# Estimating Quantile Families of Loss Distributions for Non-Life Insurance Modelling via L-Moments

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## Abstract

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## 1. Introduction: Context of Modelling Losses in General Insurance

## 2. General Families of Quantile Transform Distributions

#### 2.1. The Tukey Family of Loss Models

**Definition 2.1**(Tukey transformations).

**Definition 2.2**(Tukey’s kurtosis transformations of types h, k and j).

**Definition 2.3**(Tukey’s skewness transformation).

#### 2.2. Examples of the g-and-h, g, h, and h-h Loss Models

**Definition 2.4**(g-and-h distributional family).

**Remark 2.1.**

**Definition 2.5**(g distributional family).

**Remark 2.2.**

**Definition 2.6**(h distributional family).

**Definition 2.7**(Double h-h distributional family).

**Definition 2.8**(Generalised g-and-h distributional family).

#### 2.3. Examples of the g-and-k and g-and-j Loss Models

**Definition 2.9**(g-and-k distributional family).

**Definition 2.10**(g-and-j distributional family).

## 3. Distribution and Density Functions of the g-and-h, g, h, h-h, and g-and-k Families

**Proposition 3.1.**

**Lemma 3.1**(Super class ${T}_{hjk}$ density).

**Lemma 3.2**(g-and-h distribution and density functions (constant g and h with $h>0$)).

- the g-and-h transformation can be shown to be strictly monotonically increasing in its argument, that is, for all ${w}_{1}\le {w}_{2}$ one has ${T}_{gh}\left({w}_{1}\right)\le {T}_{gh}\left({w}_{2}\right)$;
- if $a\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0$, then the g-and-h transformation satisfies the condition ${T}_{(-g)h}\left(W\right)=-{T}_{gh}(-W)$.

**Lemma 3.3**(Generalised g-and-h distribution and density functions).

**Lemma 3.4**(g-and-k distribution and density functions).

## 4. Statistical Properties of g-and-h, g, h, h-h, and g-and-k Families Related to Claim Modelling

**Proposition 4.1**(Mode of the g-and-h, generalised g-and-h and g-and-k densities).

**Proposition 4.2**(Median of the g-and-h, generalised g-and-h and g-and-k densities).

**Remark 4.1.**

**Proposition 4.3**(Moments of the g-and-h density).

**Proposition 4.4**(Moments of the generalised g-and-h density).

**Proposition 4.5**(Moments of the g-and-k density).

**Remark 4.2.**

**Definition 4.1**(Generalised skewness and kurtosis functionals).

**Definition 4.2**(Generalised skewness and kurtosis for g-and-h and generalised g-and-h families).

#### Tail Properties of the g-and-h and g-and-k Loss Models

**Definition 4.3**(Regularly varying function).

- it is defined on some neighbourhood $[{x}_{0},\infty )$ of infinity;
- it satisfies the following limiting relationship$$\begin{array}{c}\hfill \underset{x\to \infty}{lim}\frac{f\left(\lambda x\right)}{f\left(x\right)}={\lambda}^{\alpha},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \lambda >0.\end{array}$$

**Definition 4.4**(Regularly varying random variable).

**Theorem 4.1**(Properties of regularly varying distributions).

- If ${F}_{X}\left(x\right)$ is absolutely continuous with density ${f}_{X}\left(x\right)$ such that for some $\alpha >0$ one has the limit$$\begin{array}{c}\hfill \underset{x\to \infty}{lim}\frac{x{f}_{X}\left(x\right)}{{\overline{F}}_{X}\left(x\right)}=\alpha .\end{array}$$
- If the density ${f}_{X}\left(x\right)$ for loss distribution ${F}_{X}\left(x\right)$ is assumed to be regularly varying with index $-(1+\alpha )$ for some $\alpha >0$. Then the following limit,$$\begin{array}{c}\hfill \underset{x\to \infty}{lim}\frac{x{f}_{X}\left(x\right)}{{\overline{F}}_{X}\left(x\right)}=\alpha ,\end{array}$$

**Proposition 4.6**(Index of regular variation of g-and-h distribution).

**Proposition 4.7**(h-type tail behaviour).

**Theorem 4.2**(Slow variation representation of g-and-h severity models).

**Remark 4.3.**

**Proposition 4.8**(Index of regular variation of the generalised g-and-h distribution).

## 5. Estimating the General Tukey Family Loss Model Parameters

#### 5.1. Estimating the g-and-h loss Model Parameters

#### 5.1.1. New Robust Estimation Approach for g-and-h Loss Models based on the Method of L-moments

**Remark 5.1.**

**Proposition 5.1**(g-family loss model population L-moments).

**Proposition 5.2**(h-family loss model population L-skewness and L-kurtosis).

**Proposition 5.3**(k-family loss model population L-skewness and L-kurtosis).

**Proposition 5.4**(g-and-h family loss model population L-skewness and L-kurtosis).

**Proposition 5.5**(L-moments of affine functions of random variables).

**Definition 5.1**(L-moment Tukey transforms).

- The γ-and-κ Tukey family transformation is given by$$\begin{array}{c}\hfill X={T}_{\gamma ,\kappa}\left(W\right)={\gamma}^{-1}\left(exp\left(\gamma W\right)-1\right)exp\left(\kappa \right|W\left|\right).\end{array}$$
- The ${\kappa}_{L}$-and-${\kappa}_{R}$ Tukey family transformation is given by$$\begin{array}{c}\hfill X={T}_{{\kappa}_{L},{\kappa}_{R}}\left(W\right)=\left\{\begin{array}{cc}Wexp\left({\kappa}_{L}\left|W\right|\right),\hfill & W\le 0\hfill \\ Wexp\left({\kappa}_{R}\left|W\right|\right),\hfill & W\ge 0.\hfill \end{array}\right.\end{array}$$

**Proposition 5.6**(L-moment estimators for the γ-and-κ Tukey family).

**Remark 5.2.**

**Proposition 5.7**(L-moment estimators for the ${\kappa}_{L}$-and-${\kappa}_{R}$ Tukey family).

## 6. Simulation Study: Comparison of Estimation Procedures for the Tukey Family of Claim Models

## 7. Empirical Application

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of Proposition 4.8

**Proof.**

#### Appendix A.2. Proof of Proposition 5.1

**Proof.**

#### Appendix A.3. Proof of Proposition 5.2

**Proof.**