On New Weibull Inverse Lomax Distribution with Applications

: In this paper, simulation studies and applications of the New Weibull-Inverse Lomax (NWIL) distribution were presented. In the simulation studies, different sample sizes ranging from 30, 50, 100, 200, 300, to 500 were considered. Also, 1,000 replications were considered for the experiment. NWIL is a fat tail distribution. Higher moments are not easily derived except with some approximations. However, the estimates have higher precisions with low variances. Finally, the usefulness of the NWIL distribution was illustrated by fitting two data sets.


Introduction:
In 2003, Kleiber and Kotz 1 mentioned that Inverse Lomax Distribution (ILD) belongs to the Beta-Type size distributions alongside with Dagum, Lomax, and Generalized Beta (GB) of the second kind as other members of the family. They postulated that ILD has applications in economics, actuarial sciences, and stochastic modeling. In 2004, Kleiber 2 studied Lorenz ordering relationships between order statistics from loglogistic samples of potentially different sizes. Some results extend other families including the Burr XII, Lomax, and Burr III distributions. They applied the ILD model on geophysical data, specifically on the sizes of land fires in California State of United States. Moreover, in 2013 Rahman 3 studied the ILD via the Bayesian approach by drawing inferences about the unknown shape parameter of the distribution. A comparison between Bayes estimates and Maximum Likelihood estimate was made after getting the simulated and real-data results. Therefore, it is concluded that the Bayesian method of estimation leads to better outcomes as compared to Maximum Likelihood (ML) estimates. This shows the supremacy of a Bayesian approach to classical one for parameter estimation. Other references that considered the Bayesian approach are Rahman and Aslam 4 Jan and Ahmed 5 , Rahman and Aslam 6 and Yadav 7 . All the references discussed so far, considered only the ILD.
Recently, some generalization of ILD appeared in the literature with the sole aim of increasing the flexibility of the distribution. In 2018, Falgore et. al. 8 extended the ILD with the Odd Generalized Exponential family by Tahir 9 in 2015. They derived some mathematical properties of the proposed distribution such as the quantile function, moments, order statistics and asymptotic behavior of the distribution. The estimation of the proposed distribution's parameters was conducted using the method of Maximum Likelihood Estimation (MLE) procedure, and finally, they illustrated the usefulness of the proposed model by fitting the proposed distribution using a real-life data and compared its performance with comparator distributions. In 2019, Maxwell 10  The probability density function (pdf) and cumulative distribution function (CDF) of ILD are given by the following equations as defined by Yadav 17 in 2016 as: Gx is any baseline CDF which depends on a parameter vector  .
The additional parameters caused by the New Weibull generator are only needed to increase distribution flexibility. The pdf, CDF, and quantile function of the NWIL distribution as defined by Falgore 15 et. al. (15) are given below: where = ( , , , )      and α and γ are the scale parameters while β and λ are the shape parameters, respectively. Nevertheless, some of the NWIL distribution's statistical properties and derivations such as, entropies, Moments, Moment Generating Functions, distribution of Order Statistics and estimations were given in Falgore 15 et. al.

Methods
In this section, simulation procedures and Goodness of fit criteria are to be explained.

Simulation Studies
Monte Carlo is a process or methodology that uses replicated trials (sampling) that are generated using random numbers in computer programs like R, as given in Sambridge 18 in the year 2002. However, in 1997 Mooney 19 highlighted five (5) steps on how to do a Monte Carlo simulation study as follows: 1. Clearly state the pseudo-population that can be used in generating random samples usually by writing code in a specific method. 2. Sample from the population of interest (depending on your objective).

Discussion of the Simulation Results
The estimates of the unknown parameters are relatively good as they are approaching the true parameter values as n increases (see Table 1). Moreover, the bias of the estimates of the first parameter ( ) decreases when n is small but started increasing as the sample size increases. Similarly, the bias of the estimates of the second parameter (  ) behaves as the first one. However, the bias of the third estimates (  ) reduces as the sample size (n) increases. Lastly, the bias of the fourth estimates (  ) behaves as  and  .
Generally, the variances are very small for all the estimates. The MSEs of  and  generally reduces as n increases. While that of  almost maintain its value as n increases. According to Walther 20 in 2005, Small variance shows high precision, and MSE as a measure of accuracy is good when the value is small. Also, small bias indicates a highly accurate estimator.

Shapes of the NWIL distribution
The following are the plots of the pdfs, hazard rate functions, and CDF ( Fig. 1)    Here, the applicability of the NWIL distribution to the breaking strength of 100 Yarn as reported by Duncan 21 is presented. The summary statistics for this data is presented in Table 2. The plots of the Data set 1 are presented in Fig. 4. Throughout this section, a maxLik package developed by Henningsen 22 in R Software by Team R 23 in 2014 was used. The parameters of each model were estimated by the method of ML Estimation (MLE) using the Simulated ANNealing (SANN) method. The goodness of fit statistics used in comparing the performances are the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). However, Smaller values of the AIC and BIC statistics indicate better model fittings. Throughout the analysis, we used negative log-likelihood (-ll) value to derive the AIC and BIC by using the following relations: Where R is the parameter number and n is the sample size.  Its fit is also compared with the Weibull-Lomax distribution by 9 Odd generalized exponential Inverse Lomax distribution by 8 logistic Lomax distribution by Zubair 24 Inverse Lomax distribution, as in 3 Weibull Log-logistic by Tahir (16), and Weibull exponential by Oguntunde 25 et. al. with the pdfs given below (Table 3):   Table 4. The plots of the data is presented in Fig.  5. The NWIL distribution alongside Weibull-Weibull by 17 (Table 5) was fitted, Odd Generalized exponential Inverse Lomax distribution by 8 and Inverse Lomax distribution with the pdf:

Concluding Remarks
A simulation study based on NWIL by Falgore 15 et. al. in 2019 was performed. The results indicated that the ML estimates were cosistent most of the times. The distribution was fitted to two data sets. The New Weibull-Inverse Lomax distribution (NWIL) was fitted to two datasets. Alongside the comparator distributions, the NWIL distribution outperformed its comparators, as shown in Table 3 and Table 4. For the first dataset, NWIL was compared with the Weibull Log Logistics which was fitted by Tahir 16 et. al. in 2016 using the same data set. It is clear from Table 5 that the NWIL distribution performed much better than the New Weibull-Weibull distribution. We can therefore conclude that based on these two datasets the NWIL distribution outperfomed its comparators.

Authors' declaration:
-There are no conflicts of interest, and all the Figures and Tables in the manuscript are derived from this study. -Ethical Clearance: The project was approved by the local ethical committee in University of Ahmadu Bello.