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

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.


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][7][8][9][10][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]. e 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.

Lomax Distribution.
Let a positive random variable be Y ∼ L(α, β); the CDF is given by , y > 0 and α, β > 0. (1) , y > 0. (2) 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.

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 (β/(β + 1)). e suggested distribution is called a New Weighted Lomax distribution or in short NWL distribution. e shape and scale parameter of this distribution are α and β, respectively. e 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 e corresponding PDF is given by , where α, β > 0.
(4) Figure 1 shows the behavior of the PDF and CDF of the NWL(α, β) distribution.

Survival and Hazard Function
e survival function of NWL(α, β) is defined by the expression as under Using (3), we get e hazard function or failure rate of a NWL distribution is defined by using the formula as under h(y) � (f(y)/(1 − F(y))); recalling (3) and (4) ; y > 0, α, β > 0. (7) Figure 2 delineates the capability of the suggested distribution to model the nonmonotonically hazard function.

Mode
e mode or a point by which the probability density function of a NWL will reach to its maximum point is defined as In order to find the maximum point, we have to equate this expression equal to zero and then solve for Y, and we get e mode is obtained as follows:

Quantile and Median Function
e QF is the real solution to the inverse cumulative distribution function of NWL distribution having two parameters. is function will help in providing the median but also in generating random data from NWL distribution. e QF is defined as F(y) � u where u ∼ U(0, 1). By using equation (3), we have When we solve the above function for a variable Y, we obtained 2 Mathematical Problems in Engineering 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 placing u � 0.5. Hence, the median function is obtained as follows:

Bowley Skewness (S) and Moors Kurtosis (K)
e mathematical equation of the Bowley Skewness and Moors Kurtosis [39,40] is given by

Order Statistics
. , Y n are ordered variables, then the minimum (1st) and maximum (nth) CDF of the order statistics of NWL(α, β) distribution are defined by e corresponding PDF is defined by Hence, the CDF and PDF of the 1st and nth order statistics of a new WL, respectively, take the following form:

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: After applying the log function, we obtain l � n log α αβ where α and β are estimated by partially differentiating (19) with respect to α and β and will give the following results: which finally becomes Now, differentiating (19) with respect to β, we have 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 dl e information matrix is then obtained as I � − I 11 I 12 e approximated variance-covariance matrix is defined as e approximated ml estimates are given by v � I 11 I 12 Using (26), we can easily obtain the (1 − c) 100% confidence bounds for the unknown parameters α and β in the following forms:

Mean Residual Life (MRL)
In reliability analysis or survival analysis, the mean residual life is also an important aspect of the probability model. e MRL is used to measure the remaining mean life of an object given that the object has survived until the time y. Let a random variable y represent the life expectancy of an object, then the MRL is define as follows.
; it can be expressed as where By employing these functions in (28), we get Replacing (6) and (30) result in (28), the result of the MRL is obtained:

Stress Strength Parameter
Let us consider Y 1 and Y 2 as the two IRV which follow a new WL distribution with parameters (α 1 , β) and (α 2 , β), then the stress strength of the New Weighted Lomax distribution is defined by the following expression: e solution to the first integral function in the above equation is given by Now, consider the second part of (32): Combining the result of (33) and (34) gives the stress strength parameter of a new WL:

Rank Regression on Y
CDF of a NWL distribution is defined as By comparing (36) with a simple linear regression model, we have From the least square equation method, the parameters are estimated by using the following two equations: So, in the current case, we have to replace So, the regression equations related to a new WL distribution are described as Note. ln � log and F(t) values are estimated from the median ranks.

Total Time on Test (TTT)
e TTT plot identifies various shapes of the hazard function. e 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. e general formula of the TTT plot is given by

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.

Wind Catastrophes Data.
e data set represents the losses (in millions of dollars) due to wind catastrophes recorded by Boyd [41]. e 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. e fitted line in Figure 3 shows that the data follow a constant failure function. 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). Figure 4 shows the empirical and theoretical PDF and CDF of the proposed distribution WL(α, β) and other existing distributions for the losses due to wind catastrophes.

Bladders Cancer Patients.
e data set represents the remission times (in months) of 128 bladders cancer patients and is taken from Aldeni et al. [3]. e data set values are given as follows:  Figure 5 is concave-convex type; hence, we determined that the bladder cancer patient data follow a nonmonotonic hazard function.
e 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). Figure 6 shows the empirical and theoretical CDF and CDF of the proposed distribution WL(α, β) and other existing distributions for the Bladder cancer patients.

Simulations
e 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)         result given in Table 6 declares that both the Bias and MSE are continuously decreased as the sample size increases.

Conclusion
e basic aim of this paper is to make a further contribution to the existing theory of the probability models. e 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. e first data set follows a monotonic hazard shape while the second data set (bladder cancer patients) has a nonmonotonic (bathtub) hazard shape. e 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.

Data Availability
e data sets used to support the finding of this study are taken from the literature.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this article.