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.

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. [

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 [

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 [

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 [

The paper is organized in the following sections. In

where

The probability density function (pdf) corresponding to

The hrf of the LG family becomes

According to

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

and

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

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

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

The pdf limits of the LL distribution as

Differentiating twice concerning

Note that

The hrf limits of the LL distribution as

Increasing for

Decreasing for

Unimodal for

Bathtub for

Decreasing-increasing-decreasing for

The derivative of

where

Clearly, both the hrf of the LL distribution and

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

and

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:

Actual values | Mean | Variance | Skewness | Kurtosis | Asymmetry | CV | ||
---|---|---|---|---|---|---|---|---|

0.50 | 0.5353 | 0.7821 | 16.1629 | 31.3442 | 4.0203 | 1.6522 | ||

0.50 | 1.00 | 1.0706 | 3.1285 | 16.1629 | 31.3442 | 4.0203 | 1.6522 | |

1.50 | 1.6059 | 7.0391 | 16.1629 | 31.3442 | 4.0203 | 1.6522 | ||

0.50 | 0.2775 | 0.1085 | 4.0298 | 8.4889 | 2.0074 | 1.1871 | ||

0.5 | 1.00 | 1.00 | 0.5551 | 0.4342 | 4.0298 | 8.4889 | 2.0074 | 1.1871 |

1.50 | 0.8326 | 0.9769 | 4.0298 | 8.4889 | 2.0074 | 1.1871 | ||

0.50 | 0.1986 | 0.0427 | 2.2224 | 5.5215 | 1.4908 | 1.0399 | ||

1.50 | 1.00 | 0.3973 | 0.1706 | 2.2224 | 5.5215 | 1.4908 | 1.0399 | |

1.50 | 0.5959 | 0.3839 | 2.2224 | 5.5215 | 1.4908 | 1.0399 | ||

0.50 | 0.5280 | 0.3170 | 5.7621 | 12.4969 | 2.4004 | 1.0665 | ||

0.50 | 1.00 | 1.0559 | 1.2681 | 5.7621 | 12.4969 | 2.4004 | 1.0665 | |

1.50 | 1.5839 | 2.8532 | 5.7621 | 12.4969 | 2.4004 | 1.0665 | ||

0.50 | 0.3521 | 0.0856 | 1.8671 | 5.4262 | 1.3664 | 0.8311 | ||

1.00 | 1.00 | 1.00 | 0.7042 | 0.3425 | 1.8671 | 5.4262 | 1.3664 | 0.8311 |

1.50 | 1.0563 | 0.7707 | 1.8671 | 5.4262 | 1.3664 | 0.8311 | ||

0.50 | 0.2758 | 0.0409 | 0.9781 | 3.9475 | 0.9890 | 0.7330 | ||

1.50 | 1.00 | 0.5516 | 0.1635 | 0.9781 | 3.9475 | 0.9890 | 0.7330 | |

1.50 | 0.8275 | 0.3679 | 0.9781 | 3.9475 | 0.9890 | 0.7330 | ||

0.50 | 0.5499 | 0.1879 | 2.5495 | 6.9434 | 1.5967 | 0.7882 | ||

0.50 | 1.00 | 1.0998 | 0.7515 | 2.5495 | 6.9434 | 1.5967 | 0.7882 | |

1.50 | 1.6497 | 1.6909 | 2.5495 | 6.9434 | 1.5967 | 0.7882 | ||

0.50 | 0.4093 | 0.0706 | 0.9585 | 4.1124 | 0.9790 | 0.6491 | ||

1.50 | 1.00 | 1.00 | 0.8187 | 0.2824 | 0.9585 | 4.1124 | 0.9790 | 0.6491 |

1.50 | 1.2280 | 0.6354 | 0.9585 | 4.1124 | 0.9790 | 0.6491 | ||

0.50 | 0.3358 | 0.0378 | 0.4804 | 3.3091 | 0.6931 | 0.5793 | ||

1.50 | 1.00 | 0.6716 | 0.1513 | 0.4804 | 3.3091 | 0.6931 | 0.5793 | |

1.50 | 1.0074 | 0.3405 | 0.4804 | 3.3091 | 0.6931 | 0.5793 |

The qf of the LL distribution follows, by inverting its cdf

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

Hence, the largest order statistic density follows as

The smallest order statistic pdf reduces to

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

The least squares (LS) and the weighted LS (WLS) methods are introduced by Swain et al. [

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. [

with respect to

where

Depending on the Anderson–Darling (AD) statistic, the AD method was proposed by Anderson et al. [

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

where

Luceno [

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:

Set different initial values of sample size

Generate several random samples of size

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

The three above steps are repeated 6,000 times.

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

n | Est. | Est. Par. | MLE | LSE | WLSE | CME | ADE | RADE |
---|---|---|---|---|---|---|---|---|

0.0264 | 0.0581 | 0.1284 | 0.1578 | 0.1051 | 0.1754 | |||

AB | 0.3019 | 1.4720 | 1.3029 | 1.7125 | 0.6682 | 1.1226 | ||

20 | 0.6472 | 3.1034 | 2.2290 | 2.5726 | 1.2439 | 2.0723 | ||

0.0653 | 0.0961 | 0.1917 | 0.2471 | 0.1763 | 0.2772 | |||

RMSE | 2.4905 | 3.2428 | 3.0630 | 3.8694 | 1.6943 | 2.5228 | ||

4.3459 | 5.8860 | 4.5391 | 5.1254 | 2.7913 | 4.2717 | |||

0.0030 | 0.0330 | 0.0721 | 0.0843 | 0.0427 | 0.0919 | |||

AB | 0.0091 | 0.6705 | 0.4907 | 0.6270 | 0.2050 | 0.4111 | ||

50 | 0.0243 | 1.5432 | 0.9413 | 1.0911 | 0.4128 | 0.8535 | ||

0.0185 | 0.0441 | 0.0976 | 0.1166 | 0.0778 | 0.1308 | |||

RMSE | 0.0728 | 1.4914 | 1.0426 | 1.3586 | 0.4923 | 0.7953 | ||

0.1906 | 3.1088 | 1.7544 | 2.1046 | 0.8887 | 1.5772 | |||

0.0002 | 0.0235 | 0.0495 | 0.0560 | 0.0187 | 0.0601 | |||

AB | 0.0003 | 0.3304 | 0.2686 | 0.3341 | 0.0797 | 0.2492 | ||

100 | 0.0008 | 0.7874 | 0.5549 | 0.6240 | 0.1636 | 0.5225 | ||

0.0040 | 0.0305 | 0.0641 | 0.0736 | 0.0424 | 0.0794 | |||

RMSE | 0.0080 | 0.5500 | 0.4027 | 0.5182 | 0.1830 | 0.3524 | ||

0.0231 | 1.2495 | 0.8125 | 0.9439 | 0.3792 | 0.7404 | |||

1.824E-5 | 0.0173 | 0.0337 | 0.0389 | 0.0042 | 0.0409 | |||

AB | 3.647E-5 | 0.2194 | 0.1714 | 0.2127 | 0.0178 | 0.1677 | ||

200 | 4.895E-5 | 0.5296 | 0.3555 | 0.4061 | 0.0361 | 0.3589 | ||

0.0014 | 0.0222 | 0.0434 | 0.0497 | 0.0166 | 0.0525 | |||

RMSE | 0.0028 | 0.2969 | 0.2324 | 0.2994 | 0.0700 | 0.2277 | ||

0.0038 | 0.7261 | 0.4757 | 0.5632 | 0.1452 | 0.4852 |

n | Est. | Est. Par. | MLE | LSE | WLSE | CME | ADE | RADE |
---|---|---|---|---|---|---|---|---|

0.1031 | 0.1550 | 0.1364 | 0.1744 | 0.1113 | 0.2030 | |||

AB | 1.5750 | 3.4991 | 3.0284 | 3.5275 | 1.6103 | 2.8384 | ||

20 | 0.7193 | 1.6832 | 1.4733 | 1.5649 | 0.7893 | 1.4113 | ||

0.1861 | 0.2483 | 0.2167 | 0.2836 | 0.1976 | 0.3421 | |||

RMSE | 3.7365 | 6.6284 | 5.9463 | 6.6117 | 3.8827 | 5.7166 | ||

1.6107 | 3.2113 | 2.8603 | 2.9109 | 1.8347 | 2.8283 | |||

0.0421 | 0.0898 | 0.0781 | 0.0918 | 0.0443 | 0.1018 | |||

AB | 0.3909 | 1.6592 | 1.2153 | 1.6029 | 0.4151 | 1.1408 | ||

50 | 0.2039 | 0.8158 | 0.6186 | 0.7613 | 0.2209 | 0.6092 | ||

0.0812 | 0.1207 | 0.1060 | 0.1297 | 0.0867 | 0.1455 | |||

RMSE | 0.9886 | 3.6132 | 2.6606 | 3.3993 | 0.9859 | 2.4690 | ||

0.4764 | 1.6838 | 1.2597 | 1.5159 | 0.5045 | 1.2460 | |||

0.0164 | 0.0606 | 0.0529 | 0.0621 | 0.0153 | 0.0649 | |||

AB | 0.1142 | 0.8530 | 0.6218 | 0.8429 | 0.1220 | 0.6206 | ||

100 | 0.0624 | 0.4337 | 0.3358 | 0.4209 | 0.0668 | 0.3467 | ||

0.0435 | 0.0791 | 0.0694 | 0.0816 | 0.0410 | 0.085 | |||

RMSE | 0.3236 | 1.5718 | 1.0518 | 1.6204 | 0.3309 | 1.0746 | ||

0.1719 | 0.7609 | 0.5323 | 0.7661 | 0.1812 | 0.5814 | |||

0.0036 | 0.0424 | 0.0368 | 0.0422 | 0.0031 | 0.0452 | |||

AB | 0.0201 | 0.5118 | 0.3919 | 0.5163 | 0.0221 | 0.4035 | ||

200 | 0.0114 | 0.2665 | 0.21587 | 0.2662 | 0.0121 | 0.2285 | ||

0.0179 | 0.0542 | 0.0472 | 0.0541 | 0.0157 | 0.0584 | |||

RMSE | 0.0991 | 0.7735 | 0.5505 | 0.7676 | 0.1079 | 0.5725 | ||

0.0559 | 0.3948 | 0.3019 | 0.3894 | 0.0582 | 0.3225 |

n | Est. | Est. Par. | MLE | LSE | WLSE | CME | ADE | RADE |
---|---|---|---|---|---|---|---|---|

0.4811 | 0.5322 | 0.5088 | 0.5657 | 0.4728 | 0.5389 | |||

AB | 1.1040 | 2.0602 | 1.757 | 2.0698 | 1.1153 | 1.6573 | ||

20 | 0.8345 | 1.6631 | 1.3916 | 1.5393 | 0.8873 | 1.4196 | ||

0.5817 | 0.6417 | 0.6108 | 0.6962 | 0.5707 | 0.6441 | |||

RMSE | 2.574 | 4.9099 | 4.3821 | 4.8676 | 2.8235 | 4.16114 | ||

1.9135 | 3.9267 | 3.3962 | 3.5889 | 2.2378 | 3.6226 | |||

0.33430 | 0.3964 | 0.3667 | 0.4070 | 0.3472 | 0.4116 | |||

AB | 0.4330 | 0.7562 | 0.5572 | 0.7544 | 0.4472 | 0.5720 | ||

50 | 0.3337 | 0.5948 | 0.4391 | 0.5740 | 0.3485 | 0.4710 | ||

0.4040 | 0.4713 | 0.4364 | 0.4875 | 0.4181 | 0.4846 | |||

RMSE | 0.7342 | 1.7734 | 1.0560 | 1.7807 | 0.6869 | 1.1838 | ||

0.5925 | 1.3937 | 0.8317 | 1.3576 | 0.5711 | 1.0537 | |||

0.2501 | 0.3121 | 0.2808 | 0.3158 | 0.2718 | 0.3284 | |||

AB | 0.2798 | 0.4248 | 0.3332 | 0.4202 | 0.3032 | 0.3554 | ||

100 | 0.2105 | 0.3244 | 0.2520 | 0.3109 | 0.2300 | 0.2768 | ||

0.3084 | 0.3726 | 0.3368 | 0.3785 | 0.3310 | 0.3911 | |||

RMSE | 0.3727 | 0.6582 | 0.4588 | 0.7003 | 0.4066 | 0.5013 | ||

0.2946 | 0.5332 | 0.3667 | 0.5300 | 0.3248 | 0.4301 | |||

0.1842 | 0.2391 | 0.2058 | 0.2452 | 0.2025 | 0.2500 | |||

AB | 0.1898 | 0.2877 | 0.2254 | 0.2934 | 0.2195 | 0.2535 | ||

200 | 0.1421 | 0.2098 | 0.1652 | 0.2108 | 0.1618 | 0.1903 | ||

0.2313 | 0.2897 | 0.2558 | 0.2966 | 0.2514 | 0.3054 | |||

RMSE | 0.2444 | 0.3778 | 0.2884 | 0.3877 | 0.2819 | 0.3252 | ||

0.1879 | 0.2893 | 0.2186 | 0.2915 | 0.2157 | 0.2587 |

Est. Par. | Initial values | MLE | LSE | WLSE | CME | ADE | RADE |
---|---|---|---|---|---|---|---|

First | 8 | 19 | 32 | 40 | 21 | 48 | |

Second | 12 | 34 | 24 | 38 | 12 | 48 | |

Third | 11 | 46 | 28 | 48 | 17 | 32 | |

First | 10 | 48 | 35 | 47 | 15 | 26 | |

Second | 11 | 53 | 36 | 51 | 17 | 35 | |

Third | 11 | 46 | 28 | 48 | 17 | 32 | |

First | 11 | 54 | 35 | 45 | 17 | 30 | |

Second | 8 | 49 | 29 | 43 | 16 | 29 | |

Third | 9 | 46 | 24 | 41 | 15 | 33 | |

Sum | 90 | 381 | 267 | 396 | 144 | 326 | |

Overall rank | 1 | 5 | 3 | 6 | 2 | 4 |

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

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:

The Exponentiated Lomax (EL) distribution [

The Poisson–Lomax (PoL) distribution [

The Lomax-Rayleigh (LR) distribution [

The power Lomax (PL) distribution [

The Lomax (L) distribution [

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

0.046 | 1.436 | 2.592 | 0.140 | 1.492 | 2.600 | 0.150 | 1.580 |

2.670 | 0.248 | 1.719 | 2.717 | 0.280 | 1.794 | 2.819 | 0.313 |

1.915 | 2.820 | 0.389 | 1.920 | 2.878 | 0.487 | 1.963 | 2.950 |

0.622 | 1.978 | 3.003 | 0.900 | 2.053 | 3.102 | 0.952 | 2.065 |

3.304 | 0.996 | 2.117 | 3.483 | 1.003 | 2.137 | 3.500 | 1.010 |

2.141 | 3.622 | 1.085 | 2.163 | 3.665 | 1.092 | 2.183 | 3.695 |

1.152 | 2.240 | 4.015 | 1.183 | 2.341 | 4.628 | 1.244 | 2.435 |

4.806 | 1.249 | 2.464 | 4.881 | 1.262 | 2.543 | 5.140 |

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. [

0.55 | 1.64 | 1.39 | 1.82 | 1.60 | 1.13 | 1.70 | 1.55 |

0.93 | 1.68 | 1.49 | 2.01 | 1.62 | 1.29 | 1.77 | 1.61 |

1.25 | 1.73 | 1.53 | 0.77 | 1.66 | 1.48 | 1.84 | 1.63 |

1.36 | 1.81 | 1.59 | 1.11 | 1.69 | 1.50 | 0.84 | 1.67 |

1.49 | 2.00 | 1.61 | 1.28 | 1.76 | 1.55 | 1.24 | 1.70 |

1.52 | 0.74 | 1.66 | 1.42 | 1.84 | 1.61 | 1.30 | 1.78 |

1.58 | 1.04 | 1.68 | 1.50 | 2.24 | 1.62 | 1.48 | 1.89 |

1.61 | 1.27 | 1.76 | 1.54 | 0.81 | 1.66 | 1.51 |

Data | Min | Q1 | median | Mean | Q3 | SD | Skewness | Kurtosis | Max |
---|---|---|---|---|---|---|---|---|---|

Aircraft windshield | 0.0460 | 1.1220 | 2.0650 | 2.0850 | 2.8200 | 1.2452 | 0.4292 | −0.3535 | 5.1400 |

Glass fibers | 0.5500 | 1.3750 | 1.5900 | 1.5070 | 1.6850 | 0.3241 | 0.0000 | 0.00000 | 2.2400 |

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

Models | Estimates | –ℓ | AIC | BIC | CAIC | HQIC | ||
---|---|---|---|---|---|---|---|---|

LL ( |
0.9211 | 3.9175 | 7.2869 | 98.12831 | 202.2566 | 208.6861 | 211.6861 | 204.7853 |

EL ( |
1.9012 | 228722.8 | 329853.6 | 103.5468 | 213.0936 | 219.5232 | 222.5232 | 215.6224 |

PoL ( |
216.4421 | 0.0041 | 3.3768 | 100.4224 | 206.8449 | 213.2743 | 207.2516 | 209.3736 |

LR ( |
15.99717 | 88.1644 | — | 102.4106 | 208.8212 | 213.1075 | 215.1075 | 210.5071 |

PL ( |
1.8978 | 79935.96 | 115504.9 | 103.5469 | 213.0937 | 219.5231 | 213.5005 | 215.6224 |

L ( |
17638.99 | 36780.66 | — | 109.2997 | 222.5995 | 226.8858 | 228.8858 | 224.2853 |

Models | Estimates | –ℓ | AIC | BIC | CAIC | HQIC | ||
---|---|---|---|---|---|---|---|---|

LL ( |
2.1658 | 29.8357 | 10.7791 | 14.3206 | 34.6413 | 41.0707 | 35.0481 | 37.1700 |

EL ( |
31.3556 | 23822.5418 | 9120.6519 | 31.3852 | 68.7704 | 75.1998 | 69.1772 | 71.2991 |

PoL ( |
331.0121 | 0.0081 | 34.8844 | 30.6526 | 67.3052 | 73.7347 | 67.7120 | 69.8340 |

LR ( |
2687.635 | 6379.087 | — | 49.8010 | 103.6019 | 107.8882 | 103.8019 | 105.2878 |

PL ( |
31.3664 | 7007.7084 | 2682.2111 | 31.3893 | 68.7786 | 75.2081 | 69.1854 | 71.3074 |

L ( |
26613.03 | 40103.90 | — | 88.8314 | 181.6629 | 185.9492 | 181.8629 | 183.3487 |

Models | Cr | An | K-S | K-S |
---|---|---|---|---|

LL | 0.0347 | 0.2403 | 0.0667 | 0.9241 |

EL | 0.2341 | 1.3196 | 0.1442 | 0.1321 |

PoL | 0.0992 | 0.6039 | 0.1062 | 0.4451 |

LR | 0.0757 | 1.1182 | 0.0849 | 0.7217 |

PL | 0.2035 | 1.2315 | 0.1438 | 0.1340 |

L | 0.7790 | 3.8821 | 0.2078 | 0.0073 |

Models | Cr | An | K-S | K-S |
---|---|---|---|---|

LL | 0.1710 | 0.9615 | 0.1358 | 0.1953 |

EL | 0.7862 | 4.2873 | 0.2290 | 0.0027 |

PoL | 0.7565 | 4.1374 | 0.2224 | 0.0039 |

LR | 0.4656 | 2.5544 | 0.3339 | 0.0000 |

PL | 0.7863 | 4.2878 | 0.2290 | 0.0027 |

L | 18.5583 | 121.9004 | 0.7739 | 0.0000 |

The probability-probability (P-P) plots of the fitted distributions for both data sets are provided in

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.

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

The authors received no specific funding for this study.

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.

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

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