Statistical distributions play a vital role in modeling data in applied areas such as risk management, banking, economic, financial and actuarial sciences. However, the quality of the procedures primarily depends upon the assumed probability model of the phenomenon under consideration. Among applied fields, insurance data sets are usually positive (Klugman et al., 2012), right skewed (Lane, 2000), unimodal (Cooray and Ananda, 2005) and with heavy tails (Ibragimov and Prokhorov, 2017). Right-skewness may be adequately modeled by the skewed distributions (Bernardi et al., 2012). Therefore, a number of unimodal positively skewed parametric distributions have been deployed to model such data sets (Klugman et al., 2012; Eling, 2012; Bagnato and Punzo, 2013; García et al., 2014; Bakar et al., 2015; Kazemi and Noorizadeh, 2015; Adcock et al., 2015; Landsman et al., 2016; Reynkens et al., 2017; Punzo et al., 2017; Bhati and Ravi, 2018).

Among available literature, the Pareto, Weibull, and gamma are the promising distributions used widely for modeling insurance data sets. Unfortunately, due to the monotonically decreasing shape, the Pareto distribution often does not provide a best fit to many data sets. Weibull distribution is suitable for modeling small losses, but fails to provide best fit to large losses. In addition, gamma distribution is also popular because it does not have a closed form expression of cumulative distribution function (cdf). Consequently, introducing new distributions to address these problems and cater heavy tailed data is an interesting research topic that is quite rich and growing rapidly. Therefore, researchers are often in search of finding more flexible distributions. This has been done through many different approaches such as (i) transformation method, (ii) composition of two or more distributions, (iii) compounding of distributions, and (iv) finite mixture of distributions, for details see Ahmad et al. (2019a).

Among the prominent methods, Mahdavi and Kundu (2017) recently proposed a new method for introducing statistical distributions via the cdf given by

where F(x; ξ) is the distribution function of the baseline random variable depending on the vector parameter ξ ∈ ℝ, and α is the additional parameter.

Ahmad et al. (2019b) proposed another method, called extended alpha power transformed (Ex-APT) family of distributions with cdf given by

The Ex-APT family is an extension of (1.1) by introducing the exponent term, which is a constant quantity. Because of the constant term, the Ex-APT family approach is not a flexible extension of (1.1). In this article, we introduce another new extension of (1.1), called new Ex-APT (NEx-APT) family by introducing an additional parameter β ∈ ℝ⁺. A random variable X follows the NEx-APT distribution, if its cdf is given by

where, β̄ = 1 − β. For β = 1, the NEx-APT method captures the characteristics of (1.1) and for α₁ = 1, the NEx-APT approach offers the baseline distribution characteristics. The probability density function (pdf) corresponding to (1.2) is given by

The new pdf is most tractable when F(x, ξ) and f (x, ξ) have simple analytical expressions. A random variable X with pdf (1.3) is represented by X ~ NEx-APT (x; α₁, β, ξ). Furthermore, the dependence on the vector of the parameters is omitted and G(x)=G(x; α₁, β, ξ) will be used for efficiency. The proposed method is very interesting with a closed form for the cdf and capable of modeling heavy tailed insurance data sets.

The rest of this work is as follows. In Section 2, we introduce a special sub-case of (1.3), called a NEx-APT Weibull (NEx-APTW) distribution by considering Weibull model as a parent distribution. Some mathematical properties are derived in Section 3. Certain characterizations of the proposed distribution are provided in Section 4. Maximum likelihood estimation and Monti Carlo simulation study are provided in Section 5. Section 6 offers the computation of the actuarial measures and simulation study based on these measures. A heavy tailed real-life application from the insurance sciences is discussed in Section 7. Here the proposed model is compared with some prominent distributions. Finally, the article is concluded in Section 8.

In this section, we introduce a sub-model of the (1.2) and discuss its special cases.

2.1. A new extended alpha power transformed Weibull distribution

Let F(x; ξ) and f (x; ξ) be the cdf and pdf of the two parameters Weibull distribution given by F(x; ξ) = 1 − e^−γxα, x ≥ 0, α, γ > 0, and f (x; ξ) = αγx^α−1e^−γxα, respectively, where ξ = (α, γ). Then, the cdf of the NEx-APTW distribution has the following expression

The pdf corresponding (2.1) is given by

The survival function (sf) and hazard function (hf) of NEx-APTW distribution are respectively, given by

Figure 1 sketches different plots for the density function of the NEx-APTW distribution.

2.2. Special cases of NEx-APTW distribution

Let X have the NEx-APTW distribution with parameters (α₁, β, α, γ). Then, the distribution of X reduces to:

• Alpha power transformed Weibull (APTW) distribution, if β = 1.

• Alpha power transformed exponential (APTE) distribution, if α = 1 and β = 1.

• Weibull distribution, if α₁ = 1.

• Rayleigh distribution, if α₁ = 1 and α = 2.

• Exponential distribution if α₁ = 1 and α = 1.

• One parameter Weibull distribution, if α₁ = 1 and γ = 1.

• One parameter New extended alpha power transformed Weibull, if γ = 1. (New)

• New extended alpha power transformed Rayleigh, if α = 2. (New)

• New extended alpha power transformed exponential, if α = 1. (New)

In this section, we derive some mathematical properties of the NEx-APT distribution, such as quantile function, moments and moment generating function.

3.1. Quantile function

The quantile function of the NEx-APT distribution, denoted by Q(u), can be obtained by inverting its cdf (1.2). Therefore, we have

where t is the solution of the expression ${\alpha }_1^t+\overline{\beta }t-u({\alpha }_1-\beta )-1=0$, and u ∈ (0, 1). The nonlinear expression (3.1) can be used to obtain random numbers for the NEx-APT distribution.

3.2. Moments

In this sub-section, we intend to derive the moments and the moment generating function of the NEx-APT distribution. Let X follow (1.3), then, the r^th moment of X is derived as

and using (1.3) in (3.2), we have

The function ${\alpha }_1^x$ has the Maclaurin series given by

Taking x = F (x; ξ) in (3.4), we get

Using (3.5) in (3.3), we have

where ${\mathrm{\Lambda }}_{r,n}={\int }_{-\infty }^{\infty }{x}^{r}f\hspace{0.17em}(x,\xi )\hspace{0.17em}F\hspace{0.17em}{(x;\xi )}^n\hspace{0.17em}dx$ and ${\mathrm{\Lambda }}_r={\int }_{-\infty }^{\infty }{x}^{r}f\hspace{0.17em}(x,\xi )\hspace{0.17em}dx$.

Furthermore, the moment generating function of the NEx-APT random variable X, M_X (t), is given by

In this section, we present certain characterizations of the NEx-APT distribution in the following directions: (i) based on a simple relationship between two truncated moments and (ii) in terms of the reverse hazard function. It should be mentioned that for the characterization (i) the cdf is not required to have a closed form.

We present our characterizations (i)–(ii) in two subsections.

4.1. Characterizations based on two truncated moments

In this subsection, we present characterizations of NEx-APT distribution in terms of a simple relationship between two truncated moments. The first characterization result employs a theorem due to Glänzel (1987), see Theorem 4.1 below. Note that the result holds also when the interval H is not closed. Moreover, in it could be also applied when the cdf F does not have a closed form. As shown in Glänzel (1990), this characterization is stable in the sense of weak convergence.

The reverse hazard function, r_G, of a twice differentiable distribution function, G, is defined as

In this subsection, we present a characterization of the NEx-APT distribution in terms of the reverse hazard function.

Proposition 2

Let X : Ω → ℝ be a continuous random variable. The random variable X has pdf (1.3) only if its reverse hazard function r_G (x) satisfies the following differential equation.

In this section, we estimate the parameters of the NEx-APT distribution via the method of maximum likelihood and provide Monti Carlo simulation to evaluate the performance of these estimators.

5.1. Maximum likelihood estimation

Let x₁, x₂, . . . , x_n be observed values of a random sample from NEx-APT distribution with parameters (α₁, β, ξ). The log-likelihood function of this sample is

where, Θ = (α₁, β, ξ)^T be the 3 × 1 parameter vector. The partial derivatives of (5.1) are

and

Setting (∂/∂α₁)ℓ(Θ), (∂/∂β)ℓ(Θ), and (∂/∂ξ)ℓ(Θ) equal to zero and solving numerically these expressions simultaneously, yields the maximum likelihood estimates of (α₁, β, ξ).

5.2. Asymptotic confidence bounds

In this subsection, we derive the asymptotic confidence intervals of the unknown parameters of the NEx-APT distributions. The simplest large sample approach is to assume that the maximum likelihood estimators (α̂₁, β̂, ξ̂) are approximately multivariate normal with mean (α₁, β, ξ) and covariance matrix $I_0^{-1}$, where $I_0^{-1}$ is the inverse of the observed information matrix defined by

The second partial derivatives included in $I_0^{-1}$ are given as follows

and

The above approach is used to derive the (1 − δ)100% confidence intervals for the parameters α₁, β, and ξ as in the following forms ${\widehat{\alpha }}_1\pm Z_{\delta /2}\sqrt{\text{var}({\widehat{\alpha }}_1)},\widehat{\beta }\pm Z_{\delta /2}\sqrt{\text{var}(\widehat{\beta })}$, and $\widehat{\xi }\pm Z_{\delta /2}\sqrt{\text{var}(\widehat{\xi })}$, where Z_δ/2 is the upper (δ/2)^th percentile of the standard normal distribution.

5.3. Simulation study of the NEx-APTW distribution

In order to evaluate the performances of the maximum likelihood estimators of the (α₁, β, ξ), a simulation study is conducted. The process is carried out as:

• We generate N = 750 samples of sizes n = 25, 50, . . . , 750 from the NEx-APTW distribution.

• Initial values for the parameters are selected as given in Table 1.

• Compute the biases and MSEs given by $\text{Bias}(\widehat{w})=(1/750)\hspace{0.17em}{\mathrm{\Sigma }}_{i=1}^{750}({\widehat{w}}_i-w)$ and $\text{MSE}(\widehat{w})=(1/750)\hspace{0.17em}{\mathrm{\Sigma }}_{i=1}^{750}{({\widehat{w}}_i-w)}^2$, for w = (α₁, β, ξ), respectively.

One of the most important tasks of actuaries is to evaluate the exposure to market risk in a portfolio of instruments, which arise from changes in underlying variables such as prices of equity, interest rates or exchange rates. In this section, we calculate two well-known and important risk measures Value at Risk (VaR) and Tail VaR (TVaR) for the proposed distribution, which play a crucial role in portfolio optimization under uncertainty. Furthermore, based on these measures, a simulation study is performed and we show that the proposed distribution has heavier tails the Weibull and exponentiated Weibull (EW) distributions.

6.1. Value at risk

In the context of actuarial sciences, the measure VaR is widely used by practitioners as a standard financial market risk. It is also known as the quantile risk measure or quantile premium principle. VaR of a random variable X is the q^th quantile of its cdf and specified with a given degree of confidence say q (typically 90, 95, or 99%), see Artzner (1999). If X has pdf (1.3), then the VaR of X is given by

where t is the solution of the expression ${\alpha }_1^t+\overline{\beta }t-q\hspace{0.17em}({\alpha }_1-\beta )-1=0$.

6.2. Tail value at risk

Another important measure is TVaR, also known as conditional tail expectation (CTE) used to quantify the expected value of the loss given that an event outside a given probability level has occurred. Let X follow the NEx-APTW distribution, then the TVaR of X is defined as

Using (1.3) in (6.2), we have

Using the series ${\alpha }_1^x={\mathrm{\Sigma }}_{n=0}^{\infty }{(\text{log\hspace{0.17em}}({\alpha }_1))}^n\hspace{0.17em}(x^n/n!)$,we have

or

Based on actuarial measures, a simulation study is performed for the Weibull, exponentiated (EW) and proposed model for the selected parameters values. A model with higher values of the Risk measures (VaR and TVaR) is said to have a heavier tail. The simulated results provided in Tables 2 and 3 show that the proposed model has higher values of the risk measures than the other competitive distributions.

Figure 4 displays the simulation results provided in Table 2; in addition, Figure 5 displays the simulation results provided in Table 3.

Actuaries are looking for new distributions to provide an adequate fit to heavy tailed data in actuarial, financial sciences and related areas. In this section, we analyze a real data set from insurance sciences to demonstrate the flexibility of the NEx-APTW distribution. We also calculate actual measures of the Weibull, EW, and NEx-APTW distributions using a real data set.

7.1. Hospital cost insurance data

The data set representing hospital costs in the state of Wisconsin is published by the Office of the Health Care Information, Wisconsin’s Department of Health and Human Resources. The data set is available at: https://www.dhs.wisconsin.gov/stats/index.htm. The comparison of the proposed method is made with the other ten (two, three and four parameters) well-known distributions. The cdf’s of the competitive distributions are:

• Weibull distribution

  $$G\hspace{0.17em}(x)=1-e^{-\gamma x^{\alpha }},\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }x\ge 0,\hspace{0.17em}\alpha ,\gamma >0.$$

• Ex-APTW distribution

  $$G\hspace{0.17em}(x)=\frac{{\alpha }_1^{\left(1-e^{-\gamma x^{\alpha }}\right)}-e^{\left(1-e^{-\gamma x^{\alpha }}\right)}}{{\alpha }_1-e},\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }x\ge 0,\hspace{0.17em}{\alpha }_1,\alpha ,\gamma >0,\hspace{0.17em}{\alpha }_1\ne e.$$

• Lomax distribution

  $$G\hspace{0.17em}(x)=1-{(1+\gamma x)}^{-\alpha },\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }x\ge 0,\hspace{0.17em}\alpha ,\gamma >0.$$

• Burr-XII (B-XII) distribution

  $$G\hspace{0.17em}(x)=1-{(1+x^{\gamma })}^{-\alpha },\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }x\ge 0,\hspace{0.17em}\alpha ,\gamma >0.$$

• Pareto distribution

  $$G\hspace{0.17em}(x)=1-{\left(\frac{\gamma }{x}\right)}^{\alpha },\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }x\ge 0,\hspace{0.17em}\alpha ,\gamma >0.$$

• The alpha power transformed Weibull (APTW) distribution

  $$G\hspace{0.17em}(x)=\frac{{\alpha }_1^{\left(1-e^{-\gamma x^{\alpha }}\right)}-1}{{\alpha }_1-1},\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }x\ge 0,\hspace{0.17em}{\alpha }_1,\alpha ,\gamma >0,\hspace{0.17em}{\alpha }_1\ne 1.$$

• The Marshall-Olkin (MOW) distribution

  $$G\hspace{0.17em}(x)=\frac{\left(1-e^{-\gamma x^{\alpha }}\right)}{\sigma +(1-\sigma )(1-e^{-\gamma x^{\alpha }})},\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }x\ge 0,\hspace{0.17em}\alpha ,\gamma ,\sigma >0.$$

• EW distribution

  $$G\hspace{0.17em}(x)={\left(1-e^{-\gamma x^{\alpha }}\right)}^a,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }x\ge 0,\hspace{0.17em}\alpha ,\gamma ,a>0.$$

• Kumaraswamy Weibull (Ku-W) distribution

  $$G\hspace{0.17em}(x)=1-{\left\{1-{\left(1-e^{-\gamma x^{\alpha }}\right)}^a\right\}}^b,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }x\ge 0,\hspace{0.17em}\alpha ,\gamma ,a,b>0.$$

• The beta Weibull (BW) distribution

  $$G\hspace{0.17em}(x)=I_{(1-e^{-\gamma x^{\alpha }})}(a,b),\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }x\ge 0,\hspace{0.17em}\alpha ,\gamma ,a,b>0.$$

To determine the goodness-of-fit among the applied distributions, we consider certain goodness-of-fit measures such as Cramer-Von-Messes (CM) test statistic, Anderson Darling (AD) test statistic and Kolmogorov-Simonrove (KS) test statistic with corresponding p-value. The measure are given by:

• The AD test statistic

  $$\text{AD}=-n-\frac{1}{n}\sum _{i=1}^n(2i-1)\hspace{0.17em}[\text{log\hspace{0.17em}}G(x_i)+\text{log}\{1-G\hspace{0.17em}(x_{n-i+1})\}],$$

  where n = the sample size, x_i = the i^th sample, calculated when the data is sorted in ascending order.

• The CM test statistic

  $$\text{CM}=\frac{1}{12n}+\sum _{i=1}^n{\left[\frac{2i-1}{2n}-G\hspace{0.17em}(x_i)\right]}^2.$$

• The KS test statistic is given by

  $$\text{KS}={\text{sup}}_x\hspace{0.17em}[G_n\hspace{0.17em}(x)-G\hspace{0.17em}(x)],$$

  where G_n (x) is the empirical cdf and sup_x is the supremum of the set of distances.

A distribution with lower values of these analytical measures is considered as a good candidate model among the applied distributions for the underlying data sets. Based on the considered measures, the NEx-APTW distribution has the lowest values among all fitted models for the hospital cost insurance data. Table 4 reports parameter values with standard errors in parenthesis. In support of the numerical measures provided in Table 5, the empirical cdf and sf of the NEx-APTW are plotted in Figure 6. From Figure 6, we can see that the proposed model fit empirical cdf and sf very closely. In addition, the PP plot of the NEx-APTW distribution for the respective data set is plotted in Figure 7 and shows that the proposed provide best fit to the considered data. The box plot of the data set is also sketched in Figure 7 showing that the hospital cost insurance data is skewed to the right.

7.2. Computation of actuarial measures using real data set

In this sub-section, we compute VaR and TVaR measures of Weibull, EW and the NEx-APTW distributions using estimated parameters values analyzed in Subsection 7.1. Table 6 reports the numerical results. A model with higher values of the risk measures possesses the heavier tails. The numerical results for the actuarial measures of the proposed and the other distributions show that the proposed distribution has a heavier tail than Weibull and EW distributions. In addition, it can be used as a good candidate model for modeling heavy tailed insurance data sets.

In this article, a new family of distributions called a new extended alpha power transformed family has been proposed. The proposed method examines a four-parameter special model of a new extended alpha power transformed Weibull distribution. Actuarial measures of the proposed model are also calculated and a simulation study is conducted to show the usefulness of the proposed method in actuarial sciences. A practical application to the heavy tailed insurance data is analyzed and the comparison of the proposed model with the other nine well-known competitors are presented. Actuarial measures based on a real data set is also calculated which shows that the proposed model may be a good candidate model to analyze actuarial data sets. We hope that the proposed method will attract a wider applications in actuarial sciences and related fields.