in Engineering & Sciences*,

*140*

*(1)*, 513-536. https://doi.org/10.32604/cmes.2024.046533

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*Comput. Model. Eng. Sci.*, vol. 140, no. 1, pp. 513-536. 2024. https://doi.org/10.32604/cmes.2024.046533

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