SINE-LOMAX DISTRIBUTION: PROPERTIES AND APPLICATIONS TO REAL DATA SETS

In this study, a novel distribution called the two-parameter Sine Lomax distribution was introduced. The distribution was developed by combining the Sine generalized family of distributions with the Lomax distribution. Various statistical properties of this new distribution were investigated, including the survival function, hazard function, quantile function, rth moment, entropy, moment generating function, and order statistics. The probability density function (PDF) plot indicated that the distribution is skewed to the right. Additionally, the hazard plot of the Sine Lomax distribution showed both monotonic increase and monotonic decrease. To estimate the parameters of the newly proposed distribution, the maximum likelihood approach was employed. A simulation study was conducted to evaluate the consistency of the estimators. The simulation results indicated that the estimators are consistent, as the bias and mean square error decrease with increasing sample sizes. The performance of the Sine Lomax distribution was compared to other extensions of Lomax distributions and the baseline distribution which is the Lomax distribution using various evaluation criteria, including the Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), and Hannan-Quinn Information Criterion (HQIC). The proposed distribution demonstrated the lowest scores among the competing models, indicating its potential for accurately modeling real-world data sets. Based on the results, the proposed Sine Lomax distribution is recommended as a superior alternative to the competing models for modeling certain real-world data sets.


INTRODUCTION
The accuracy of parametric statistical inference and data set modeling relies heavily on the goodness of fit between the probability distribution and the given data sets, assuming all distributional assumptions are met. Numerous studies have been conducted to develop distributions with more desirable and flexible properties to effectively model real-world data sets of varying density and failure rate functions. Currently, researchers are focused on creating new hybrid distributions that generalize existing ones, aiming to achieve better data modeling capabilities. These hybrid distributions are formed by combining a baseline distribution with a family distribution. Several authors have extensively reviewed different families of distributions (Hamedani et al., 2018). The main objective behind constructing this distribution family is to enhance the flexibility of classical distributions, enabling them to provide improved fits for survival data sets compared to other candidate distributions with the same number of parameters. This family should be capable of modeling various types of failure rates, including monotonic and non-monotonic patterns. The Lomax distribution, also known as the Pareto distribution of the second kind with two parameters (α, λ), has attracted significant attention from theoretical and statistical researchers due to its applications in reliability and lifetime testing studies. Lomax first introduced and studied this distribution in 1954, and it has since been utilized for analyzing business failures, as well as in economic, behavioral, scientific, and traffic modeling. Researchers such as Falgore and Doguwa (2020) employed the Lomax distribution to model firm size data, while Hassan and Al-Ghamdi (2009) used it for reliability and life testing, while Zweig and Cambell (1993) applied it to analyze receiver operating characteristic (ROC) curves. Ijaz et al. (2019) suggested this distribution as a heavy-tailed alternative to the exponential, Weibull, and gamma distributions.
Statistical distributions are widely utilized to describe real-life phenomena, and as a result, the field of statistical distribution theory is extensively explored, leading to the development of novel distributions. Statisticians often seek distributions that offer greater flexibility. This has led to a demand for generalized or extended distributions capable of simulating lifetime data with monotonically increasing, decreasing, constant, or more importantly, unimodal bathtub-shaped failure rates. Consequently, there is a necessity to extend traditional distributions or construct new ones, as certain distributions can only accommodate monotonically increasing or decreasing failure rates. Therefore, in this study, we propose the Sine-Lomax distribution, which can effectively handle datasets exhibiting monotonically or non-monotonically increasing, decreasing, constant, and unimodal bathtub-shaped failure rates. The Lomax distribution, with its heavy-tailed nature, serves as the foundation for this two-parameter distribution.

MATERIALS AND METHODS Lomax Distribution
Given a non-negative random variable which follows the Lomax distribution with parameters and , with the CDF is given as: ( , , ) = 1 − (1 + ) − , > 0, > 0, > 0 (1) and the corresponding PDF is expressed as: (2) where, > 0 and > 0 are the shape and scale parameters respectively The Lomax distribution has been applied to some real-world data sets by researchers in recent times. It was applied to breaking stress of carbon fibers data by Ijaz et al. (2019), in losses due to wind catastrophes recorded in 1977 by Ijaz et al. (2020), remission times of bladder cancer patients by Chesneau and Jamal (2020). Some extensions of the Lomax distribution that exists in the literature include: Transmuted Lomax distribution by Ashour and Eltehiwy (2013), Power Lomax distribution by Rady et al. (2016), Type II half logistic Exponentiated Lomax by Bukoye et al. (2021) and Slashed Lomax distribution by Li and Tian (2022). The rapid progress and accessibility of processing power has spurred recent advancements in stochastic modeling. They have permitted direct applications of existing continuous distributions with some functional complexity for a variety of statistical purposes. Additionally, these have accelerated the development of new and flexible distribution families. Kumar et al. (2015) pioneered the sinusoidal transformation that leads to the sine generated (S-G or Sin-G) family. Some trigonometric families of distribution include Cos-G family by Souza (2015), Polyno-Expo-Trigonometric distributions by Jamal and Chesneau (2019), New Sine-G family by Mahmood et al. (2019), Sine Topp-Leone-G family by Al-Babtain et al. (2020), Hyperbolic Tan-X family by Ampadu (2021), Arcsine family of distribution by Rahman (2021) and Tangent Topp-Leone-G family by Nanga et al. (2022). In this paper, we focused on Sine G family proposed by Kumar et al. (2015) to develop a new probability distribution called the Sine-Lomax Distribution. Some compounding of baseline distribution with the Sine G family proposed by Kumar et al. (2015) includeSine Power Lomax by Nagarjuna et al. (2021), Sine Modified Lindley distribution by Tomy et al. (2021),Sine-Exponential distribution by Isa et al. (2022a), and Sine Burr XII by Isa et al. (2022b) among others.

Sine Family of Distribution
The CDF and PDF of the Sine G family of distributions proposed by Kumar et al. (2015) are defined by the following equations: Where G( ; ξ) and g( ; ξ) in equation (3) and (4) above are the CDF and PDF of the base line distribution with parameter(s) vector denoted by , respectively.

Development of the Proposed Sine-Lomax Distribution
The cumulative density function (CDF) of the newly proposed probability distribution (Sine Lomax) and the probability density function (PDF) are presented in equation (5) and (6): and the pdf is given by The survival function S(x), the hazard function h(x), the reverse hazard function r(x), the cumulative hazard function H(x) and the quantile function are given in equation (7) to (10): The hazard plot of the sine Lomax distribution shows that the distribution has an inverted bathtub shape

Parameter Estimation
In this section, we consider maximum likelihood estimation (MLE) to estimate the involved parameters.

Method of Maximum Likelihood Estimation
Let 1 , 2 , … , be a random sample of size n from the Sine-Lomax distribution with pdf given in equation (6), the loglikelihood function ( , ) of the Sine-Lomax distribution is given by: (12) Differentiating with respect to gives: (13) Differentiating with respect to gives the following expression: (14) Equation (13) and equation (14) gives the maximum likelihood estimators of the parameters and .

Useful Expansion
The pdf of the proposed Sine-Lomax Distribution can be expanded as follows: , we will have the following: (1 + ) Therefore, the pdf will be given by: The pdf will be reduced to: Again, The pdf is also reduced to: Therefore, the pdf can be expressed as: The expansion of the cdf is also given below: Therefore, the cdf will be reduced to: Equation (15) and equation (16) gives the reduced form of the PDF and the CDF of the Sine-Lomax Distribution and they were used to derive some of the mathematical properties of the newly developed distribution.

Mathematical Properties
Some of the mathematical properties such as the rth moment, moment generating function, the entropy and order statistics are derived.

r th Moment
Moments are required and vital in any statistical study, particularly in applications. It can be used to investigate some properties of a distribution such as skewness, kurtosis and measures of dispersion (Halid and Sule, 2022). The r th moment of the random variable X with PDF f (x) is expressed as: The rth moment of the Sine-Lomax Distribution is given by: Therefore, the rth moment is given by:

Moment Generating Function
The moment generating function of a random variable X is the expected value of that is:

Entropy
Entropy is used as a measure of information or uncertainty, which present in a random observation of its actual population. There will be the greater uncertainty in the data if the value of entropy is large. The entropy for the true continuous random variable X is defined as: Order Statistics (OS) Let 1 , 2 , … , be a random sample of size from a continuous distribution having a PDF, ( ) and CDF, F(x), Let 1: ≤ 2: ≤ : be the corresponding order statistics (OS). The ℎ OS is given by: The order statistics of the Sine-Lomax Distribution is given by: Equation (20) gives the order statistics of the newly developed Sine-Lomax Distribution

Simulation study
The modeling process heavily relies on assumptions that are associated with uncertainty. Monte Carlo simulation is a valuable tool for assessing the impact of risk and uncertainty in prediction and forecasting models. In this study, we employ Monte Carlo simulation to evaluate our proposed distribution in terms of its ability to model lifetime data while considering the associated risks. To evaluate the performance of the newly proposed Sine-Lomax distribution, we conduct a simulation study using the Monte Carlo Simulation method. The objective is to compute the mean, bias, and mean square error of the estimated parameters derived from the maximum likelihood estimates. The simulation generates synthetic data by utilizing the quantile function defined in equation (11) for various sample sizes, including n=20, 50, 100, 250, 500, and 1000. For each sample size, α is set to 1.0 and λ is set to 1.2. Table 1 presents the estimation results, bias, and mean square error obtained from the new distribution. The results obtained from the Monte Carlo Simulations are presented in Table 1. These findings provide evidence that as the sample size increases, both the bias and mean square error (MSE) tend to approach zero. This indicates that the suggested distribution exhibits favorable characteristics and a low level of risk when applied for modeling lifetime datasets.

Application
Two data sets were used to illustrate the applicability and the practicability of the proposed model.

CONCLUSION
This study introduces an extension of the Lomax distribution known as the Sine-Lomax distribution (SL). The Sine-Lomax Distribution model was developed by incorporating the Sine-G family of distributions proposed by Kumar et al. (2015), resulting in a new trigonometric distribution. Various properties of the proposed distribution, including the survival function, hazard rate function, reverse hazard function, cumulative hazard function, moments, moment generating function, quantile function, and order statistics, were derived.
The parameters of the proposed distribution were estimated using the maximum likelihood method. Additionally, a simulation study was conducted to evaluate the performance of the maximum likelihood estimators (MLEs) for the distribution parameters. Furthermore, the proposed distribution was applied to two different real-life datasets to compare its effectiveness with other well-known standard distributions such as the Lomax distribution, Exponential Lomax distribution, Sine-Inverse Weibull distribution, and Inverse Lomax distribution. The results demonstrated that the new Sine-Lomax Distribution (SLD) outperformed its competitors in terms of fitting the two datasets.