The actuaries always look for heavy-tailed distributions to model data relevant to business and actuarial risk issues. In this article, we introduce a new class of heavy-tailed distributions useful for modeling data in financial sciences. A specific sub-model form of our suggested family, named as a new extended heavy-tailed Weibull distribution is examined in detail. Some basic characterizations, including quantile function and raw moments have been derived. The estimates of the unknown parameters of the new model are obtained via the maximum likelihood estimation method. To judge the performance of the maximum likelihood estimators, a simulation analysis is performed in detail. Furthermore, some important actuarial measures such as value at risk and tail value at risk are also computed. A simulation study based on these actuarial measures is conducted to exhibit empirically that the proposed model is heavy-tailed. The usefulness of the proposed family is illustrated by means of an application to a heavy-tailed insurance loss data set. The practical application shows that the proposed model is more flexible and efficient than the other six competing models including (i) the two-parameter models Weibull, Lomax and Burr-XII distributions (ii) the three-parameter distributions Marshall-Olkin Weibull and exponentiated Weibull distributions, and (iii) a well-known four-parameter Kumaraswamy Weibull distribution.

Modelling insurance risk data using a heavy tailed distribution has obtained more importance and interest for actuaries. Mostly the Insurance risk data sets are positively skewed, more peaked than meso-kurtic, unimodal and owns thick right tail; for detail, we refer to [

Realising the significance of these types of data modelling, researchers have shown a great interest in proposing new statistical models appropriate for modelling such data. A few of such models used for modelling insurance risk data and risk returns are Weibull, Pareto, Lognormal and gamma distributions, see for detail [

Studying the above literature, we are inspired to develop more dynamic probability models that are flexible in data fittings. Henceforth, in this article, our main objective is to suggest a new family of heavy-tailed (for short, NEFHT) models for modelling heavy tailed data. Several characterizations of the NEFHT distributions will be discussed here. Our research focuses on the special sub case of the NEFHT distributions, named as, a new heavy-tailed Weibull (NEHTW) distribution. Moreover, the most widely used maximum likelihood method of estimation is taken into conderation for estimation of the unknown model parameters. Furthermore, value at risk (VaR) and tail value at risk (TVaR) also computed. At last, we are concentrating our contemplations on the conclusions obtained from the NEHTW distribution fitted to insurance data.

The cumulative distribution function (cdf) of the NEFHT distributed random variable say

where

Henceforth, representing

The main objective of the present work is to develop and examine the proposed family in order to get new models appropriate for modelling financial data sets. Its key advantage is that it offers more flexibility to the resulting models by inserting just one extra parameter instead of including two or three parameters as appeared in other methods. Based on the NEFHT family of distributions, we introduce a three-parameter NEHTW model and give a comprehensive description of some of its mathematical properties so that it will attract the wider applications in insurance sciences and other related areas of research.

The rest of this article is structured in the different sections illustrated as: In Section 2, we have incorporated the NEHTW model and several plots displayed for its density. Section 3 contains mathematical properties including quantile function and moments. Section 4 focuses on the estimation and simulation studies of the recently recommended family. Actuarial measures VaR and TVaR of the NEHTW model are derived and based on these measures, a simulation study is conducted in Section 5. Section 6 offers insurance data modeling, While the Section 7 presented the final conclusion of the paper.

This portion of the article presents a particualr sub case of NEHT family by using the cdf of Weibull distribution with scale and shape parameters

with density function

The pdf plots of the NEHTW model for selected parameter values are presented in

This section presents some important characterizations of the NEFHT family.

Quantile function is extensively utilized for collecting samples from a specific model. The quantile function of X, represented by

where

Skewness

and the Moor’s kurtosis is

Kurtosis

Usually, these measures are slightly influenced by the extreme observations. For

Suppose a random variable X has the NEFHT distribution. Then, its

Using the series, we have

Replacing

Using

where,

where,

The moment generating function (mgf), say

Using

The following sub-section provides a well-known approach for estimation of unknown model parameters, named as maximum likelihood method estimation. Moreover, for assessing the nature of the maximum likelihood estimators (MLEs), a comprehensive analysis is performed.

This section focuses on the estimation of model parameters of the NEFHT family via the ML method of estimation. Let

By differentiating

and

Equating

In this portion, a comprehensive Monte Carlo simulation analysis is considered for assessing the performance of the ML estimates. The simulation study is conducted using the NEHTW distribution. The generation of random numbers is successfully performed using the inverse cdf procedure from the NEHTW model through R software. The major steps taken while performing simulation study are given below:

We produced different samples of sizes

MLEs of the parameters are derived.

MSEs and biases are calculated as

The numerical results of the simulation study are displayed in

One of the major role of financial science organizations is to determine the market loss. This portion contains the computation of some essential risk measures named as, VaR and TVaR for suggested model, which plays a key role in portfolio optimisation under the unpredictable situations.

The VaR is most widely considered by the professionals with in the field of insurance and finance to determine risk factor. The measure VaR is mostly specified with 90, 95 and 99% of the confidence level, representing the risk probability equal or greater than X percent of the time. The VaR measure of

The TVaR is an essential technique used for the computation of the estimated value of the risk provided that an event turned out beyond a determined significance level has occurred. Let

Using

We presented a computational analysis of these risk measures using two parameter Weibull and proposed models for various combination of parameters values with in this section. This process is carried out as:

From the Weibull and NEHTW models, random samples of sizes n = 100 and 150 are obtained.

The parameters are estimated via the MLE approach.

The process is replicated 1000 times to acquire the numerical figures for VaR and TVaR for comparing the competitive models.

The TVaR and TVaR measures are reported in

Distribution | Parameter | Significance level | VaR | TVaR |
---|---|---|---|---|

Weibull | 0.700 | 1.4316 | 5.7850 | |

0.750 | 1.8980 | 6.6115 | ||

0.800 | 2.5582 | 7.7125 | ||

0.850 | 3.5545 | 9.2770 | ||

0.900 | 5.2362 | 11.7596 | ||

0.950 | 8.8633 | 16.7558 | ||

0.990 | 13.4393 | 22.7010 | ||

0.999 | 47.1262 | 62.7459 | ||

NEHTW | 0.700 | 28.1151 | 39.0563 | |

0.750 | 29.8637 | 41.0757 | ||

0.800 | 32.0718 | 43.6145 | ||

0.850 | 35.0287 | 46.9963 | ||

0.900 | 39.4058 | 51.9697 | ||

0.950 | 47.4531 | 61.0298 | ||

0.990 | 56.2074 | 70.7893 | ||

0.999 | 105.9726 | 125.1375 |

Distribution | Parameter | Significance level | VaR | TVaR |
---|---|---|---|---|

Weibull | 0.700 | 1.0959 | 1.4754 | |

0.750 | 1.1803 | 1.5430 | ||

0.800 | 1.2768 | 1.6219 | ||

0.850 | 1.3922 | 1.7184 | ||

0.900 | 1.5417 | 1.8461 | ||

0.950 | 1.7707 | 2.0468 | ||

0.990 | 1.9757 | 2.2305 | ||

0.999 | 2.7486 | 2.9459 | ||

NEHTW | 0.700 | 1.1861 | 2.3344 | |

0.750 | 1.3857 | 2.5447 | ||

0.800 | 1.6336 | 2.8047 | ||

0.850 | 1.9586 | 3.1436 | ||

0.900 | 2.4251 | 3.6278 | ||

0.950 | 3.2421 | 4.4707 | ||

0.990 | 4.0795 | 5.3300 | ||

0.999 | 8.1585 | 9.4824 |

The comprehensive simulation study is conducted for suggested and Weibull model. A model is considered to be a heavy tailed, if the risk assessment values are higher. The results given in

The heavy tailed models are prominently used for measuring the risk values of the data. We have considered an insurance loss data, in order to assess the performance of the proposed model. Moreover, the study provides simplified calculations of the actuarial measurements while using the existing data set for the Weibull and NEHTW models.

The link given in this subsection, provides the insurance loss data available at

The maximum likelihood estimates of the model parameters are presented in

Distribution | ||||||
---|---|---|---|---|---|---|

NEHTW | 0.7333 | 0.1101 | 0.6971 | |||

Weibull | 0.759 | 0.106 | ||||

MOW | 1.153 | 0.085 | 3.098 | |||

EW | 0.176 | 3.389 | 9.120 | |||

Ku-W | 0.4387 | 2.800 | 31.00 | 0.175 | ||

Lomax | 1.690 | 19.139 | ||||

B-XII | 0.109 | 3.984 |

Distribution | AIC | BIC | CIAC | HQIC |
---|---|---|---|---|

NEHTW | 1380.098 | 1387.129 | 1381.650 | 1384.491 |

Weibull | 1432.698 | 1441.406 | 1432.719 | 1436.094 |

MOW | 1410.814 | 1423.877 | 1410.856 | 1415.909 |

EW | 1400.419 | 1413.482 | 1397.440 | 1405.514 |

Ku-W | 1397.006 | 1414.423 | 1397.076 | 1403.799 |

Lomax | 1418.450 | 1427.158 | 1418.471 | 1421.846 |

B-XII | 1467.001 | 1475.710 | 1467.022 | 1470.397 |

The researchers always interested in a smaller values resulted by the aforesaid measures.

Here we have considered an insurance data set already used in Section 6.1, in order to compute the numerical values of VaR and TVaR and to compare the Weibull and NEHTW distribtuions. The obtained results of the VaR and TVaR, while considering different intervals of significance levels are illustrated in

Distribution | Parameter | Level of significance | VaR | TVaR |
---|---|---|---|---|

Weibull | 0.700 | 6.0099 | 7.8876 | |

0.750 | 6.6154 | 9.9982 | ||

0.800 | 8.0933 | 9.0543 | ||

0.850 | 9.2902 | 12.0655 | ||

0.900 | 12.5648 | 13.0724 | ||

0.950 | 12.0919 | 14.7247 | ||

0.975 | 16.5602 | 17.1298 | ||

0.995 | 19.9980 | 21.9845 | ||

NEHTW | 0.700 | 8.5092 | 9.1209 | |

0.750 | 8.6907 | 10.9540 | ||

0.800 | 11.7630 | 12.9234 | ||

0.850 | 14.9879 | 16.0912 | ||

0.900 | 15.9009 | 18.0990 | ||

0.950 | 15.0532 | 21.5432 | ||

0.975 | 21.9120 | 28.7895 | ||

0.995 | 29.9099 | 32.4359 |

From the above discussion, it is clearly shown that while modelling data, as the risk value of a model increases, the model becomes heavier tailed. From the calculated values given in

In this article, we have provided the most flexible and prominent family, named as, new extended family of heavy tailed distributions. A specific three parameter form of the NEFHT class of distributions, named as, NEHTW distribution is studied, which has the capability to model heavy tailed data sets. Various basic statistical characterization have been studied. The estimates of the unknown model parameters are estimated via the most widely used ML method. A detailed evaluation of the of the simulation study is done to investigate the efficiency of the estimators. Moreover, the significance of the NEHTW model is illustrated via a practical application of the insurance loss data set. The practical application demonstrates that the NEHTW model is a prominent alternate model for modelling insurance losses. We expect that the new techniques will motivate the researchers for applications in actuarial sciences and many more different fields of research.