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The uses of statistical distributions for modeling real phenomena of nature have received considerable attention in the literature. The recent studies have pointed out the potential of statistical distributions in modeling data in applied sciences, particularly in financial sciences. Among them, the two-parameter Lomax distribution is one of the prominent models that can be used quite effectively for modeling data in management sciences, banking, finance, and actuarial sciences, among others. In the present article, we introduce a new three-parameter extension of the Lomax distribution via using a class of claim distributions. The new model may be called the Lomax-Claim distribution. The parameters of the Lomax-Claim model are estimated using the maximum likelihood estimation method. The behaviors of the maximum likelihood estimators are examined by conducting a brief Monte Carlo study. The potentiality and applicability of the Lomax claim model are illustrated by analyzing a dataset taken from financial sciences representing the vehicle insurance loss data. For this dataset, the proposed model is compared with the Lomax, power Lomax, transmuted Lomax, and exponentiated Lomax distributions. To show the best fit of the competing distributions, we consider certain analytical tools such as the Anderson–Darling test statistic, Cramer–Von Mises test statistic, and Kolmogorov–Smirnov test statistic. Based on these analytical measures, we observed that the new model outperforms the competitive models. Furthermore, a bivariate extension of the proposed model called the Farlie–Gumble–Morgenstern bivariate Lomax-Claim distribution is also introduced, and different shapes for the density function are plotted. An application of the bivariate model to GDP and export of goods and services is provided.

The Lomax or Pareto II (the shifted Pareto) distribution was proposed by Lomax in the mid of the last century to model business failure data. This model has a wide range of applications in a variety of fields, particularly, in income and wealth inequality, size of cities, and financial and actuarial sciences. Furthermore, it has been applied to model income and wealth data, the size distribution of computer files on servers, reliability, life testing, and curve analysis [

A random variable say

The Lomax model can be obtained in a number of ways. It can be obtained as a special form of the Pearson type VI distribution. It is also considered a mixture of the exponential and gamma distributions. In the context lifetime scenario, the Lomax distribution falls under the domain of decreasing failure rate distributions. This distribution has been proved as a significant alternative to the exponential, Weibull, and gamma distributions to model heavy-tailed data sets. Due to the importance and applicability of the Lomax distribution, it has been extensively generalized and modified to obtain a more flexible extension of the Lomax distributions, for example, power Lomax distribution [

In this article, we focus on proposing a new three-parameter modification of the Lomax distribution called the Lomax-Claim (L-Claim) distribution for modeling financial data. The L-Claim distribution is introduced by adopting the approach of a class of claim distributions of Ahmad et al. [

In this article, the L-Claim distribution along with its statistical properties will be given intensive statistical treatment. However, the flexibility and applicability of the L-Claim distribution are examined by an application to the insurance loss insurance data.

The rest of this paper is carried out as follows. In Section

In the following section, we study a three-parameter L-Claim distribution and investigate the shapes of its pdf. The cdf and pdf of the L-Claim distribution can be obtained by substituting expressions (

A random variable

The pdf corresponding to (

Some possible behaviors of the pdf of the L-Claim distribution are shown in Figure

Plots for the pdf of the L-Claim distribution for selected parameter values.

From Figure

The sf and hrf of the L-Claim distributed random variable are given by

Plots for the hrf of the L-Claim distribution for selected parameter values.

As we stated above, the two-parameter traditional Lomax distribution belongs to the class of decreasing failure rate distributions. However, from the plots provided in Figure

Copula functions are used to represent the joint cdf of the two marginal univariate distributions. If

The random variables say

Different plots for the FGMBL-Claim distribution are provided in Figures

Different plots for the pdf of the FGMBL-Claim distribution for

Different plots for the pdf of the FGMBL-Claim distribution for

Plots for the cdf and sf of the FGMBL-Claim for

The quantile function of distribution is very useful to apply for generating random numbers by Monte Carlo simulation. Suppose

The nonlinear expression provided in (

Furthermore, the effects of the shape parameters on the skewness and kurtosis can be detected on quantile measures. We obtain skewness and kurtosis measures of the L-Claim distribution using (

These measures are less sensitive to outliers. Moreover, they do exist for distributions without moments. Some possible plots for the mean, variance, skewness, and kurtosis of the L-Claim distribution are provided in Figures

Plots for the mean, variance, skewness, and kurtosis of the L-Claim distribution for

Plots for the mean, variance, skewness, and kurtosis of the L-Claim distribution for

Several approaches for the estimation of the unknown parameters have been studied and applied in the literature. Among them, the method of maximum likelihood estimation is the most widely applied approach. The estimators obtained via this method possess several desirable properties and can be used quite effectively for constructing confidence bounds. In this section, we adopt this method to obtain the estimators of the model parameters. Let

The expression provided in (

This section deals with assessing the performance of the maximum likelihood estimators of the L-Claim distribution by the Monte Carlo simulation study. The simulation is performed for two different sets of parameters of the L-Claim distribution. The simulation study is carried out as follows:

Random samples of different sizes

The model parameters have been estimated via the maximum likelihood method.

750 repetitions are made to calculate the biases, absolute biases, and mean square errors (MSEs) of these estimators.

The formulas for obtaining the biases and MSEs are given by

respectively.

Step (4), is also repeated for the parameters

For the simulated dataset 1, the box plot and Kernel density estimator are presented in Figure

Box plot and Kernel density estimator plot of the simulated dataset 1.

Box plot and Kernel density estimator plot of the simulated dataset 2.

The simulation results of the L-Claim distribution are presented in Tables

Simulation results of the L-Claim distribution for

Parameters | MLE | Biases | MSEs | |
---|---|---|---|---|

25 | 0.740063 | 0.040063 | 0.023912 | |

1.975112 | 0.675111 | 2.348829 | ||

0.646899 | 0.146899 | 0.218631 | ||

100 | 0.708882 | 0.008882 | 0.007295 | |

1.645087 | 0.345087 | 0.953070 | ||

0.561791 | 0.061791 | 0.077792 | ||

200 | 0.708633 | 0.008633 | 0.003408 | |

1.411091 | 0.111091 | 0.280088 | ||

0.508988 | 0.008988 | 0.034374 | ||

400 | 0.705445 | 0.005445 | 0.001236 | |

1.325511 | 0.025511 | 0.044632 | ||

0.498914 | 0.007085 | 0.013414 | ||

600 | 0.705628 | 0.005628 | 0.000693 | |

1.301424 | 0.001424 | 0.011400 | ||

0.490299 | 0.004700 | 0.006831 | ||

750 | 0.705378 | 0.005378 | 0.000528 | |

1.299216 | 0.000783 | 0.007576 | ||

0.491804 | 0.002195 | 0.004624 |

Simulation results of the L-Claim distribution for

Parameters | MLE | Biases | MSEs | |
---|---|---|---|---|

25 | 1.407703 | 0.007702 | 0.089969 | |

1.841798 | 0.741797 | 2.329233 | ||

1.817713 | 0.917713 | 2.571805 | ||

100 | 1.381831 | -0.018168 | 0.027254 | |

1.308675 | 0.208675 | 0.469191 | ||

1.237221 | 0.337221 | 0.629859 | ||

200 | 1.398295 | -0.001705 | 0.012032 | |

1.144440 | 0.044439 | 0.015172 | ||

1.051417 | 0.151417 | 0.223189 | ||

400 | 1.395308 | -0.004691 | 0.006124 | |

1.118567 | 0.018566 | 0.004892 | ||

0.956584 | 0.056584 | 0.095525 | ||

600 | 1.398676 | -0.001323 | 0.004355 | |

1.113499 | 0.013499 | 0.002874 | ||

0.951820 | 0.051820 | 0.069850 | ||

750 | 1.398821 | -0.001178 | 0.003376 | |

1.110305 | 0.010305 | 0.002152 | ||

0.934313 | 0.034313 | 0.054993 |

Plots of MLEs, MSEs, biases, and absolute biases for

Plots of MLEs, MSEs, biases, and absolute biases for

The main applications of the heavy-tailed distributions are the extreme value theory or insurance loss phenomena. In this section, we illustrate the applicability of the L-Claim distribution by analyzing the vehicle insurance loss data. The data are available at

Table

The estimated values of the parameters of the fitted distributions.

Model | |||||
---|---|---|---|---|---|

L-Claim | 1.879 | 0.039 | 0.017 | — | — |

Lomax | 1.845 | 0.002 | — | — | — |

T-Lomax | 1.850 | 0.874 | — | — | 0.793 |

P-Lomax | 1.732 | 0.852 | — | 31.713 | — |

E-Lomax | 1.493 | 0.006 | — | 1.882 | — |

The analytical measures of the fitted models.

Model | CM | AD | KS | |
---|---|---|---|---|

L-Claim | 0.056 | 0.320 | 0.086 | 0.951 |

Lomax | 0.062 | 0.387 | 0.147 | 0.443 |

T-Lomax | 0.059 | 0.374 | 0.198 | 0.407 |

P-Lomax | 0.075 | 0.470 | 0.229 | 0.087 |

E-Lomax | 0.108 | 0.671 | 0.150 | 0.386 |

The fitted pdf, cdf, PP, and Kaplan–Meier survival and QQ plots of the L-Claim distribution along with the box plot of the data.

The economy is an important sector in many developed and developing countries. Therefore, the government and other responsible institutions are always interested in GDP growth and exports of goods and services. To demonstrate the effectiveness of the proposed FGMBL-Claim distribution, we consider the GDP growth and exports of goods and services. The summary measures of the considered data based on the response variable such as exports of goods and services

Summary measures of the economics data.

Variables | Min. | 1st qu. | Median | Mean | 3rd qu. | Max. |
---|---|---|---|---|---|---|

15.070 | 19.140 | 21.170 | 23.310 | 27.610 | 32.800 | |

2.990 | 4.220 | 5.130 | 5.510 | 6.660 | 9.210 |

An assessment of the normality of data is a requirement for many statistical tests because normal data are a basic assumption in parametric testing. There are two main methods of assessing normality such as numerically and graphically. In this section, we check the normality of both the variables

The SW normality test can be performed as follows:

Graphical display of the

The Anderson–Darling (AD) test is another prominent approach to check the normality of the data. In this subsection, we perform the AD normality test to check the normality of the data. Under this test, the null hypothesis and alternative hypothesis can be defined as follows:

Graphical display of the

The prime interest of the proposed FMGBL-Claim distribution is to be applied for data analysis purposes, making it useful in many applied fields. Here, this aspect is illustrated by considering an economics dataset based on the GDP growth and exports of goods and services. The total time on test (TTT) plot is an empirical plot and used for model identification purposes. The TTT plots of the economics dataset are presented in Figure

The TTT plot of the economics data.

In this subsection, we use the economics data to illustrate the proposed model. We prove that the fit power of the FGMBL-Claim model is better than the Farlie–Gumbel–Morgenstern bivariate Lomax (FGMBL) distribution [

First, we estimate the unknown parameters of the fitted models using the maximum likelihood approach using the

MLEs of the competing models for the economics data.

Fitted models | |||||||
---|---|---|---|---|---|---|---|

FGMBL-Claim | 1.097 | 0.909 | 1.298 | 1.170 | 0.983 | 1.306 | 1.205 |

FGMBL | 1.256 | 1.287 | 1.696 | 1.953 | 0.854 | — | — |

The comparison is done based on some discrimination measures such as the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). After performing the analysis, the discrimination measures of the fitted models are presented in Table

Discrimination measures of the competing models for the economics data.

Fitted models | AIC | BIC |
---|---|---|

FGMBL-Claim | 191.656 | 194.909 |

FGMBL | 224.549 | 229.075 |

In this study, a new extension of the Lomax distribution called the Lomax-Claim model is introduced. The proposed distribution is very flexible and possess heavy tails. The maximum likelihood estimators of the parameters are obtained. Furthermore, a Monte Carlo simulation study is provided to evaluate the behavior of the estimators. The proposed L-Claim model is illustrated via analyzing a heavy-tailed vehicle insurance loss dataset, and the comparison is made with some well-known competitive models. From the real application, we showed that the L-Claim model provides a better fit to the heavy-tailed vehicle insurance loss data than the other distributions. Furthermore, a bivariate extension of the L-Claim distribution called Farlie–Gumble–Morgenstern bivariate Lomax-Claim distribution is introduced. Finally, the bivariate extension of the L-Claim distribution is illustrated by analyzing economic data related to the GDP and export of goods and services.

library (AdequacyModel)

data

data = data [,6]

data = data [!is.na(data)]

hist (data)

########################################################################### Probability density function######################################################

pdf_pm

{

(^{∧}(−^{∧}(−^{∧}(−^{∧}(−

}

########################################################################### Probability density function######################################################

cdf_pm

{

(s∗((1−((1^{∧}(−^{∧}2))/(1−(1−^{∧}(−a))))

}

set.seed (0)

goodness.fit (pdf = pdf_pm, cdf = cdf_pm,

starts =

method = ”Nelder-Mead,” domain =

This work is mainly a methodological development and has been applied on secondary data, but if required, data will be provided.

The authors declare that they have no conflicts of interest to report regarding the present study.

The study was funded by the department of statistics, Yazd University, Yazd, Iran.