New Weighted Lomax (NWL) Distribution with Applications to Real and Simulated Data

Abstract

The rationale of the paper is to present a new probability distribution that can model both the monotonic and nonmonotonic hazard rate shapes and to increase their flexibility among other probability distributions available in the literature. The proposed probability distribution is called the New Weighted Lomax (NWL) distribution. Various statistical properties have been studied including with the estimation of the unknown parameters. To achieve the basic objectives, applications of NWL are presented by means of two real-life data sets as well as a simulated data. It is verified that NWL performs well in both monotonic and nonmonotonic hazard rate function than the Lomax (L), Power Lomax (PL), Exponential Lomax (EL), and Weibull Lomax (WL) distribution.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

From the last few years, it is usual practice to make a contribution to the existing theory of probability due to its wide application in different fields of sciences, for example, in reliability analysis, signal processing, survival analysis, and so on. Due to the advanced computer technology and statistical software, many researchers have developed new probability distributions to improve the goodness of fit measures. For example, Lemonte et al. [1] introduced the additive Weibull distribution by adding the two Weibull distributions, Al-Aqtash et al. [2] presented the new family of distribution with a logit function, Aldeni et al. [3, 4] explored by employing the quantile function, Alzaatreh et al. investigated the gamma-normal distribution [5], and references [6–11] presented new probability distributions using transmutation technique. Alzaghal et al. [12] introduced an exponentiated T-X family of distribution. Extended Lomax distribution was introduced by Lemonte and Cordeiro [13].

The fundamental goal of this paper is to present a new life-time probability distribution that improves the flexibility of the model and also provides a better fit in monotonic and nonmonotonic hazard function than other existing probability models.

1.1. Lomax Distribution

Let a positive random variable be $Y\sim L\alpha ,\beta$; the CDF is given by$$\begin{array}{}\text{(1)}& Fy=1-{1+\frac{y}{\beta }}^{-\alpha },\hspace{1em}y>0\text{\hspace{0.17em}and\hspace{0.17em}}\alpha ,\beta >0.\end{array}$$

The PDF related to (1) is defined as$$\begin{array}{}\text{(2)}& fy=\frac{\alpha }{\beta }{1+\frac{y}{\beta }}^{-\alpha +1},\hspace{1em}y>0.\end{array}$$

Equation (2) is one of the right skewed distributions and has been applied by many researchers to real data sets found in business science, engineering, computer, survival analysis, and some others.

To increase the flexibility of the model, modification of this distribution has been done by many researchers; for example, Ashour and Eltehiwy [10] introduced transmuted Lomax distribution, Ashour and Eltehiwy [11] transmuted Exponentiated Lomax distribution, Lemonte and Cordeiro [13] explored the extended Lomax, Cordeiro et al. [14] defined gamma-Lomax, Ghitany et al. [15] presented Marshall–Olkin extended Lomax and discussed their applications to censored data, Al-Zahrani and Sagor [16] modified Poisson Lomax distribution. El-Bassiouny et al. [17] defined Exponential Lomax, and Shams [18] presented Kumaraswamy-generalized Lomax distribution. Ijaz et al. [19] worked on the Flexible Lomax distribution, ZeinEldin et al. [20] presented Alpha power transformed inverse Lomax distribution, Almetwally and Gamal [21] defined Alpha Power Inverse Lomax, ul Haq et al. [22] discussed Marshall–Olkin power Lomax distribution, Lemonte and Cordeiro [13] explored an Extended Lomax, and Cordeiro et al. [14] worked on gamma-Lomax distribution. Kilany [23] worked on the Weighted Lomax distribution, and Ahmad et al. [24] derived a Length-Biased Weighted Lomax distribution. For other modifications, refer [25–38].

1.2. A New Weighted Lomax (NWL) Distribution

In this paper, we developed a highly flexible Lomax distribution by replacing a Lomax random variable $y$ by $e^y$ and using the inner product of $\beta /\beta +1$. The suggested distribution is called a New Weighted Lomax distribution or in short NWL distribution. The shape and scale parameter of this distribution are $\alpha$ and $\beta$, respectively. The proposed distribution in this paper provides more flexibility and provides the best fit than other existing distributions.

Definition 1.

. Considering a continuous random variable $Y$, the CDF of a New Weighted Lomax distribution is defined by$$\begin{array}{}\text{(3)}& Fy=1-{\frac{\beta }{\beta +1}1+\frac{e^y}{\beta }}^{-\alpha },\hspace{1em}y>0\text{\hspace{0.17em}and\hspace{0.17em}}\alpha ,\beta >0.\end{array}$$

The corresponding PDF is given by$$\begin{array}{}\text{(4)}& Fy=\frac{\alpha }{\alpha \beta }{\frac{\beta +1}{\beta }}^{\alpha }{e}^y{1+\frac{e^y}{\beta }}^{-\alpha +1},\hspace{1em}\text{where\hspace{0.17em}}\alpha ,\beta >0.\end{array}$$

Figure 1 shows the behavior of the PDF and CDF of the $\text{NWL}\alpha ,\beta$ distribution.

Figure 1 shows the probability and distribution function of the New Weighted Lomax distribution with different parameter values.

[figure omitted; refer to PDF]

2. Survival and Hazard Function

The survival function of $\text{NWL}\alpha ,\beta$ is defined by the expression as under$$\begin{array}{}\text{(5)}& Sy=PY>y,\hspace{1em}y>0.\end{array}$$

Using (3), we get$$\begin{array}{}\text{(6)}& Sy=1-1-{\frac{\beta }{\beta +1}1+\frac{e^y}{\beta }}^{-\alpha }={\frac{\beta }{\beta +1}1+\frac{e^y}{\beta }}^{-\alpha }.\end{array}$$

The hazard function or failure rate of a NWL distribution is defined by using the formula as under $hy=fy/1-Fy$; recalling (3) and (4), we have$$\begin{array}{}\text{(7)}& hy;\alpha ,\beta =\frac{\alpha }{\alpha \beta }{e}^y{1+\frac{e^y}{\beta }}^{-1};\hspace{1em}y>0,\alpha ,\beta >0.\end{array}$$

Figure 2 delineates the capability of the suggested distribution to model the nonmonotonically hazard function.

[figure omitted; refer to PDF]

Table 2 reflects the values of ml estimates, and their corresponding standard error is attached in the parentheses. Table 3 defines the values of the goodness of fit measures, and it has been observed that the values of AIC, CAIC, BIC, and HQIC are less while W and A statistics are larger for the New Weighted Lomax distribution than other probability models. Hence, a new WL leads to a better fit than Lomax (L), Power Lomax (PL), Exponential Lomax (EL), and Weibull Lomax (WL).

Table 2

ML estimates.

Model	Estimates
	$\widehat{\alpha }$	$\widehat{\beta }$	$\widehat{\gamma }$	$\widehat{\vartheta }$
NW-Lomax	0.1115898 (0.0178886)	24.7376349 (4.6011156)
Lomax	2.259102 (0.9034286)	13.107217 (6.7006737)
P-Lomax	0.1612794 (0.0677728)	5.4172791 (1.7104091)	20.3051382 (13.2356863)
$W$-Lomax	2.8345778 (1.10337502)	1.9742578 NaN	1.0284592 NaN	0.2073842 (0.03295629)
E-Lomax	28.842426 (36.1915908)	1.481920 (0.2297605)	2.482791 (2.5815765)

Table 3

Goodness of fit measures for losses due to wind catastrophes.

Model	AIC	CAIC	BIC	HQIC	−log	$W$	A
NW-Lomax	52.2338	52.56714	55.56093	53.42755	24.1169	0.5933578	3.367013
Lomax	252.6833	253.016	256.0104	253.877	124.3416	0.2949964	1.946171
P-Lomax	235.6173	236.303	240.608	237.4079	114.8086	0.1678204	1.406701
$W$-Lomax	249.5339	250.7104	256.1881	251.9214	120.7669	0.2982084	1.942425
E-Lomax	237.7877	238.4734	242.7784	239.5783	115.8939	0.1700244	1.383728

Figure 4 shows the empirical and theoretical PDF and CDF of the proposed distribution $\text{WL}\alpha ,\beta$ and other existing distributions for the losses due to wind catastrophes.

[figure omitted; refer to PDF]

The ml estimates and their standard error in braces are given in Table 4. Table 5 explains the goodness of fit measures, and it has been noted that the proposed model provides a better fit to these data as compared with other probability models including Lomax (L), Power Lomax (PL), Exponential Lomax (EL), and Weibull Lomax (WL).

Table 4

ML estimates.

Model	Estimates
	$\widehat{\alpha }$	$\widehat{\beta }$	$\widehat{\gamma }$	$\widehat{\vartheta }$
NW-Lomax	0.10613 (0.009386)	31.98979 (3.188287)
Lomax	3.8661 (1.1079)	28.4134 (9.4998)
P-Lomax	1.404926 (0.4085148)	1.438151 (0.1522463)	19.819986 (5.4992568)
$W$-Lomax	5.4732949 (0.0506327)	1.5096438 NaN	4.7310404 NaN	0.2643016 NaN
E-Lomax	10.0160681 (2.41150223)	9.7029282 (1.84701652)	0.1310464 (0.03225501)

Table 5

Goodness of fit measures for bladder cancer patient.

Model	AIC	CAIC	BIC	HQIC	−log	$W$	A
NW-Lomax	100.39	100.49	106.10	102.71	48.199	0.56604	3.7004
Lomax	835.54	835.64	841.25	837.86	415.77	0.034874	0.224
P-Lomax	827.8986	828.0921	836.4547	831.375	410.9493	0.02481589	0.1860483
$W$-Lomax	828.6928	829.018	840.1009	833.328	410.3464	0.03741088	0.2432686
E-Lomax	305.2265	305.4765	313.0421	308.3896	149.6133	0.3014044	1.615242

Figure 6 shows the empirical and theoretical CDF and CDF of the proposed distribution $\text{WL}\alpha ,\beta$ and other existing distributions for the Bladder cancer patients.

[figure omitted; refer to PDF]

13. Simulations

The simulation study also plays an important role in making a decision that whether the given model provides a better fit or not. In order to get random data from the New Weighted Lomax distribution, equation (12) would be considered. The random experiment is replicated 100 times with different samples of sizes n with different values of parameters. The result given in Table 6 declares that both the Bias and MSE are continuously decreased as the sample size increases.

Table 6

Bias and MSE of $\text{NWL}\alpha ,\beta$ distribution.

α	Β	N	MSE (α)	MSE (β)	Bias (α)	Bias (β)
0.033	16.87	30	3.996406e − 05	15.18753	0.00570363	3.460371
		60	3.967258e − 05	2.213798	0.005476957	0.9058611
		80	6.484166e − 06	0.183196	0.002486657	0.3842137
	17.1	30	3.170174e − 05	14.62279	0.00233932	3.752701
		60	1.079283e − 05	4.312617	0.002267292	1.901363
		80	4.80265e − 06	2.457985	0.0009147354	0.8291876
	17.5	30	2.453543e − 05	16.9866	0.003788296	3.993388
		60	1.591147e − 05	8.288684	0.003596319	2.036911
		80	3.825583e − 06	3.029128	0.0007795378	0.8864978
0.036	21.29	30	4.484847e − 05	54.86977	0.005403825	7.251341
		60	2.466488e − 05	28.85079	0.004804561	5.304583
		90	6.124085e − 06	10.57544	0.002321392	3.138554
	21.35	30	4.484847e − 05	55.74353	0.005403825	7.311341
		60	2.466488e − 05	30.46467	0.004804561	5.454583
		90	6.085762e − 06	10.95351	0.002315325	3.198279
	22.45	30	5.25542e − 05	57.16769	0.005964773	7.383071
		60	1.331295e − 05	36.26379	0.003408052	6.00667
		90	5.233742e − 06	7.576437	0.002065704	2.621302
0.04	23.2	90	3.126413e − 05	32.53668	0.005583154	5.696644
		120	1.527273e − 05	7.007146	0.003620997	2.272011
		150	1.384378e − 05	2.284185	0.003689726	1.413837
	24.2	90	1.818826e − 05	36.94084	0.004142978	5.972791
		120	1.527273e − 05	12.55117	0.003620997	3.272011
		150	1.426254e − 05	6.008216	0.003729036	2.364439
	23.4	90	3.126413e − 05	32.53668	0.005583154	5.696644
		120	1.527273e − 05	7.007146	0.003620997	2.272011
		150	1.384378e − 05	2.88972	0.003689726	1.613837

14. Conclusion

The basic aim of this paper is to make a further contribution to the existing theory of the probability models. The paper presents a New Weighted Lomax (NWL) distribution model with two parameters, which is very versatile than others. Various statistical properties are discussed like hazard function, mean residual life function, and stress strength function. To make a comparison with other existing distributions, we have considered two real data sets. The first data set follows a monotonic hazard shape while the second data set (bladder cancer patients) has a nonmonotonic (bathtub) hazard shape. The results demonstrated in both data sets that a new WL model is too much better and provides an adequate fit than the Lomax, P-Lomax, W-Lomax, and E-Lomax distribution.

Acknowledgments

This research was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program.

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