The application of accumulation tests in peaks over threshold modeling with fire insurance data sets

Abstract

Modeling excesses remains to be an important topic in insurance data modeling. Among the alternatives to modeling excess, the Peaks Over Threshold (POT) framework with Generalized Pareto distribution (GPD) is regarded as an efficient approach due to its flexibility. However, selecting an appropriate threshold for such a framework is a major difficulty. To address such difficulty, we applied several accumulation tests along with Anderson-Darling test to determine an optimal threshold. Limited simulations were conducted to assess the performance of accumulation tests. Based on the selected thresholds, the fitted GPD with the estimated quantiles can be found. We applied the procedure to the wellknown Norwegian fire insurance data and AON Re Belgian fire loss data. With the selected thresholds, confidence intervals for the Value-at-Risks (VaR) were constructed correspondingly. The accumulation test approach provides satisfactory performance in modeling the VaR of both data sets compared to the previous graphical methods.

Keywords

generalized pareto distribution sequential selection procedure peaks over threshold modeling tail risk estimation accumulation tests

1. Introduction

The selection of a proper risk modeling approach is a major topic in the insurance industry. Among all existing approaches, parametric modeling is preferred for its flexibility and interpretability. Within parametric methods, the Peaks Over Threshold (POT) approach with Generalized Pareto Distribution (GPD) remains popular due to the support of the Pickands-Balkema-De Haan theorem (McNeil, 1997). However, the performance of such approach is mainly determined by the selection of thresholds. If the threshold is chosen too low, the excess distribution might not be well approximated by GPD; If the threshold is chosen too high, number of excesses would not be enough to establish a useful model.

To identify an appropriate threshold in GPD modeling, many methods have been proposed in the literature (Scarrott & MacDonald, 2012; Langousis et al., 2016). The Gertensgarbe-Werner (GW) plot was first developed to identify the change point that separates non-extreme and extreme parts of the data (Gerstengarbe & Werner, 1989). However, the uniform assumption on the data in this method is an essential limitation. Therefore, this method should be avoided in real applications (Langousis et al., 2016). Mean Residual Life (MRL) plot was also widely used to determine GPD thresholds (Davison & Smith, 1990; Lang et al., 1999; Coles, 2001). This method aims to identify the threshold by looking for the linear trend of the candidate thresholds in GPD parameters. However, the selection of the threshold is usually done by a visual inspection, which is subjective. Compared to the methods mentioned above, threshold selection procedures using Goodness-of-Fit (GoF) tests can be interpreted with better support from statistical theory.

The objective of GPD threshold selection procedures based on the GoF tests is to identify the lowest threshold such that the exceedances above the threshold fit a GPD well. For a single fixed threshold, the GoF test can be easily carried out (Choulakian & Stephens, 2001). However, a candidate set of possible thresholds is usually involved in finding the 'optimal' threshold. Thus, error control is necessary since the candidate thresholds are tested simultaneously, which generates a multipletesting problem. Because of the natural ordering of the candidate thresholds, the hypotheses for the GoF tests should also be ordered. Therefore, False Discovery Rate (FDR) control procedures such as the Benjamini-Hochberg (Benjamini & Hochberg, 1995) or Benjamini-Yekutieli (Benjamini & Yekutieli, 2001) cannot be used without modifications. To adapt FDR control procedures to the ordered hypotheses testing structure, ForwardStop was proposed (G'Sell et al., 2016). ForwardStop provides a way to choose a stopping point for ordered hypotheses such that the hypotheses occur before the stopping point can be rejected with adequate error control. (Bader et al., 2018) Then, this procedure was adapted to the selection of GPD thresholds and applied to rainfall data.

The accumulation tests (Li & Barber, 2017) can be seen as a generalization of the ForwardStop procedure. Any functions that satisfy certain regularity conditions can be utilized in the ordered hypotheses testing procedure with FDR control at a desired level (Li & Barber, 2017). In addition to ForwardStop, procedures like SeqStep (Barber & Candès, 2015) and HingeExp (Li & Barber, 2017) can also control FDR under different scenarios.

In this paper, we attempt to adapt the accumulation test procedures to GPD threshold selection problems and propose a generalized framework of GPD threshold selection based on accumulation tests. We aim to assess the performances of accumulation test threshold selection methods with numerical simulations. To investigate the applicability of the methods with real data and provide useful insights to decision-makers in the field of insurance, we also apply the methods to two fire insurance data sets and assess the performances of the methods in terms of estimated Value-at-Risk (VaR).

The rest of the paper is organized as follows: In section 2, an introduction to GPD model and POT method is given, with a brief introduction of accumulation tests for sequential hypotheses testing. Section 3 provides simulations to assess the performance of different accumulation tests in selecting GPD thresholds. The real data analysis of the fire insurance data is given in Section 4. The final section concludes with a discussion and our thoughts on the future directions.

2. Methodology

2.1. POT Modeling and Threshold Selection Procedure

A common practice in extreme value modeling is to apply the Peaks Over Threshold (POT) approach to exceedances beyond a certain threshold $μ$ . This approach is built upon the second theorem of extreme value theory (the Pickands-Balkema-De Haan theorem). Under appropriate regularity conditions, the conditional excess function $F_{u}$ of a sequence of i.i.d. random variables ${X_{1}, X_{2}, \dots}$ can be well approximated by a Generalized Pareto Distribution with location parameter u. A Generalized Pareto distribution is defined with the following cumulative distribution function (CDF):

F (x | γ, σ, μ) = {\begin{array}{ll} 1 - {(1 + \frac{γ (x - μ)}{σ})}^{- \frac{1}{γ}} & γ \neq 0 \\ 1 - e^{- \frac{x - μ}{σ}} & γ = 0 \end{array}

(1)

where

x > μ

. When

γ = 0

, GPD distribution is defined as an exponential distribution with parameter

σ

The choice of $μ$ is important in establishing a valuable model when applying the POT method. By the second theorem of extreme value theory, the exceedances over the sufficiently high threshold $μ$ follow a GPD approximately. However, if the threshold is chosen to be too high, the corresponding fitted GPD model would have a poor ability to describe the data distribution.

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample of size n. Assume this random sample is taken such that the regularity conditions for the Pickands-Balkema-De Haan theorem hold. Then, for an appropriate threshold $μ$ , the exceedances $Y_{i} = X_{i} - μ$ approximately follow a GPD. Thus, the objective is to identify the lowest possible $μ$ such that the exceedances approximately follow GPD.

2.2. MLE of

σ

and

γ

when

μ

is known

Consider $X_{1}, X_{2}, \dots, X_{n}$ to be a random sample of size n and assume the regularity conditions for the Pickands-Balkema-De Haan theorem hold for this sample. Suppose the threshold $μ$ is known and both $σ$ and $γ$ are unknown. Suppose $μ$ is fixed, let $Y_{i} = X_{i} - μ$ for $i = 1, 2, \dots, n$ . The log-likelihood of $Y_{1}, Y_{2}, \dots, Y_{n}$ is as follows:

l (σ, γ | Y_{1}, Y_{2}, \dots, Y_{n}) = - n l o g (σ) - (1 + \frac{1}{γ}) \sum_{i = 1}^{n} l o g (1 + \frac{γ y_{i}}{σ}) .

(2)

Essentially, the maximum of the likelihood function can be obtained:

{\begin{array}{l} \frac{\partial l}{\partial σ} = 0 \\ \frac{\partial l}{\partial γ} = 0. \end{array}

Davison showed that the solution of the above system of equations can be reduced to finding the solution of equation with one variable, with the following steps (Davison & Smith, 1990):

Let $\frac{γ}{σ} = θ$ . The log-likelihood function becomes:

l (θ, γ | Y_{1}, Y_{2}, \dots, Y_{n}) = - n l o g (γ) + n l o g (θ) - (1 + \frac{1}{γ}) \sum_{i = 1}^{n} l o g (1 + θ y_{i})

(3)

For any fixed value of $θ$ , the ML estimate of the $γ$ given $θ$ is determined by $\frac{\partial l}{\partial γ} = 0$ :

\hat{γ} = γ \frac{\sum_{i = 1}^{n} \log (1 + θ y_{i})}{n}

(4)

Plug Equation (4) back to Equation (3). The optimization becomes a onedimensional search of $θ$ :

l (θ) = - n - n l o g (- \frac{1}{n} \sum_{i = 1}^{n} log (1 + θ y_{i})) - \sum_{i = 1}^{n} l o g (1 + θ y_{i})

(5)

Eventually, the estimates of $γ$ and $σ$ can be expressed as follows, given the estimate of $θ$ has been determined by maximizing Equation (5) numerically:

{\begin{array}{l} \hat{γ} = \frac{1}{n} \sum_{i = 1}^{n} log (1 - \hat{θ} y_{i}) \\ \hat{σ} = \frac{\hat{γ}}{\hat{θ}} . \end{array}

(6)

Grimshaw showed that the MLE for

γ

and

σ

might not exist in some situations (Grimshaw, 1993). However, Choulakian pointed out that this is not likely to occur especially when the sample size is large enough (Choulakian & Stephens, 2001). Under most of the scenarios, the MLE of

γ

and

σ

can be obtained with the procedure introduced earlier in this subsection.

2.3. GoF Tests for GPD with Anderson-Darling Statistics

In order to assess the performance of GPD fit, several tests were developed generally based on the GoF tests such as Anderson-Darling (AD) test. The details for the AD test are provided as follows: Given a sample z is arranged in increasing order as $z_{(1)} < z_{(2)} < \dots < z_{(n)}$ and suppose we want to test if z follows a standard uniform distribution, the AD test statistic for the uniform distribution is defined as:

A^{2} = - n - \frac{1}{n} \sum_{i = 1}^{n} (2 i - 1) (log (z_{(i)}) - log (1 - z_{(n + 1 - i)}))

(7)

Therefore, to use the AD test for assessing the GoF of any distribution with the CDF

F (x | θ)

properly defined, one could always use the probability integral transformation

(z_{i} = F (x_{i} | \hat{θ}))

on the ordered sample first and apply the AD test for the uniform distribution. Notice

\hat{θ}

is the MLE of the parameter

θ

under

H_{0}

, where,

H_{0}

is defined as:

H_{0} : x_{1}, x_{2}, \dots, x_{n} is an i .i .d . sample drawn with the CDF F (x | θ) .

In practice, the p-values are usually obtained by using the tables of critical values generated based on Monte Carlo simulations. The distribution assumption is rejected when p-value is small.

2.4. Existing method: Automated Threshold Selection Procedure

The automated threshold selection for GPD was developed by utilizing an ordered hypothesis testing procedure named ForwardStop (G'Sell et al., 2016; Bader et al., 2018). The selection procedure for GPD could be summarized as follows:

Pick l candidate thresholds as $μ_{1} < μ_{2} < \dots < μ_{l}$ .

For each $μ_{j} (j \in {1, 2, \dots, l})$ , shift the data by applying the transformation $Y_{i} = X_{i} - μ_{j} (i \in {1, 2, \dots, n}) . n$ is the sample size. After the shift, only the non-negative $Y_{i} \; s$ are reserved since only the data values beyond the threshold $μ_{j}$ are used in the fitting of a GPD model. Denote the non-negative $Y_{i}$ s as follows:

Y_{(j)} = {Y_{i} | Y_{i} \geq 0}

Therefore, with

Y_{(j)}

determined by each

μ_{j}

, The corresponding MLE of

γ

and

σ

(

γ_{j}

and

σ_{j}

) can be obtained by using the method introduced in section 2.2,

Then, for each $μ_{j}$ , the null hypothesis:

H_{j} : Y_{(j)} is drawn from a GPD distribution,

could be examined by using GoF tests. l different p -values are obtained correspondingly.

4.
The largest index for the candidate thresholds to be rejected is determined by applying the ForwardStop procedure (G'Sell et al., 2016) as follows:
$\hat{k} = \underset{k \in 1, \dots, l}{argmax} {\frac{1}{k} \sum_{i = 1}^{k} - log (1 - p_{i}) \leq α},$
(8)
where $α$ is pre-specified before the selection procedure. $μ_{\hat{k} + 1}$ is then the optimal threshold.

The above procedure could produce good estimates for the thresholds of GPD if the candidates are chosen properly. For instance, when modeling the rainfall data, (Bader et al., 2018) selected the thresholds within the $70^{th\;}$ and the $98^{th\;}$ percentile of the rainfall data and obtained a good geographical explanation based on the results obtained from the automated threshold selection procedure.

2.5. Accumulation Tests

Li and Barber (Li & Barber, 2017) later generalized the results from different ordered hypothesis testing procedures (G'Sell et al., 2016; Barber & Candès, 2015) including ForwardStop. A family of accumulation tests were developed to choose a cutoff point k such that first k hypotheses are rejected, with a modified false discovery rate (FDR) being controlled. Assume n hypotheses are ordered sequentially as $H_{1}, H_{2}, \dots, H_{n}$ with corresponding p -values $p_{1}, p_{2}, \dots, p_{n}$ . An accumulation function is defined as an integrable nondecreasing function h that maps $[0, 1]$ to $[0, \infty]$ . Then an accumulation test associated with the function h chooses the cutoff $\hat{k}$ as:

\hat{k} = \underset{k \in 1, \dots, l}{argmax} {\frac{1}{k} \sum_{i = 1}^{k} h (p_{i}) \leq α},

Therefore, if we use any function h that satisfies the conditions for the accumulation functions, a generalized GPD threshold selection procedure can be implemented along with the automated threshold selection procedure developed by Bader (Bader et al., 2018). The flow chart for the generalized GPD threshold selection procedure is summarized in Figure 1.

Figure 1.

The flow chart of the accumulation test threshold selection framework where the hypotheses $H_{1}, H_{2}, \dots, H_{\hat{k}}$ can be rejected at the pre-specified FDR level $α$ . Therefore, ForwardStop is essentially a special case of the accumulation test procedure when $h (p) = - log (1 - p)$ .

2.6. Var and the Interval Estimation of var for GPD

The estimation of VaR is important for the insurance data modeling. For a loss random variable, VaR at the level of p is defined as:

P (X < Va R_{p} (X)) = p,

In the insurance industry, VaR is the amount of capital that one insurance company needs to maintain to prevent the company from bankruptcy due to very extreme claims. Given that a random variable X follows a GPD given CDF in (1), the VaR of X could be easily derived in the following form:

Va R_{p} (X) = μ + \frac{σ}{γ} [{(1 - p)}^{- γ} - 1] .

(9)

Therefore, for given

μ

\hat{σ}

and

\hat{γ}

, the ML estimate of

V a_{p} (X)

is:

\hat{Va R_{p} (X)} = \hat{μ} + \frac{\hat{σ}}{\hat{γ}} [{(1 - p)}^{- \hat{γ}} - 1] .

As we explained in section 2.2, the MLE of

σ

and

γ

could be obtained numerically for a given

μ

. (Smith, 1984) proved that when

σ > - \frac{1}{2}

, the MLE of

σ

and

γ

are asymptotically normal and efficient:

[\begin{array}{l} \hat{σ} \\ \hat{γ} \end{array}] \sim AN ([\begin{array}{l} σ \\ γ \end{array}], Σ),

where

Σ = n^{- 1} [\begin{array}{ll} 2 σ^{2} (1 + γ) & σ (1 + γ) \\ σ (1 + γ) & {(1 + γ)}^{2} \end{array}]

. With the MLE of

σ

and

γ

, the asymptotic variance of

\hat{Va R_{p} (X)}

could be derived from the multivariate Delta method. Consider

Va R_{p} (X) = g (σ, γ | μ) = μ + \frac{σ}{γ} [{(1 - p)}^{- γ} - 1]

, the gradient is:

\nabla g = {[\frac{{(1 - p)}^{- γ} - 1}{γ}, \frac{σ {(1 - p)}^{- γ} [{(1 - p)}^{γ} - γ log (1 - p) - 1]}{σ^{2}}]}^{T} .

By Delta method,

\hat{Va R_{p} (X})

has the asymptotic distribution

N (Va R_{p} (x), ω^{2})

, where

ω^{2} = \nabla g^{T} Σ \nabla g = 2 \frac{{(1 - p)}^{- γ} - 1}{γ^{2}} σ^{2} (1 + γ) + \frac{σ {(1 - p)}^{- γ} {[{(1 - p)}^{γ} - γ log (1 - p) - 1]}^{2}}{σ^{2}} {(1 + γ)}^{2}

(10)

By utilizing Delta method, the asymptotic

100 (1 - α) \%

CI for

VaR (X)

is given as:

\hat{Va R_{p} (X}) \pm z_{α / 2} \hat{ω},

(11)

where

z_{α}

represents the

α^{th\;}

quantile of a standard Gaussian random variable.

3. Simulation

3.1. Simulation Settings

To assess the performance of sequential testing procedures, we conducted limited simulations. Three different accumulation tests were assessed for our simulations. The accumulation functions of these three accumulation tests are provided as follows:

ForwardStop: $h (p) = - log (1 - p)$

SeqStep: $h (p) = C \cdot 1_{{p > 1 - \frac{1}{C}}}$

HingeExp: $h (p) = C \cdot 1_{{p > 1 - \frac{1}{C}}} log (C (1 - p))$

For the value of C involved in SeqStep and HingExp, we selected

C = 2

as Li and Barber (Li & Barber, 2017) recommended in their work of accumulation tests. For the significance threshold

α

, we selected

α = 0.01

and

α = 0.05

in the simulations.

3.2. Scenarios of Composite Density with GPD Tails

To demonstrate the ability of accumulation tests in selecting the thresholds for a GPD distribution in a POT model, we generated samples from four parametric composite distributions. The composite distributions were widely used in the modeling of insurance claim sizes (Cooray & Ananda, 2005; Scollnik, 2007; Scollnik & Sun, 2012; Brazauskas & Kleefeld, 2016; Grün & Miljkovic, 2019; Liu & Ananda, 2023, 2022; Mutali & Vernic, 2022; Deng & Aminzadeh, 2022; Aminzadeh & Deng, 2019; Nadarajah, 2005; CalderínOjeda, 2018). The details of the simulation scenarios are listed in Table 1. For each scenario, $r = 1, 000$ replicates were generated.

Table 1.
Simulation Results for Three Different Tests Under Different Simulation Scenarios $(α = 0.01)$ _.

Forward Stop $\begin{array}{l} SeqStep \\ (C = 2) \end{array}$ $\begin{array}{l} HingeExp \\ (C = 2) \end{array}$

Scenario True Threshold Head Weight Tail Weight Sample Size Mean RMSE Mean RMSE Mean RMSE

$GPD (μ = 0.5, σ = 1, γ = 1)$ 0.5 – – $n = 100$ 0.52 0.05 2.22 1.04 5.27 1.69

$n = 200$ 0.51 0.04 2.01 0.88 4.97 1.47

$n = 500$ 0.51 0.01 1.84 0.52 4.55 1.03

$GPD (μ = 2, σ = 1, γ = 1)$ 2 – – $n = 100$ 2.02 0.01 3.33 0.67 8.66 2.12

$n = 200$ 2.02 0.01 3.31 0.49 8.18 1.76

$n = 500$ 2.01 0.003 3.26 0.18 7.91 0.78

$\begin{array}{l} lognormal (μ = 2, σ = 0.5) \\ GPD (μ = 2, σ = 0.8, γ = 0.8) \end{array}$ 2 0.3 0.7 $n = 100$ 1.94 0.08 5.58 1.13 9.50 1.55

$n = 200$ 1.95 0.07 4.89 0.98 8.77 1.38

$n = 500$ 1.98 0.09 4.14 0.58 7.64 1.06

$\begin{array}{l} lognormal (μ = 1, σ = 1) \\ GPD (μ = 1, σ = 1.3, γ = 1.5) \end{array}$ 1 0.2 0.8 $n = 100$ 0.90 0.11 9.90 4.16 43.74 23.40

$n = 200$ 0.91 0.09 9.04 3.57 35.68 17.77

$n = 500$ 0.95 0.04 8.15 2.15 29.18 12.38

					Forward Stop	$\begin{array}{l} SeqStep \\ (C = 2) \end{array}$	$\begin{array}{l} HingeExp \\ (C = 2) \end{array}$
$GPD (μ = 0.5, σ = 1, γ = 1)$	0.5	–	–	$n = 100$	0.52	0.05	2.22	1.04	5.27	1.69
$n = 200$	0.51	0.04	2.01	0.88	4.97	1.47
$n = 500$	0.51	0.01	1.84	0.52	4.55	1.03
$GPD (μ = 2, σ = 1, γ = 1)$	2	–	–	$n = 100$	2.02	0.01	3.33	0.67	8.66	2.12
$n = 200$	2.02	0.01	3.31	0.49	8.18	1.76
$n = 500$	2.01	0.003	3.26	0.18	7.91	0.78
$\begin{array}{l} lognormal (μ = 2, σ = 0.5) \\ GPD (μ = 2, σ = 0.8, γ = 0.8) \end{array}$	2	0.3	0.7	$n = 100$	1.94	0.08	5.58	1.13	9.50	1.55
$n = 200$	1.95	0.07	4.89	0.98	8.77	1.38
$n = 500$	1.98	0.09	4.14	0.58	7.64	1.06
$\begin{array}{l} lognormal (μ = 1, σ = 1) \\ GPD (μ = 1, σ = 1.3, γ = 1.5) \end{array}$	1	0.2	0.8	$n = 100$	0.90	0.11	9.90	4.16	43.74	23.40
$n = 200$	0.91	0.09	9.04	3.57	35.68	17.77
$n = 500$	0.95	0.04	8.15	2.15	29.18	12.38

The mean and the RMSE were used to assess the performances of three tests under all scenarios. The formula for RMSE is provided as follows:

R M S E = \sqrt{\frac{\sum_{i = 1}^{r} {({\hat{μ}}_{i} - μ)}^{2}}{r}}

where

{\hat{μ}}_{i}

denotes the chosen threshold for the

i

-th replicate,

μ

stands for the true threshold under each scenario, and r is the number of replicates.

3.3. Simulation Results

The simulation results are presented in Table 1 $(α = 0.01)$ and Table 2 $(α = 0.05)$ . For all scenarios, ForwardStop showed great performance, while SeqStep and HingeExp provided unsatisfactory performance. Notice for all sequential testing procedures, the RMSE decrease as the sample size increases.

Table 2.
Simulation Results for Three Different Tests Under Different Simulation Scenarios $(α = 0.05)$ .

Scenario Forward Stop $\begin{array}{l} SeqStep \\ (C = 2) \end{array}$ $\begin{array}{l} HingeExp \\ (C = 2) \end{array}$

Scenario True Threshold Head Weight Tail Weight Sample Size Mean RMSE Mean RMSE Mean RMSE

$GPD (μ = 0.5, σ = 1, γ = 1)$ 0.5 – – $n = 100$ 0.66 0.06 3.13 1.24 7.13 1.72

$n = 200$ 0.60 0.05 2.12 0.77 6.17 1.35

$n = 500$ 0.57 0.01 1.94 0.57 5.81 1.00

$GPD (μ = 2, σ = 1, γ = 1)$ 2 – – $n = 100$ 2.51 0.02 3.99 0.79 12.18 2.38

$n = 200$ 2.42 0.02 3.69 0.51 10.05 1.89

$n = 500$ 2.29 0.01 3.49 0.20 9.91 0.75

$\begin{array}{l} lognormal (μ = 2, σ = 0.5) \\ GPD (μ = 2, σ = 0.8, γ = 0.8) \end{array}$ 2 0.3 0.7 $n = 100$ 2.24 0.125 6.91 1.25 11.50 1.92

$n = 200$ 2.17 0.08 6.00 0.99 10.77 1.57

$n = 500$ 2.10 0.09 5.24 0.69 9.65 1.34

$\begin{array}{l} lognormal (μ = 1, σ = 1) \\ GPD (μ = 1, σ = 1.3, γ = 1.5) \end{array}$ 1 0.2 0.8 $n = 100$ 1.20 0.15 10.89 3.93 33.61 15.46

$n = 200$ 1.13 0.10 10.14 3.26 31.44 13.62

$n = 500$ 1.07 0.05 9.35 2.65 27.38 11.45

Scenario					Forward Stop	$\begin{array}{l} SeqStep \\ (C = 2) \end{array}$	$\begin{array}{l} HingeExp \\ (C = 2) \end{array}$
$GPD (μ = 0.5, σ = 1, γ = 1)$	0.5	–	–	$n = 100$	0.66	0.06	3.13	1.24	7.13	1.72
$n = 200$	0.60	0.05	2.12	0.77	6.17	1.35
$n = 500$	0.57	0.01	1.94	0.57	5.81	1.00
$GPD (μ = 2, σ = 1, γ = 1)$	2	–	–	$n = 100$	2.51	0.02	3.99	0.79	12.18	2.38
$n = 200$	2.42	0.02	3.69	0.51	10.05	1.89
$n = 500$	2.29	0.01	3.49	0.20	9.91	0.75
$\begin{array}{l} lognormal (μ = 2, σ = 0.5) \\ GPD (μ = 2, σ = 0.8, γ = 0.8) \end{array}$	2	0.3	0.7	$n = 100$	2.24	0.125	6.91	1.25	11.50	1.92
$n = 200$	2.17	0.08	6.00	0.99	10.77	1.57
$n = 500$	2.10	0.09	5.24	0.69	9.65	1.34
$\begin{array}{l} lognormal (μ = 1, σ = 1) \\ GPD (μ = 1, σ = 1.3, γ = 1.5) \end{array}$	1	0.2	0.8	$n = 100$	1.20	0.15	10.89	3.93	33.61	15.46
$n = 200$	1.13	0.10	10.14	3.26	31.44	13.62
$n = 500$	1.07	0.05	9.35	2.65	27.38	11.45

4. Real Data Application: Threshold Selection

In this section, the well-known Norwegian fire insurance data and AON Re Belgian fire loss data are used to assess the performance of different methods of GPD threshold selection.

4.1. Selection of Accumulation Functions and Methods for Comparison

The accumulation tests that we chose in the simulations were applied to the Norwegian fire insurance data set. In addition, we selected two most commonly-used methods (GW plot and MRL plot) for comparison purposes. We utilized the R packages "tea" and "eva" to construct the plots and select GPD thresholds.

4.2. Case 1: Norwegian Fire Insurance Data

Data

Norwegian fire insurance data contains the fire insurance claims from a Norwegian company from year 1972 to 1992. The dataset is available via the R package 'ReIns’ (Reynkens & Verbelen, 2020). No information regarding inflation adjustments was provided so we chose the claims from year 1985 to 1989. Moreover, only the damages over 500,000 Norwegian Krones (NKK) are available. For analysis concern, we scaled the data by $1, 000, 000$ . The summary statistics of the claims from year 1985 to 1989 are presented in Table 3.

Table 3.
Summary Statistics for Norwegian Fire Insurance Claims (1985–1989).

Year Sample Size Mean SD Q1 Q2 Q3 Maximum

1985 607 2.55 8.01 0.68 1.00 1.71 135.08

1986 647 2.48 9.70 0.70 0.99 1.55 188.27

1987 767 2.06 3.64 0.76 1.14 1.85 44.93

1988 827 3.18 17.68 0.76 1.18 2.05 465.37

1989 718 2.40 7.10 0.75 1.18 2.00 145.16

Year	Sample Size	Mean	SD	Q1	Q2	Q3	Maximum
1985	607	2.55	8.01	0.68	1.00	1.71	135.08
1986	647	2.48	9.70	0.70	0.99	1.55	188.27
1987	767	2.06	3.64	0.76	1.14	1.85	44.93
1988	827	3.18	17.68	0.76	1.18	2.05	465.37
1989	718	2.40	7.10	0.75	1.18	2.00	145.16

Result: Threshold selection

Table 4 presents the selection of GPD thresholds $μ$ , the corresponding MLE of the scale parameter $σ$ and the shape parameter $γ$ for the Norwegian fire insurance data with the GW plot, MRL plot, ForwardStop selection, the SeqStep selection, and HingeExp selection. Notice the GW plot and the MRL plot methods provided higher estimates for $μ$ compared to the selection methods based on the accumulation tests.

Table 4.

Selection of Thresholds $(μ)$ with MLEs of Scale $(σ)$ and Shape $(γ)$ Parameters for norwegian Fire Insurance Claims (1985–1989).

Year	Method	$α$	Chosen Threshold $(μ)$	Scale $(σ)$	Shape $(γ)$
1985	GW	–	2.57	2.76	0.84
	MRL	–	3.50	3.85	0.81
	ForwardStop	0.01	0.54	0.56	0.77
		0.05	0.58	0.52	0.82
	SeqStep $(C = 2)$	0.01	0.86	0.68	0.87
		0.05	1.19	0.88	0.92
	HingeExp $(C = 2)$	0.01	1.21	0.89	0.93
		0.05	1.38	0.89	1.00
1986	GW	–	3.36	2.46	1.01
	MRL	–	4.50	4.36	0.84
	ForwardStop	0.01	0.68	0.54	0.80
		0.05	1.08	0.66	0.97
	SeqStep $(C = 2)$	0.01	1.08	0.66	0.97
		0.05	1.15	0.69	1.00
	HingeExp $(C = 2)$	0.01	1.15	0.69	1.00
		0.05	6.97	3.52	1.15
1987	GW	–	3.50	3.01	0.54
	MRL	–	3.00	2.00	0.72
	ForwardStop	0.01	0.66	0.76	0.55
		0.05	0.97	0.77	0.66
	SeqStep $(C = 2)$	0.01	1.09	0.82	0.68
		0.05	1.10	0.82	0.68
	HingeExp $(C = 2)$	0.01	1.10	0.82	0.68
		0.05	1.19	0.83	0.71
1988	GW	–	3.33	3.01	0.8
	MRL	–	5	4.39	0.77
	ForwardStop	0.01	0.75	0.77	0.77
		0.05	0.97	0.87	0.81
	SeqStep $(C = 2)$	0.01	1.25	0.92	0.91
		0.05	1.59	1.39	0.85
	HingeExp $(C = 2)$	0.01	1.59	1.39	0.85
		0.05	1.66	1.53	0.81
1989	GW	–	3.06	2.53	0.73
	MRL	–	8.50	10.80	0.44
	ForwardStop	0.01	0.53	0.76	0.57
		0.05	0.56	0.75	0.58
	SeqStep $(C = 2)$	0.01	1.14	0.90	0.70
		0.05	1.17	0.87	0.73
	HingeExp $(C = 2)$	0.01	1.14	0.90	0.70
		0.05	1.18	0.90	0.72

Notice based on Table 4, the selected thresholds with ForwardStop are significantly lower compared to the selected thresholds with HingeExp or SeqStep. This could be relevant since, as mentioned in Section 4.2.1, only claims with very high values (over $500, 000 NKKs$ ) are available. Given the data is obtained in this way, it is likely that the distribution of most of the claims over 500,000 NKKs could be modeled appropriately using GPD, based on the second theorem of extreme value theory. Since the ForwardStop selection provided the lowest thresholds among all of the methods mentioned in this study, it obviously includes more claims for GPD modeling in comparison to other methods that chose higher thresholds.

Result: Measuring tail risk using var

Table 5 provides the estimates of VaRs at the $90^{th\;}$ percent level and the $95^{th\;}$ percent level, along with the asymptotic $95 \%$ CIs of the VaRs. Similar to what we described in section 4.2.2, the accumulation tests (ForwardStop, SeqStep, and HingeExp) generally produced closer estimates to the empirical VaRs compared to the previous graphical methods (GW and MRL) over all the years. In addition, notice when using the previous graphical methods, all the asymptotic $95 \%$ CIs for the VaRs did not cover the corresponding empirical estimates. This implies the graphical methods generally could not provide satisfactory fittings for the high quantiles.

Table 5.

Var of the Fitted GPD Models with Norwegian Fire Insurance Claims (1985–1992).

Year	Method	$α$	VaR(0.90)	$VaR (0.95)$
1985	Empirical		3.5	7.15
	GW	–	22.13 (18.92, 25.35)	40.26 (33.89, 46.84)
	MRL	–	29.46 (25.40, 33.53)	52.60 (44.81, 66.39)
	ForwardStop	0.01	4.07 (2.34, 5.81)	7.08 (2.93, 11.23)
		0.05	4.18 (1.92, 6.44)	7.43 (1.77, 13.10)
	SeqStep $(C = 2)$	0.01	5.88 (3.50, 8.26)	10.71 (4.68, 16.74)
		0.05	8.17 (5.63, 19.71)	15.26 (8.84, 21.68)
	HingeExp $(C = 2)$	0.01	8.32 (5.74, 10.89)	15.57 (9.04, 22.10)
		0.05	9.47 (6.10, 12.85)	18.51 (9.41, 27.61)
1986	Empirical		3.34	6.97
	GW	–	25.85 (21.84, 29.86)	51.12 (42.09, 60.15)
	MRL	–	35.10 (30.45, 39.75)	63.29 (54.26, 72.33)
	ForwardStop	0.01	$4.24 (2.31, 6.18)$	$7.32 (2.63, 12.13)$
		0.05	6.71 (3.29, 10.12)	12.75 (3.41, 22.08)
	SeqStep $(C = 2)$	0.01	6.71 (3.29, 10.12)	12.75 (3.41, 22.08)
		0.05	7.32 (3.65, 10.98)	14.14 (3.96, 24.32)
	HingeExp $(C = 2)$	0.01	7.32 (3.65, 10.98)	14.14 (3.96, 24.32)
		0.05	47.32 (40.25, 54.40)	100.38 (83.36, 117.39)
1987	Empirical		3.52	6.04
	GW	–	17.19 (15.46, 18.93)	25.89 (23.02, 28.75)
	MRL	–	14.72 (13.09, 16.51)	24.04 (20.88, 27.21)
	ForwardStop	0.01	4.19 (3.54, 4.84)	6.48 (5.26, 7.70)
		0.05	5.15 (4.22, 6.08)	8.26 (6.33, 10.19)
	SeqStep $(C = 2)$	0.01	5.61 (4.63, 6.59)	9.04 (7.00, 11.08)
		0.05	5.67 (4.67, 6.67)	9.16 (7.08, 11.24)
	HingeExp $(C = 2)$	0.01	5.67 (4.67, 6.67)	9.16 (7.08, 11.24)
		0.05	6.00 (4.91, 7.08)	9.78 (7.48, 12.07)
1988	Empirical		4.55	7.72
	GW	–	23.16 (20.46, 25.86)	40.52 (35.35, 45.69)
	MRL	–	32.69 (29.06, 36.32)	56.11 (49.35, 62.86)
	ForwardStop	0.01	5.61 (4.30, 6.91)	9.73 (6.76, 12.70)
		0.05	6.85 (5.36, 8.34)	12.09 (8.76, 15.52)
	SeqStep $(C = 2)$	0.01	8.43 (6.37, 10.49)	15.62 (10.51, 20.73)
		0.05	11.46 (9.70, 13.21)	20.64 (16.84, 24.43)
	HingeExp $(C = 2)$	0.01	11.46 (9.70, 13.21)	20.64 (16.84, 24.43)
		0.05	12.05 (10.35, 13.74)	21.35 (17.84, 24.86)
1989	Empirical		3.86	6.19
	GW	–	18.31 (16.10, 20.52)	30.71 (26.60, 34.82)
	MRL	–	51.60 (46.25, 56.96)	75.77 (67.40, 84.13)
	ForwardStop	0.01	4.16 (3.45, 4.86)	6.55 (5.22, 7.88)
		0.05	4.20 (3.46, 4.94)	6.67 (5.24, 8.09)
	SeqStep $(C = 2)$	0.01	6.26 (5.17, 7.36)	10.25 (7.98, 12.53)
		0.05	6.43 (5.15, 7.55)	10.52 (7.95, 13.08)
	HingeExp $(C = 2)$	0.01	6.26 (5.17, 7.36)	10.25 (7.98, 12.53)
		0.05	6.44 (5.27, 7.61)	10.63 (8.17, 13.09)

Text in italic indicates the CI does not cover the corresponding empirical VaR estimate.

The $95 \%$ asymptotic CIs for VaRs were computed based on the Delta method described in section 2.6. We labeled a CI in bold font if the CI does not cover the corresponding empirical VaR estimate. We noticed that the CIs produced from the ForwardStop selection covered almost all the empirical VaR estimates. This implies the ForwardStop selection has the best ability to model the high quantiles of the claim size distributions in comparison with the SeqStep and HingeExp selections.

4..3 Case 2: AON re Belgian Fire Loss Data Set

Data

AON Re Belgian data contains the 1823 fire losses collected by the reinsurance company AON. The building type of the associated fire losses and the loss amount in thousands of Danish Krones (DKK) are provided in this data set. We selected building type 'C' and 'D' for our analysis. The summary statistics of the losses are presented in Table 6. Notice the fire losses for both types of the buildings are characterized with right-skewed distributions.

Table 6.
Summary Statistics for AON re Belgian Fire Loss Claims (Building C and D).

Building Type Sample Size Mean SD Q1 Q2 Q3 Maximum

C 53 850.55 1939.94 16.67 58.30 243.40 9006.60

D 801 279.86 1946.66 3.19 10.57 43.77 45656.52

Building Type	Sample Size	Mean	SD	Q1	Q2	Q3	Maximum
C	53	850.55	1939.94	16.67	58.30	243.40	9006.60
D	801	279.86	1946.66	3.19	10.57	43.77	45656.52

Result: Threshold selection

Table 7 demonstrates the selected GPD thresholds $μ$ from different procedures and corresponding MLE for the shape $(σ)$ and scale parameter $(γ)$ when fitting to AON $Re$ Belgian fire loss data. With the estimates, we were able to construct the estimated CDF plots presented in Figure 2.

Figure 2.

Comparison of the CDFs using different GPD modeling methods for AON re Belgian fire insurance loss data.

Table 7.

Chosen Thresholds $(μ)$ with MLEs of Scale $(σ)$ and Shape $(γ)$ Parameters in Different GPD Models.

Building Type	Method	$α$	Chosen Threshold $(μ)$	Scale $(σ)$	Shape $(γ)$
	GW	–	173.65	2062.32	0.21
	MRL	–	243.40	3417.20	−0.19
		ForwardStop	0.01	0.90	35.96
1.98
C		0.05	0.90	35.96	1.98
	SeqStep (C=2)	0.01	0.90	35.96	1.98
		0.05	0.90	35.96	1.98
	HingeExp $(C = 2)$	0.01	0.90	35.96	1.98
	0.05	0.90	35.96	1.98
	GW	–	61.50	160.59	1.29
	MRL	–	0.27	8.63	1.63
	ForwardStop	0.01	0.72	7.85	1.75
		0.05	1.08	7.94	1.71
	SeqStep (C=2)	0.01	116.36	346.24	1.02
		0.05	4909.21	5943.92	0.34
	HingeExp $(C = 2)$	0.01	116.36	346.24	1.02
	0.05	4909.21	5943.92	0.34

For building Type C, all the accumulation test procedure selected the same threshold as 0.9 while GW and MRL procedure selected significantly higher thresholds in comparison. The comparison of CDF plots (Figure 2) suggest that the accumulation test procedure provided closer fit to the empirical CDF, while the estimated CDF based on GW and MRL procedure cannot provide satisfactory performance.

For building Type D, SeqStep and HingeExp made the same selection regardless of chosen alpha level, while ForwardStop selected significantly lower threshold in comparison. The selection based on MRL plot, however, indicates that the mean excess graph is characterized with linear trend from the minimum of the data set. Thus, the chosen threshold based on MRL is the minimum of the data set. The CDF plots (Figure 2) indicate that GW, SeqStep, and HingeExp failed to provide satisfactory performance, while ForwardStop and MRL procedure can produce the estimated CDFs closer to the empirical one.

Result: Measuring tail risk using Var

The estimated $90^{th\;}$ and $95^{th\;}$ percent VaR and corresponding $95 \%CI$ are provided in Table 8. If a CI does not cover the empirical VaR estimate, then the CI is labeled in bold font.

Table 8.

Var of the Fitted GPD Models with AON Re Belgian Fire Loss Data.

Method	$α$	VaR (0.90)	VaR (0.95)
Empirical	–	2377.76	5665.16
GW	–	$6259.01 (3713.34, 8804.68)$	$8736.15 (5154.22, 12318.07)$
MRL	–	$6615.737 (4432.30, 8799.17)$	$8048.26 (5373.98, 10722.54)$
	0.01	$1725.28 (588.26, 2862.30)$	$6869.38 (2335.74, 11403.01)$
CorwardStop	0.05	$1725.28 (588.26, 2862.30)$	$6869.38 (2335.74, 11403.01)$
	0.01	$1725.28 (588.26, 2862.30)$	$6869.38 (2335.74, 11403.01)$
SeqStep $(C = 2)$	0.05	$1725.28 (588.26, 2862.30)$	$6869.38 (2335.74, 11403.01)$
	0.01	$1725.28 (588.26, 2862.30)$	$6869.38 (2335.74, 11403.01)$
HingeExp $(C = 2)$	0.05	$1725.28 (588.26, 2862.30)$	$6869.38 (2335.74, 11403.01)$
Gmpirical	–	299.48	827.95
MRL	–	$2362.17 (2021.22, 2703.12)$	$5865.06 (5004.93, 6725.08)$
ForwardStop	0.01	$234.24 (194.91, 273.56)$	$688.05 (573.84, 802.26)$
	0.05	$251.28 (208.63, 293.93)$	$856.44 (642.65, 908.22)$
SeqStep $(C = 2)$	0.01	$3333.58 (2885.71, 3781.46)$	$6991.60 (6034.49, 7948.712)$
	0.05	$25545.34 (23209.99, 27880.69)$	$35581.17 (32110.09, 39052.26)$
HingeExp $(C = 2)$	0.01	$3333.58 (2885.71, 3781.46)$	$6991.60 (6034.49, 7948.712)$
0.05	$25545.34 (23209.99, 27880.69)$	$35581.17 (32110.09, 39052.26)$

Text in italic indicates the CI does not cover the corresponding empirical VaR estimate.

For building type C, the accumulation test procedures produced closer estimate to the empirical estimate, compared to GW and MRL procedures. The asymptotic 95% CI created based on threshold selected with accumulation tests covered the empirical VaR while the CIs created using thresholds selected with GW and MRL procedure failed to cover the empirical estimate.

Among all the procedures, GPD model selected based on ForwardStop procedure produced the closest estimates to both $90^{th\;}$ and $95^{th\;}$ percent emipircal VaR, regardless the choice of $α$ level. For $90^{th\;}$ percent VaR, all procedures failed to produce a estimated CI that covers empirical estimate. However, the $95 \%$ CIs for $95^{th\;}$ percent VaR constructed based on Forward procedure and MRL procedure are able to cover the empirical VaR while the CIs created based on other procedure failed to perform so.

5. Conclusion

In this article, three different accumulation tests have been used to generate GPD models for claim size distributions and VaR estimates over the Norwegian fire insurance data and AON Re Belgian fire loss data. Among the accumulation tests, the ForwardStop utilizes a smooth logarithmic function as the accumulation function while the SeqStep and the HingeExp employ a discrete step function with a pre-specified parameter C as the accumulation function. We also used the previous graphical methods including GW-Plot and MRL-Plot to generate the models for comparison purposes. Among all the models, the ForwardStop selection demonstrates the best performance as it produces the closest fits to the empirical CDFs and the closest estimates to the empirical VaRs

at the $90 \%$ and $95 \%$ level. The SeqStep and the HingeExp also performed better in comparison with previous graphical methods. Therefore, regarding both fire insurance data sets, the threshold selection methods based on the accumulation tests have better ability to capture the features of the claim size distributions compared to the previous widely-used methods.

The threshold selection based on the accumulation tests is attractive due to their good performance fitting both data sets. However, one needs to take precautions in terms of the following: the quality of the modeling for the accumulation tests are affected by the chosen FDP $(α)$ . For instance, the ForwardStop selection with $α = 0.01$ generally overperformed the same procedure with $α = 0.05$ . Hence, before using the accumulation tests, a reasonable selection on $α$ is necessary in order to obtain a better model, especially for modeling high quantiles.

It is quite interesting to see that ForwardStop demonstrated great performance for all the chosen data sets in comparison to other two accumulation test procedures. In fact, the accumulation test procedure for GPD threshold selection can be described as accumulating 'evidence' till the lowest GPD threshold is detected, which 'evidence' is quantified as GoF test p-values for GPD. Based on the accumulation functions we chose in 3.1, ForwardStop procedure collects "evidence" from all candidate thresholds, while SeqStep and HingeExp procedure only collects "evidence" when p-value is greater than $1 - 1 / C$ at the candidate thresholds. In both simulation and real data example, we chose $C = 2$ and thus only p -values that are greater than $\frac{1}{2}$ were used in the accumulation procedure. This leads to the conservative selection for the threshold, especially compared to ForwardStop procedure that accumulates all p -values from the candidate thresholds.

Thus, pre-specifying $C = 2$ for both SeqStep and HingeExp may not be a great idea based on our simulations and real data examples. A limitation of this study is that we did not test the effect of C on the selection of GPD thresholds. In order to use the SeqStep and HingeExp efficiently for GPD threshold selection, future researchers may explore a good framework to specify the value of C first, then apply the accumulation test procedure. Another limitation is that we only investigated VaR. Future researchers are warranted to assess the performances of the accumulation test procedures with other risk measures, such as conditional tail expectation (CTE).

Finally, we conclude that the accumulation tests have great ability to choose the thresholds in GPD models for Norwegian fire insurance data and AON Re Belgian fire loss data. More extended uses and improvements for such method in GPD threshold selection are warranted, especially designing a useful framework to select C for both SeqStep and HingeExp.

Footnotes

ORCID iD

Bowen Liu

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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					Forward Stop		$\begin{array}{l} SeqStep \\ (C = 2) \end{array}$		$\begin{array}{l} HingeExp \\ (C = 2) \end{array}$
Scenario	True Threshold	Head Weight	Tail Weight	Sample Size	Mean	RMSE	Mean	RMSE	Mean	RMSE
$GPD (μ = 0.5, σ = 1, γ = 1)$	0.5	–	–	$n = 100$	0.52	0.05	2.22	1.04	5.27	1.69
				$n = 200$	0.51	0.04	2.01	0.88	4.97	1.47
				$n = 500$	0.51	0.01	1.84	0.52	4.55	1.03
$GPD (μ = 2, σ = 1, γ = 1)$	2	–	–	$n = 100$	2.02	0.01	3.33	0.67	8.66	2.12
				$n = 200$	2.02	0.01	3.31	0.49	8.18	1.76
				$n = 500$	2.01	0.003	3.26	0.18	7.91	0.78
$\begin{array}{l} lognormal (μ = 2, σ = 0.5) \\ GPD (μ = 2, σ = 0.8, γ = 0.8) \end{array}$	2	0.3	0.7	$n = 100$	1.94	0.08	5.58	1.13	9.50	1.55
				$n = 200$	1.95	0.07	4.89	0.98	8.77	1.38
				$n = 500$	1.98	0.09	4.14	0.58	7.64	1.06
$\begin{array}{l} lognormal (μ = 1, σ = 1) \\ GPD (μ = 1, σ = 1.3, γ = 1.5) \end{array}$	1	0.2	0.8	$n = 100$	0.90	0.11	9.90	4.16	43.74	23.40
				$n = 200$	0.91	0.09	9.04	3.57	35.68	17.77
				$n = 500$	0.95	0.04	8.15	2.15	29.18	12.38

The application of accumulation tests in peaks over threshold modeling with fire insurance data sets

Abstract

Keywords

1. Introduction

2. Methodology

2.1. POT Modeling and Threshold Selection Procedure

3.1. Simulation Settings

3.2. Scenarios of Composite Density with GPD Tails

4.1. Selection of Accumulation Functions and Methods for Comparison

4.2. Case 1: Norwegian Fire Insurance Data

Data

Result: Threshold selection

Result: Measuring tail risk using var

Data

Table 6. Summary Statistics for AON re Belgian Fire Loss Claims (Building C and D). Building Type Sample Size Mean SD Q1 Q2 Q3 Maximum C 53 850.55 1939.94 16.67 58.30 243.40 9006.60 D 801 279.86 1946.66 3.19 10.57 43.77 45656.52

Result: Threshold selection

Result: Measuring tail risk using Var

Footnotes

ORCID iD

Funding

Declaration of Conflicting Interests

References

Table 6.
Summary Statistics for AON re Belgian Fire Loss Claims (Building C and D).

Building Type Sample Size Mean SD Q1 Q2 Q3 Maximum

C 53 850.55 1939.94 16.67 58.30 243.40 9006.60

D 801 279.86 1946.66 3.19 10.57 43.77 45656.52