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

Authors: Huda M. Alshanbari; Muhammad Ijaz; Syed Muhammad Asim; Abd Al-Aziz Hosni El-Bagoury; Javid Gani Dar
Journal: Mathematical Problems in Engineering
 (2021)

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## 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.


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## Body

## 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 beY∼Lα,β; the CDF is given by(1)Fy=1−1+yβ−α,y>0andα,β>0.The PDF related to (1) is defined as(2)fy=αβ1+yβ−α+1,y>0.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 variabley by ey and using the inner product of β/β+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 α and β, 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 variableY, the CDF of a New Weighted Lomax distribution is defined by(3)Fy=1−ββ+11+eyβ−α,y>0andα,β>0.
The corresponding PDF is given by(4)Fy=ααββ+1βαey1+eyβ−α+1,whereα,β>0.
Figure1 shows the behavior of the PDF and CDF of the NWLα,β distribution.
Figure1 shows the probability and distribution function of the New Weighted Lomax distribution with different parameter values.Figure 1
The PDF and CDF of theNWLα,β distribution.


## 1.1. Lomax Distribution

Let a positive random variable beY∼Lα,β; the CDF is given by(1)Fy=1−1+yβ−α,y>0andα,β>0.The PDF related to (1) is defined as(2)fy=αβ1+yβ−α+1,y>0.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 variabley by ey and using the inner product of β/β+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 α and β, 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 variableY, the CDF of a New Weighted Lomax distribution is defined by(3)Fy=1−ββ+11+eyβ−α,y>0andα,β>0.
The corresponding PDF is given by(4)Fy=ααββ+1βαey1+eyβ−α+1,whereα,β>0.
Figure1 shows the behavior of the PDF and CDF of the NWLα,β distribution.
Figure1 shows the probability and distribution function of the New Weighted Lomax distribution with different parameter values.Figure 1
The PDF and CDF of theNWLα,β distribution.


## 2. Survival and Hazard Function

The survival function ofNWLα,β is defined by the expression as under(5)Sy=PY>y,y>0.Using (3), we get(6)Sy=1−1−ββ+11+eyβ−α=ββ+11+eyβ−α.The hazard function or failure rate of a NWL distribution is defined by using the formula as underhy=fy/1−Fy; recalling (3) and (4), we have(7)hy;α,β=ααβey1+eyβ−1;y>0,α,β>0.Figure2 delineates the capability of the suggested distribution to model the nonmonotonically hazard function.Figure 2
The hazard function ofNWLα,β distribution.


## 3. Mode

The mode or a point by which the probability density function of a NWL will reach to its maximum point is defined as(8)f′y=ddyααββ+1βαey1+eyβ−α+1.In order to find the maximum point, we have to equate this expression equal to zero and then solve forY, and we get(9)αey−β=0.The mode is obtained as follows:(10)ym=logβα.


## 4. Quantile and Median Function

The QF is the real solution to the inverse cumulative distribution function of NWL distribution having two parameters. This function will help in providing the median but also in generating random data from NWL distribution. The QF is defined asFy=u where u∼U0,1.By using equation (3), we have(11)1−ββ+11+eyβ−α=u.When we solve the above function for a variableY, we obtained(12)yQ=logββ/β+1α1−u1/α−β.Now, if we are interested to find the median of the data understudy, we can easily measure the median value using the above equation by just placingu=0.5. Hence, the median function is obtained as follows:(13)yM=logββ/β+1α0.51/α−β.


## 5. Bowley Skewness (S) and Moors Kurtosis (K)

The mathematical equation of the Bowley Skewness and Moors Kurtosis [39, 40] is given by(14)SK=Q3/4+Q1/4−2Q2/4Q3/4−Q1/4,KM=Q7/8+Q3/8−Q5/8−Q1/8Q3/4−Q1/4,where Q represents different quartile values. The numerical values of Skewness and Kurtosis using different parameter values are given in Table 1.Table 1
Skewness and Kurtosis ofNWLα,β.
αΒSK0.90.60.212635461.2626440.50.50.248447801.2988830.20.40.261777111.3063110.40.30.259194981.3052870.60.20.255553461.3015931.12.10.127637671.2172101.22.20.117678831.2101751.32.30.108627791.2041491.42.40.100362781.1989881.52.50.092781021.194565


## 6. Order Statistics

IfY1,Y2,Y3,…,Yn are ordered variables, then the minimum (1st) and maximum (nth) CDF of the order statistics of NWLα,β distribution are defined by(15)FY1y=1−1−Fyn,FYny=Fyn.The corresponding PDF is defined by(16)fY1y=n1−Fyn−1fy,fYny=nFyn−1fy.Hence, the CDF and PDF of the 1st and nth order statistics of a new WL, respectively, take the following form:(17)FY1y=1−ββ+11+eyβ−αn,FYny=ββ+11+eyβ−αn,f1;nyi=n!n−1!ααβββ+1αey1+eyβ−α+1ββ+11+eyβ−αn−1,fn;nyi=n!n−1!ααββ+1βαey1+eyβ−α+11−ββ+11+eyβ−αn−1.


## 7. Parameter Estimation

In statistical inference, the estimation of the unknown parameters of the model is an important phase. In general, the parameters are unknown constant; we obtain their representative value through sample data. Under this section, we have considered the following likelihood function to estimation the parameters of NWL distribution:(18)L=∏i=1nααβββ+1αey1+eyβ−α+1.After applying the log function, we obtain(19)l=nlogααβββ+1α+∑yi−α+1log1+e∑yiβ,where α and β are estimated by partially differentiating (19) with respect to α and β and will give the following results:(20)dldα=dldαnlogααββ+1βα+∑yi−α+1log1+e∑yiβ,which finally becomes(21)=n1α+logβ+1β−logβ−∑1+eyiβ.Now, differentiating (19) with respect to β, we have(22)dldβ=dldβnlogααββ+1βα+∑yi−α+1log1+e∑yiβ,dldβ=n−αβ+2ββ+1+∑α+1+eyiβ2.Since the two expressions (21) and (22) are not in closed form, we can obtain the asymptotic confidence bounds for the population parameter of a new WL distribution. To achieve the asymptotic confidence bounds, we need the second time partial derivative of the parameters, and we have(23)dldα2=I11=n−1α2+0,dldβ2=I22=n−β+2ββ+1+∑eyiβ2,dldαβ=I12=n−1β+β2−1β+∑0−eyiβ2.The information matrix is then obtained as(24)I=−I11I12I21I22.The approximated variance-covariance matrix is defined as(25)V=v11v12v21v22=I11I12I21I22−1.The approximated ml estimates are given by(26)v^=I^11I^12I^21I^22−1.Using (26), we can easily obtain the 1−γ 100% confidence bounds for the unknown parameters α and β in the following forms:(27)α±Zγ/2varα,β±Zγ/2varβ.


## 8. Mean Residual Life (MRL)

In reliability analysis or survival analysis, the mean residual life is also an important aspect of the probability model. The MRL is used to measure the remaining mean life of an object given that the object has survived until the timey. Let a random variable y represent the life expectancy of an object, then the MRL is define as follows.MNWLy=EY−y/Y>y; it can be expressed as(28)=1Sy;α,β∫y∞tft;α,βdt−y,where(29)SY;α,β=ββ+11+eyiβ−α.By employing these functions in (28), we get(30)∫y∞tft;α,βdt=∫y∞tααββ+1βαet1+etβ−α+1dt=−tαβα2ββ+1βα1+eyβ−α.Replacing (6) and (30) result in (28), the result of the MRL is obtained:(31)MNWLy=1β/β+11+eyi/β−α−tαβα2ββ+1βα1+eyβ−α−y=−tαβα2β−y.


## 9. Stress Strength Parameter

Let us considerY1 and Y2 as the two IRV which follow a new WL distribution with parameters α1,β and α2,β, then the stress strength of the New Weighted Lomax distribution is defined by the following expression:(32)Sstress,strengthy=∫0∞f1yF2ydy=∫0∞α1α1ββ+1βα1ey1+eyβ−α1+11−ββ+11+eyβ−α2dy=∫0∞α1α1ββ+1βα1ey1+eyβ−α1+1dy−∫0∞α1α1ββ+1βα1ey1+eyβ−α1+1ββ+11+eyβ−α2dy.The solution to the first integral function in the above equation is given by(33)=∫0∞1ββ+1βα1ey1+eyβ−α1+1dy=β+1βα1∫0∞1βey1+eyβ−α1+1dy=1α1.Now, consider the second part of (32):(34)=∫0∞α1α1ββ+1βα1ey1+eyβ−α1+1ββ+11+eyβ−α2dy=α1α1ββ+1βα1+α2∫0∞ey1+eyβ−α1−α2+1dy=α1α1ββ+1βα1+α21β−α1−α2+1∑n=0∞βn−α1−α2n1α1+α2.Combining the result of (33) and (34) gives the stress strength parameter of a new WL:(35)Sstress,strengthy=1α1−α1α1ββ+1βα1−α21β−α1−α2+1∑n=0∞βn−α1−α2n1α1+α2.


## 10. Rank Regression onY

CDF of a NWL distribution is defined as(36)1−Ft=β+1β1+etβ−α,log1−Ft=−αlogβ+1β=−αlog1+etβ.By comparing (36) with a simple linear regression model, we have(37)y=log1−Ft,a=−logβ+1β,b=\α.From the least square equation method, the parameters are estimated by using the following two equations:(38)a^=∑iyiN−b^∑ixiN,b^=∑ixiyi−∑ixi∑iyi/N∑ixi2−∑ixi2/N.So, in the current case, we have to replace(39)y=log1−Ft,x=log1+etr.So, the regression equations related to a new WL distribution are described as(40)a^=∑ln1−Ft−b^ln1+etr,b^=∑iln1+et/rln1−Ft−∑ln1+et/r∑ln1−Ft/N∑ln1+et/r2−∑ln1+et/r2/N.Note. ln = log and Ft values are estimated from the median ranks.


## 11. Total Time on Test (TTT)

The TTT plot identifies various shapes of the hazard function. The TTT plot exhibits a straight line (diagonal) for a constant failure rate. For nonmonotonic failure rates, this plot would first decrease and then increase or vice versa. For monotonic failure rates, the TTT plot will be decreased if it is convex and increases if it is concave. The general formula of the TTT plot is given by(41)Grn=∑i=1rxi:n+n−rxi:n∑i=1nxi:n,r=xi:n=1,2,3,…,n,where xi:n are the order statistics.


## 12. Applications

Under this section, we provided applications to the proposed probability model using two real-lifetime data sets. To decide the best among other models, we considered goodness of fit statistics including AIC, CAIC, BIC, HQIC, W (Cramer-von Mises), and A (Anderson Darling). It is noted that a probability model with less value of AIC, CAIC, BIC, and HQIC and with a greater value of W and A will be considered the best one among others.


### 12.1. Wind Catastrophes Data

The data set represents the losses (in millions of dollars) due to wind catastrophes recorded by Boyd [41]. The data set consists of the following information:2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 8, 8, 9, 15, 17, 22, 23, 24, 25, 27, 32, 43.The fitted line in Figure3 shows that the data follow a constant failure function.Figure 3
TTT plot of the losses due to wind catastrophes using WL distribution.Table2 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.
ModelEstimatesα^β^γ^ϑ^NW-Lomax0.1115898 (0.0178886)24.7376349 (4.6011156)Lomax2.259102 (0.9034286)13.107217 (6.7006737)P-Lomax0.1612794 (0.0677728)5.4172791 (1.7104091)20.3051382 (13.2356863)W-Lomax2.8345778 (1.10337502)1.9742578 NaN1.0284592 NaN0.2073842 (0.03295629)E-Lomax28.842426 (36.1915908)1.481920 (0.2297605)2.482791 (2.5815765)Table 3
Goodness of fit measures for losses due to wind catastrophes.
ModelAICCAICBICHQIC−logWANW-Lomax52.233852.5671455.5609353.4275524.11690.59335783.367013Lomax252.6833253.016256.0104253.877124.34160.29499641.946171P-Lomax235.6173236.303240.608237.4079114.80860.16782041.406701W-Lomax249.5339250.7104256.1881251.9214120.76690.29820841.942425E-Lomax237.7877238.4734242.7784239.5783115.89390.17002441.383728Figure4 shows the empirical and theoretical PDF and CDF of the proposed distribution WLα,β and other existing distributions for the losses due to wind catastrophes.Figure 4
The empirical and theoretical PDF and CDF of theWLα,β distribution.


### 12.2. Bladders Cancer Patients

The data set represents the remission times (in months) of 128 bladders cancer patients and is taken from Aldeni et al. [3]. The data set values are given as follows:0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.20, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.40, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.90, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.40, 3.02, 4.34, 5.71, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69.The fitted line in Figure5 is concave-convex type; hence, we determined that the bladder cancer patient data follow a nonmonotonic hazard function.Figure 5
TTT plot of the bladders cancer patients using NWL distribution.The ml estimates and their standard error in braces are given in Table4. 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.
ModelEstimatesα^β^γ^ϑ^NW-Lomax0.10613 (0.009386)31.98979 (3.188287)Lomax3.8661 (1.1079)28.4134 (9.4998)P-Lomax1.404926 (0.4085148)1.438151 (0.1522463)19.819986 (5.4992568)W-Lomax5.4732949 (0.0506327)1.5096438 NaN4.7310404 NaN0.2643016 NaNE-Lomax10.0160681 (2.41150223)9.7029282 (1.84701652)0.1310464 (0.03225501)Table 5
Goodness of fit measures for bladder cancer patient.
ModelAICCAICBICHQIC−logWANW-Lomax100.39100.49106.10102.7148.1990.566043.7004Lomax835.54835.64841.25837.86415.770.0348740.224P-Lomax827.8986828.0921836.4547831.375410.94930.024815890.1860483W-Lomax828.6928829.018840.1009833.328410.34640.037410880.2432686E-Lomax305.2265305.4765313.0421308.3896149.61330.30140441.615242Figure6 shows the empirical and theoretical CDF and CDF of the proposed distribution WLα,β and other existing distributions for the Bladder cancer patients.Figure 6
The empirical and theoretical PDF and CDF of theWLα,β distribution.


## 12.1. Wind Catastrophes Data

The data set represents the losses (in millions of dollars) due to wind catastrophes recorded by Boyd [41]. The data set consists of the following information:2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 8, 8, 9, 15, 17, 22, 23, 24, 25, 27, 32, 43.The fitted line in Figure3 shows that the data follow a constant failure function.Figure 3
TTT plot of the losses due to wind catastrophes using WL distribution.Table2 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.
ModelEstimatesα^β^γ^ϑ^NW-Lomax0.1115898 (0.0178886)24.7376349 (4.6011156)Lomax2.259102 (0.9034286)13.107217 (6.7006737)P-Lomax0.1612794 (0.0677728)5.4172791 (1.7104091)20.3051382 (13.2356863)W-Lomax2.8345778 (1.10337502)1.9742578 NaN1.0284592 NaN0.2073842 (0.03295629)E-Lomax28.842426 (36.1915908)1.481920 (0.2297605)2.482791 (2.5815765)Table 3
Goodness of fit measures for losses due to wind catastrophes.
ModelAICCAICBICHQIC−logWANW-Lomax52.233852.5671455.5609353.4275524.11690.59335783.367013Lomax252.6833253.016256.0104253.877124.34160.29499641.946171P-Lomax235.6173236.303240.608237.4079114.80860.16782041.406701W-Lomax249.5339250.7104256.1881251.9214120.76690.29820841.942425E-Lomax237.7877238.4734242.7784239.5783115.89390.17002441.383728Figure4 shows the empirical and theoretical PDF and CDF of the proposed distribution WLα,β and other existing distributions for the losses due to wind catastrophes.Figure 4
The empirical and theoretical PDF and CDF of theWLα,β distribution.


## 12.2. Bladders Cancer Patients

The data set represents the remission times (in months) of 128 bladders cancer patients and is taken from Aldeni et al. [3]. The data set values are given as follows:0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.20, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.40, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.90, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.40, 3.02, 4.34, 5.71, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69.The fitted line in Figure5 is concave-convex type; hence, we determined that the bladder cancer patient data follow a nonmonotonic hazard function.Figure 5
TTT plot of the bladders cancer patients using NWL distribution.The ml estimates and their standard error in braces are given in Table4. 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.
ModelEstimatesα^β^γ^ϑ^NW-Lomax0.10613 (0.009386)31.98979 (3.188287)Lomax3.8661 (1.1079)28.4134 (9.4998)P-Lomax1.404926 (0.4085148)1.438151 (0.1522463)19.819986 (5.4992568)W-Lomax5.4732949 (0.0506327)1.5096438 NaN4.7310404 NaN0.2643016 NaNE-Lomax10.0160681 (2.41150223)9.7029282 (1.84701652)0.1310464 (0.03225501)Table 5
Goodness of fit measures for bladder cancer patient.
ModelAICCAICBICHQIC−logWANW-Lomax100.39100.49106.10102.7148.1990.566043.7004Lomax835.54835.64841.25837.86415.770.0348740.224P-Lomax827.8986828.0921836.4547831.375410.94930.024815890.1860483W-Lomax828.6928829.018840.1009833.328410.34640.037410880.2432686E-Lomax305.2265305.4765313.0421308.3896149.61330.30140441.615242Figure6 shows the empirical and theoretical CDF and CDF of the proposed distribution WLα,β and other existing distributions for the Bladder cancer patients.Figure 6
The empirical and theoretical PDF and CDF of theWLα,β distribution.


## 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 ofNWLα,β distribution.
αΒNMSE (α)MSE (β)Bias (α)Bias (β)0.03316.87303.996406e − 0515.187530.005703633.460371603.967258e − 052.2137980.0054769570.9058611806.484166e − 060.1831960.0024866570.384213717.1303.170174e − 0514.622790.002339323.752701601.079283e − 054.3126170.0022672921.901363804.80265e − 062.4579850.00091473540.829187617.5302.453543e − 0516.98660.0037882963.993388601.591147e − 058.2886840.0035963192.036911803.825583e − 063.0291280.00077953780.88649780.03621.29304.484847e − 0554.869770.0054038257.251341602.466488e − 0528.850790.0048045615.304583906.124085e − 0610.575440.0023213923.13855421.35304.484847e − 0555.743530.0054038257.311341602.466488e − 0530.464670.0048045615.454583906.085762e − 0610.953510.0023153253.19827922.45305.25542e − 0557.167690.0059647737.383071601.331295e − 0536.263790.0034080526.00667905.233742e − 067.5764370.0020657042.6213020.0423.2903.126413e − 0532.536680.0055831545.6966441201.527273e − 057.0071460.0036209972.2720111501.384378e − 052.2841850.0036897261.41383724.2901.818826e − 0536.940840.0041429785.9727911201.527273e − 0512.551170.0036209973.2720111501.426254e − 056.0082160.0037290362.36443923.4903.126413e − 0532.536680.0055831545.6966441201.527273e − 057.0071460.0036209972.2720111501.384378e − 052.889720.0036897261.613837


## 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.



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*Source: 8558118-2021-09-30.xml*