4.2.1 EVA for Pareto‐type Tails

In order to model large claims, Pareto tail modelling is probably the most common approach. Here we use the subset of models with tails heavier than exponential for which the EVI γ is positive, as discussed in Section 3.3.2.2, which in fact equals the set of Pareto‐type models that can be defined through tail functions 1 − F, quantile functions Q, or tail quantile functions U(x) = Q(1 − 1/x). Indeed

(4.2.3)

where γ = 1/α > 0 and ℓ_U is a slowly varying function. Also (3.2.7) is equivalent to

(4.2.4)

for every u > 0. In this section we discuss the estimation of γ = 1/α, large quantiles Q(1 − p) = U(1/p), and small tail probabilities . We assume in this and the next subsection that the data are independent and identically distributed (i.i.d.). Moreover mathematical approximations of variances (AVar), bias (ABias), mean squared error (AMSE), and distributions of estimators using k largest observations of the n data will hold when and k/n → 0.

4.2.1.1 Estimating a Positive EVI

The most popular estimator for γ is given by the Hill estimator H_{k, n} defined in (4.1.2), see [442]. In the preceeding section this estimator was retrieved through regression on the Pareto QQ‐plot. Here, we also show how the maximum likelihood method based on the so‐called POT approach leads to the same estimation method in the Pareto‐type case.

• In Section 4.1 the Hill estimator was motivated as an estimator of the slope of a linear Pareto QQ‐plot to the right of an anchor point . In fact, this interpretation can be carried over to the general case of Pareto‐type distributions since then ultimately for the Pareto QQ‐plot is still linear with slope γ for a small enough k or, equivalently, for a large enough X_{n−k, n}. Indeed, under (4.2.3),
  
  It can now be shown that for every slowly varying function ℓ
  
  as . Hence, whereas Pareto QQ‐plots are hardly ever completely linear, they are ultimately linear at some set of largest values. The speed at which the linearity sets in depends on the underlying slowly varying function. Like many publications, following Hall [419], we assume here that
  (4.2.5)
  for some C, β > 0, and D a real constant. This can, however, be generalized to
  
  as with b essentially a power function or, more correctly, a regularly varying function with index − β, and . Under (4.2.5)
  (4.2.6)
  from which H_{k, n} follows taking x = (n + 1)/j (j = 1, …, k), estimating U((n + 1)/j) by X_{n−j+1, n} (j = 1, …, k + 1), and taking the average of both sides of (4.2.6) over j = 1, …, k after deleting the last term on the right‐hand side. Omitting this final term (or, equivalently, assuming a strict Pareto distribution with constant slowly varying function) causes a bias which will be more important with smaller β. Adverse situations for the Hill estimator are logarithmic slowly varying functions ℓ_U, as in case of the log‐gamma distribution. Such cases exhibit β = 0.
• Alternatively, the Hill estimator is also a maximum likelihood estimator based on (4.2.4). Indeed, extreme value methodology proposes fitting the limiting Pareto distribution with distribution function 1 − x^−1/γ to the POT values Y = X/t over a high threshold t conditionally on X > t. Note that the use of the mathematical limit in (4.2.4) to fit the exceedance data introduces an approximation error that leads to estimation bias. Let N_t denote the number of exceedances over t. Then the log‐likelihood equals
  
  with
  
  leading to the maximum likelihood estimator
  
  Choosing an upper order statistic X_{n−k, n} for the threshold t (so that N_t = k) we obtain H_{k, n}.
• From Section 4.1 it also follows that the Hill statistic can be interpreted as an estimator of the mean excess function of the log‐transformed data, that is, , with the threshold value t substituted by X_{n−k, n}. As in (3.4.10) we here find
  
  Estimating F(u) using the empirical distribution function
  (4.2.7)
  with value over the interval [X_{n−j, n}, X_{n−j+1, n}) we are led to the estimator
  (4.2.8)
  Using summation by parts one observes that this final expression equals the Hill estimator:
  (4.2.9)
  with
  
  (with ).

To deduce approximate expressions for the variance and bias of the Hill estimator it is helpful to consider the preceding interpretation in terms of the scaled log‐spacings Z_j. Thanks to the Rényi representation j(E_{n−j+1, n} − E_{n−j, n}) = _d E_j (j = 1, …, n) concerning order statistics E_{1, n} ≤ E_{2, n} ≤ … ≤ E_{n, n} from a random sample E₁, E₂, …, E_n of n independent standard exponential random variables, we have in case of a strict Pareto distribution (i.e., with ℓ_U constant), that

(4.2.10)

This representation is based on the memoryless property of the exponential distribution and the fact that nE_{1, n} is standard exponentially distributed. From (4.2.9) and (4.2.10) we expect that, as ,

Concerning the bias due to the approximation error, we confine ourselves to the model (4.2.5). Then the theoretical analogue of the Hill estimator is given by

with . Hence, the approximate mean squared error is given by

while in order to construct confidence bounds we have that, as with k/n → 0,

(4.2.11)

4.2.1.2 Estimating Large Quantiles and Small Tail Probabilities

One of the most important applications of EVA is the estimation of extreme quantiles q_p = Q(1 − p) with p small, also termed Value‐at‐Risk (VaR) in risk applications. Alternatively, the return period for a high claim amount x given by is another measure describing extreme risks.

The estimation of a high quantile under Pareto‐type modelling can be performed by extrapolating along a fitted regression line on the Pareto QQ‐plot through the point with slope H_{k, n}. Following (4.2.6) with x = 1/p, estimating U((n + 1)/(k + 1)) by X_{n−k, n} and γ by H_{k, n}, and omitting the second term on the right‐hand side, that is, using

we arrive at the estimator

(4.2.12)

which was first proposed by Weissman [777]. The estimator can also be retrieved from (4.2.3), leading to the approximation U(vx)/U(x) ≈ v^γ for large values of x. Setting vx = 1/p, x = (n + 1)/(k + 1) so that v = (k + 1)/((n + 1)p), and estimating U((n + 1)/(k + 1)) by X_{n−k, n} and γ by H_{k, n}, we obtain again.

Estimation of return periods can be obtained using the inverse relationship on the Pareto QQ‐plot:

(4.2.13)

The expression for can also be deduced from (4.2.4), leading to the approximation for large values of t. Setting tu = x, t = X_{n−k, n} so that u = x/X_{n−k, n}, and estimating by (k + 1)/(n + 1) we obtain .

Approximate confidence bounds for such parameters have been derived based on asymptotic distributions of the estimators. In the case of the tail probability estimator we find with

when , k/n → 0, np_x/k → τ ∈ [0, 1) and , while with q_p = Q(1 − p),

when , k/n → 0, np/k → τ ∈ [0, 1) and .

4.2.1.3 Bias Reduction

When constructing confidence intervals for risk measures such as p_x and q_p, again the approximation of the underlying conditional distribution by the simple Pareto distribution entails a bias for all the existing estimators, next to the bias induced by estimating γ. One approach to reduce the bias is to construct estimators based on regression models of the values Z_j. Indeed, under (4.2.5), using the approximation and with the mean value theorem on x^−β at the points j/(n + 1) and (j + 1)/(n + 1), the theoretical analogue of a Z_j random variable can be approximated by

(4.2.14)

(4.2.15)

An alternative representation, using 1 + u ≈ e^u for small values of u, is then

The more accurate approximation

where E_j denotes a sequence of independent standard exponentially distributed random variables, was derived in an asymptotic sense in Beirlant et al. [97]. For each k, model (4.2.15) can be considered as a non‐linear regression model in which one can estimate the intercept γ, the slope b_{k, n}, and the power β with the covariates j/(k + 1). One can estimate these parameters jointly, or by using an external estimate for β, or using external estimation for β and on the regression model

(4.2.16)

Gomes et al. found that external estimation for B and β should be based on k₁ extreme order statistics where k = o(k₁) and . Such an estimator for β was presented, for example, in Fraga Alves et al. [35]. Given an estimator for β, an estimator for B was given in Gomes and Martins [399]:

When the three parameters are jointly estimated for each k, the asymptotic variance turns out to be γ²((1 + β)/β)⁴, which is to be compared with the asymptotic variance γ² for the Hill estimator. Performing linear regression on importing an external estimator for β, the asymptotic variance drops down to γ²((1 + β)/β)². The original variance γ² is retained when using the external estimators for B and β in (4.2.16).

Bias reduction of the extreme quantile estimator should not be based solely on replacing H_{k, n} by a bias‐reduced estimator for γ. Here we use the fact that X_{n−k, n} = _dU(1/U_{k+1, n}), where U_{k+1, n} denotes the (k + 1)th smallest order statistic from a uniform (0,1) sample of size n. Then we obtain from (4.2.3) and (4.2.5) with x = 1/p, approximating U_{k+1, n} by its expected value (k + 1)/(n + 1), and using 1 + u ≈ e^u for u small, that

so that a bias‐reduced version of is given by

where , and are bias‐reduced estimators based on the regression model (4.2.15).

Bias‐reduced estimators can also be obtained by improving on the approximation (4.2.4) of the POT distribution by the simple Pareto distribution, using an extension of the Pareto distribution as introduced in Beirlant et al. [102]. Indeed, when ℓ_U satisfies (4.2.5), then

(4.2.17)

The distribution of the POT’s X/t (X > t) can then be approximated using the expansion (1 + u)^b ≈ 1 + bu for u small:

This leads to the extended Pareto distribution (EPD) with distribution function

(4.2.18)

() with δ = δ_t = DC^β/γt^−β/γ and τ = −β/γ. Note that for an EPD random variable Y with τ = −1 and , it follows that Y − 1 is GPD distributed with parameters γ and σ.

Using the density of the EPD g_{γ, δ, τ}(y) = γ⁻¹y^−1/γ−1{1 + δ(1 − y^τ)}^−1/γ−1[1 + δ{1 − (1 + τ)y^τ}], maximum likelihood estimators are then derived through maximization of

with respect to γ, δ using an external estimator of τ through estimates of β and γ, where the values denote the POT values over the threshold t.

Bias‐reduced estimation of return periods is then obtained using X_{n−k, n} again as a threshold t:

4.2.1.4 Estimating the Scale Parameter

Finally, note that the scale parameter C in (4.2.5), or A = C^1/γ in (4.2.17), can be estimated with

(4.2.19)

(4.2.20)

which follows, for instance, from (4.2.17) replacing x by X_{n−k, n} and estimating by the empirical probability (k + 1)/(n + 1).

The estimator Ĉ_k,n can also be retrieved using least squares regression on the k top points of the log–log plot

minimizing

(4.2.21)

Substituting H_{k, n} for γ and taking the derivative with respect to log C indeed gives

In Beirlant et al. [104] it is shown that Â_k,n is asymptotically normally distributed with asymptotic variance and asymptotic bias .

A bias‐reduced estimator of the scale parameter A is then given by

where is a bias‐reduced estimator of γ, and estimators of b_{k, n} and β.

4.2.2 General Tail Modelling using EVA

In order to allow tail modelling with log‐normal or Weibull tails, one has to incorporate the case where the EVI γ can be 0, next to positive values. Estimation of γ, extreme quantiles and return periods under the max‐domains of attraction conditions C_γ in (3.2.5) or (3.2.6), with as few restrictions on the value of γ as possible, is the next step in tail modelling. Again we have two possible approaches: using quantile plotting or using a likelihood approach on POT values.

• Here, several existing estimators start from the following condition, which follows from (3.2.5): for all u ≥ 1 as
  (4.2.22)
  From this it follows with , that as , k/n → 0
  (4.2.23)

  Hence estimating EH_{k, n} by H_{k, n}, and by X_{n−k, n}, we find that for any estimator of γ

  (4.2.24)

  leads to an estimator for .

  Since a regularly varies with index , it also follows from (4.2.23) that U((n + 1)/(k + 1))EH_{k, n} = ((n + 1)/(k + 1))^γℓ((n + 1)/(k + 1)) for some slowly varying function ℓ. Hence the approach using linear regression and extrapolation on linear tail patterns on a QQ‐plot can be generalized to the case of a real‐valued EVI using the generalized QQ‐plot

  

  which ultimately for smaller values of k will be linear with slope γ, whatever the sign or values of γ. Hence if a generalized QQ‐plot is ultimately horizontal, then tail modelling using a distribution in the Gumbel domain of attraction is appropriate. An ultimately decreasing generalized QQ‐plot indicates a negative EVI, which can occur, for instance, for truncated heavy‐tailed distributions.

  A generalized Hill estimator of γ estimating the slopes at the last k points on the generalized QQ‐plot is then given by

  

  where UH_{j, n} = X_{n−j, n}H_{j, n}.

  Another generalization of the Hill estimator to real‐valued EVI was given in Dekkers et al. [268], termed the moment estimator:

  

  where

  
• Condition (3.2.6) for a distribution to belong to a domain of attraction of an extreme value distribution means that the generalized Pareto law is the limit distribution of the distribution of POT values X − t given X > t when t → x₊: setting h(t) = σ_t
  (4.2.25)
  Hence, we are led to modelling the tail function of POT values Y = X − t with X > t using the GPD with survival function . Denoting the number of exceedances over t again by N_t, the log‐likelihood is given by
  
  Using a reparametrization (γ, σ) → (γ, τ) with τ = γ/σ, leads to the likelihood equations
  
  Replacing t by an intermediate order statistic X_{n−k, n} again gives
  
  In order to assess the goodness‐of‐fit of the GPD when modelling the POT values Y = X − t for a given threshold t, one can use the transformation
  
  so that R is standard exponentially distributed if the POT values do follow a GPD, and the fit can be validated inspecting the overall linearity of the exponential QQ‐plot
  
  where (i = 1, …, N_t) denote the ordered values of
  

For all these estimators is asymptotically normal under some regularity conditions on the underlying distributions when and k/n → 0, with mean 0 if k is not too large (or, equivalently, if the threshold t = x_{n−k, n} is not too small), and asymptotic variances (or covariance matrix for ) given by

Estimators for small tail probabilities or return periods can easily be constructed from the POT approach. In fact, the approximation of P(X − t > y|X > t) by (1 + (γ/σ)y)^−1/γ for y > 0, setting t + y = x, leads to

Inversion leads to an extreme quantile estimator

The estimators for high quantiles based on the approach used in the construction of the moment estimator are defined by

with â defined in (4.2.24). Note that can be seen as an alternative estimator for σ when comparing the expressions of and . This then in turn leads to a moment tail probability estimator

The asymptotic distributions of these tail estimators have been derived in the literature. For instance, for we have under some regularity conditions that with a_n = (k + 1)/(p(n + 1)) as np_n → c ≥ 0, one has

• for γ > 0
  
• for γ < 0
  

For further details see de Haan and Ferreira [258].

4.2.3 EVA under Upper‐truncation

Practical problems can arise when using the strict Pareto distribution and its generalization to the Pareto‐type model because some probability mass can still be assigned to loss amounts that are unreasonably large or even impossible. With respect to tail fitting of an insurance claim data, upper‐truncation is of interest and can be due to the existence of a maximum possible loss. Such truncation effects are sometimes visible in data, for instance when an overall linear Pareto QQ‐plot shows non‐linear deviations at only a few top data. Let W be an underlying non‐truncated distribution with distribution function F_W, quantile function Q_W and tail quantile function U_W. Upper‐truncation of the distribution of W at some value T was defined in the preceding chapter through the conditioning operation W|W < T. Let F_T and U_T be the distribution and tail quantile function of this truncated distribution. In practice one does not always know if the data X₁, …, X_n come from a truncated or non‐truncated distribution, and hence the behavior of estimators should be evaluated under both cases, and a statistical test for upper‐truncation is useful. This section is taken from Beirlant et al. [95].

Upper‐truncation of the distribution of W at some truncation point T yields

The corresponding quantile function Q_T is then given by

while the tail function U_T satisfies

(4.2.26)

or

(4.2.27)

with the odds of the truncated probability mass under the untruncated distribution W. Note also that for a fixed T, upper‐truncation models are known to exhibit an EVI γ = −1. This follows from verifying (3.2.5) for U_T as given in (4.2.27). For instance when U_W(x) = x^γ, we find as . This final expression satisfies (3.2.5) with γ = −1.

4.2.3.1 EVA for Upper‐truncated Pareto‐type Distributions

We restrict attention to tail estimation for upper‐truncated Pareto‐type distributions:

where ℓ_F is a slowly varying function at infinity or, with W/t denoting the peaks over a threshold t when W > t,

Upper‐truncation of a Pareto‐type distribution at a high value T then necessarily requires and

One can now consider two cases as :

• Rough upper‐truncation with the threshold t when T/t → β > 1 and
  (4.2.28)
  This corresponds to situations where the deviation from the Pareto behavior due to upper‐truncation at a high value will be visible in the data from the threshold t onwards, and an adaptation of the above Pareto tail extrapolation methods appears appropriate.
• Light (or no) upper‐truncation with the threshold t: when
  (4.2.29)
  and hardly any truncation is visible in the data from the threshold t onwards, and the Pareto‐type model without truncation and the corresponding extreme value methods for Pareto‐type tails appear appropriate when restricted to excesses over t.

Under rough upper‐truncation we have

with density

Estimating T by X_{n, n} and taking t = X_{n−k, n} so that , we obtain the following log‐likelihood:

Now

which leads to the defining equation for the likelihood estimator :

This estimator was first proposed in Aban et al. [6]. Beirlant et al. [95] showed that with κ = β^1/γ − 1

From (4.2.27) it is clear that the estimation of D_T is an intermediate step in important estimation problems following the estimation of γ, namely of extreme quantiles and of the endpoint T. When U_W satisfies (4.2.3) it follows from (4.2.27) that as and T/t → β

(4.2.30)

so that

(4.2.31)

Motivated by (4.2.31) and estimating Q_T(1 − (k + 1)/(n + 1))/Q_T(1 − 1/(n + 1)) by R_{k, n}, one arrives at

(4.2.32)

as an estimator for D_T. In practice one makes use of the admissible estimator

to make it useful for truncated and non‐truncated Pareto‐type distributions.

For D_T > 0, in order to construct estimators of T and extreme quantiles q_p = Q_T(1 − p), as in (4.2.31) we find that

(4.2.33)

Then taking logarithms on both sides of (4.2.33) and estimating Q_T(1 − (k + 1)/(n + 1)) by X_{n−k, n} we find an estimator of q_p:

(4.2.34)

which equals the Weissman estimator when . An estimator of T follows from letting p → 0 in the above expressions for :

(4.2.35)

Here we take the maximum of log X_n,n and the value following from (4.2.34) with p → 0 in order for this endpoint estimator to be admissible. It has been shown that is superefficient under rough upper‐truncation, which means that the asymptotic variance is o(1/k) and the asymptotic bias is also smaller than, for instance, that of the moment quantile estimator .

However, is not a consistent estimator for q_p under light upper‐truncation and when np_n → 0. In that case one should use

(4.2.36)

The estimation of tail probabilities p_x = P(X > x) can be based directly on (4.2.28) using R_{k, n} as an estimator for 1/β:

(4.2.37)

Of course, in order to decide between (4.2.34) and (4.2.36) one should use a statistical test for deciding between rough and light upper‐truncation.

4.2.3.2 Testing for Upper‐truncated Pareto‐type Tails

Aban et al. [6] proposed a test for versus under the strict Pareto model, rejecting H₀ at level q ∈ (0, 1) when

(4.2.38)

for some 1 < k < n with A the scale parameter in the Pareto model. In (4.2.38), γ is estimated by H_{k, n}, the maximum likelihood estimator under H₀, while A is estimated using Â_k,n from (4.2.19). Note that the rejection rule (4.2.38) can be rewritten as

(4.2.39)

and the P‐value is given by .

Considering the testing problem

under the upper‐truncated Pareto‐type model, Beirlant et al. [95] propose to reject when an appropriate estimator of (n + 1)D_T/(k + 1) is significantly different from 0. Here we construct such an estimator generalizing with an average of ratios (X_{n−k, n}/X_{n−j+1, n})^1/γ, j = 1, …, k, which then possesses an asymptotic normal distribution under the null hypothesis. Observe that with (4.2.30) under as

Estimating Ē_k,n by

leads now to

(4.2.40)

as an estimator of (n + 1)D_T/(k + 1), with an appropriate estimator of γ. Under , the Hill estimator H_{k, n} is an appropriate estimator of γ. Moreover, it can be shown that under some regularity assumptions on the underlying Pareto‐type distribution, we have under for and k/n → 0, that is asymptotically normal with mean 0 and variance 1/12. It is then also shown under rough upper‐truncation as , k/n → 0 that L_{k, n}(H_{k, n}) tends to a negative constant so that an asymptotic test based on L_{k, n}(H_{k, n}) rejects on level q when

(4.2.41)

with . The P‐value is then given by .

4.3.1 Tail‐mixed Erlang Splicing

The Erlang distribution has a gamma density

where r is a positive integer shape parameter. Following Lee and Lin [534] we consider mixtures of M Erlang distributions with common scale parameter 1/λ having density

and tail function

where the positive integers r = (r₁, …, r_M) with r₁ < r₂ < … < r_M are the shape parameters of the Erlang distributions, and α = (α₁, …, α_M) with α_j > 0 and are the weights in the mixture. Tijms [745] showed that the class of mixtures of Erlang distributions with a common scale 1/λ is dense in the space of positive continuous distributions on . Moreover this class is also closed under mixtures, convolution and compounding. Hence aggregate risks calculations are simple, and XL premiums and risk measures based on quantiles can also be evaluated in a rather straightforward way.For instance, a composite ME generalized Pareto distribution can be built using (3.5.14), that is, a two‐component spliced distribution with density

If a continuity requirement at t were imposed, this would lead to

The survival function of this spliced distribution is given by

Alternatively, one can take , where k_* is an appropriate number of top order statistics corresponding to an extreme value threshold .

Fitting ME distributions through direct likelihood maximization is difficult. A first algorithm was proposed by Tijms [745], but it turns out to be slow and can lead to overfitting. Lee and Lin [534] use the expectation‐maximization (EM) algorithm proposed by Dempster et al. [271] to fit the ME distribution. Model selection criteria, such as the Akaike information criterion (AIC) and Bayesian information criterion (BIC) information criteria, are then used to avoid overfitting. Verbelen et al. [758] extend this approach to censored and/or truncated data. The need for the EM algorithm follows from the data incompleteness due to mixing and censoring.

The EM algorithm is used to compute the maximum likelihood estimator (MLE) for incomplete data where direct maximization is impossible. It consists of two steps that are put in an iteration until convergence:

• E‐step: Compute the conditional expectation of the log‐likelihood given the observed data and previous parameter estimates.
• M‐step: Determine a subsequent set of parameter estimates in the parameter range through maximization of the conditional expectation computed in the E‐step.

Rather than proposing a data‐driven estimator of the splicing point t, we use an expert opinion on the splicing point t based on EVA as outlined above. Then, π can be estimated by the fraction of the data not larger than t. Similarly, T is deduced from the EVA. The extreme value index γ is estimated in the algorithm, starting from the value obtained from the EVA at the threshold t. The final estimates for γ always turned out to be close to the EVA estimates. Next, the ME parameters (α, λ) are estimated using the EM algorithm as developed in Verbelen et al. [758]. The number of ME components M is estimated using a backward stepwise search, starting from a certain upper value, whereby the smallest shape is deleted if this decreases an information criterion such as AIC or BIC. Moreover, for each value of M, the shapes r are adjusted based on maximizing the likelihood starting from r = (s, 2 s, …, …, M × s), where s is a chosen spread factor.

Of course tail splicing of an ME can also be performed using a simple Pareto fit, or an EPD fit, whether or not adapted for truncation. For instance, splicing an ME with an upper‐truncated Pareto approximation leads to

(4.3.42)

4.3.2 Tail‐mixed Erlang Splicing under Censoring and Upper‐truncation

In reinsurance data left truncation appears at some point, denoted here by t^l, which can be a deductible or a percentage of the retention u from an XL contract. Claims leading to a cumulative payment below t^l at a given stage during development are then left truncated. Such a claim constitutes an IBNR claim. As discussed above, an upper‐truncation mechanism at some point T can appear.

We denote the ME density and distribution function by f_ME and F_ME, and similarly f_EV and F_EV for the EVA distribution. We then define, omitting the model parameters from the notation for the moment,

with 0 ≤ t^l < t < T where T can be equal to . The densities f₁ and f₂ are then valid densities on the intervals [t^l, t] and [t, T], respectively. For the first density, this means that it is lower truncated at t^l and upper truncated at t, and the second density is lower truncated at t and upper truncated at T. The corresponding distribution functions are

We consider the splicing density and distribution function

and

(4.3.43)

Next to truncation, censoring mechanisms occur in reinsurance¹ :

• right censoring occurs for instance when a claim has not been settled at the evaluation date (RBNS claims). See Chapter 1 for the case of motor liability data. The final claim amount x_i will be larger than the lower censoring value l_i
• left censoring occurs when only an upper bound u_i to the claim x_i is given
• interval censoring means that the final claim value x_i is only known to be inside an interval [l_i, u_i] ⊂ [t^l, T].

In the splicing context with an EVA component from a threshold t on, we have the following five classes of observations:

1. Uncensored observations with t^l ≤ l_i = u_i = x_i ≤ t < T.
2. Uncensored observations with t^l < t < l_i = u_i = x_i ≤ T.
3. Interval censored observations with t^l ≤ l_i < u_i ≤ t < T.
4. Interval censored observations with t^l < t ≤ l_i < u_i ≤ T.
5. Interval censored observations with t^l ≤ l_i < t < u_i ≤ T.

These classes are shown in Figure 4.12. In the conditioning argument in the E‐step of the algorithm, the fifth case is split into x_i ≤ t and x_i > t, as indicated in Figure 4.12.

For the Erlang mixture, the number M and the integer shapes r are fixed when estimating Θ₁ = (α, λ). Also, Θ₂ denotes the extreme value parameter γ (together with σ when using the GPD tail fit). The idea behind the EM algorithm in this context is to consider the censored sample in contrast to the complete data which is not fully observed. Given a complete version of the data, we can construct a complete likelihood function as

where is the indicator function for the event {X_i ≤ t}. The corresponding complete data log‐likelihood function is

As we do not fully observe the complete version of the data sample, it is not possible to optimize the complete data log‐likelihood directly. The intuitive idea for obtaining parameter estimates in the case of incomplete data is to compute the expectation of the complete data log‐likelihood and then use this expected log‐likelihood function to estimate the parameters. However, taking the expectation of the complete data log‐likelihood requires the knowledge of the parameter vector, and so the algorithm has to run iteratively. Starting from an initial guess for the parameter vector, the EM algorithm iterates between two steps. In the hth iteration of the E‐step the expected value of the complete data log‐likelihood is computed with respect to the unknown data given the observed data and using the current estimate of the parameter vector Θ^(h−1) as true values,

In the M‐step, one maximizes the expected value of the complete data log‐likelihood obtained in the E‐step with respect to the parameter vector:

Both steps are iterated until convergence.

In the E‐step we distinguish the five cases of data points again to determine the contribution of a data point to this expectation:

Note that the event {t^l ≤ l_i = u_i ≤ t < T} indicates that we know t^l, l_i = u_i, t and T, and that the ordering t^l ≤ l_i = u_i ≤ t < T holds. Similar reasonings hold for the other conditional arguments in the expectations. Then, using the law of total probability, the final case can be rewritten as

where {t^l ≤ l_i < X_i ≤ t < u_i ≤ T} denotes that t^l, l_i, t, u_i and T are known, that the ordering t^l ≤ l_i < t < u_i ≤ T holds, and that {X_i ≤ t}. Using (4.3.43) we find that the probability in the first term is then given by

and similarly for the second term. The M‐step with maximization with respect to π, Θ₁ and Θ₂, and the choice of the initial values, is discussed in detail in Reynkens et al. [647].

EVA is not available in the literature for interval censored data. The role of the empirical survival and quantile functions in the construction of a tail analysis for complete data (i.e., setting X_{n−j+1, n} as an estimator of Q(1 − j/(n + 1)), j = 1, …, n) is taken over by the Turnbull [747] estimator . The Turnbull estimator is an extension to interval censoring of the Kaplan–Meier estimator or product‐limit estimator [482], that is, when .

• The Kaplan–Meier estimator of 1 − F is defined as follows: letting 0 = τ₀ < τ₁ < τ₂ < … < τ_N (with N < n) denote the observed possible censored data, N_j the number of observations X_i ≥ τ_j, and d_j the number of values l_i equal to τ_j, then
  
  This expression is motivated from the fact that
  
• Turnbull’s algorithm is then constructed as follows: Let 0 = τ₀ < τ₁ < … < τ_m denote here the grid of all points l_i, u_i, i = 1, 2, …, n. Define δ_ij as the indicator whether the observation in the interval (l_i, u_i] could be equal to τ_j, j = 1, …, m. δ_ij equals 1 if (τ_j−1, τ_j] ⊂ (l_i, u_i] and 0 otherwise. Initial values are assigned to 1 − F(τ_j) by distributing the mass 1/n for the ith individual equally to each possible τ_j ∈ (l_i, u_i]. The algorithm is given as:
  1. Compute the probability p_j that an observation equals τ_j by p_j = F(τ_j) − F(τ_j−1), j = 1, …, m.
  2. Estimate the number of observations at τ_j by
     
  3. Compute the estimated number of data with l_i ≥ τ_j by .
  4. Update the product‐limit estimator using the values of d_j and N_j found in the two preceding steps. Stop the iterative process if the new and old estimate of 1 − F for all τ_j do not differ too much.

In case of interval censored data we can then estimate the mean excess function e (see (3.4.10)) substituting 1 − F by the Turnbull estimator :

(4.3.44)

As discussed in Section 4.2.1, the mean excess function based on the log‐data leads to an estimator of a positive extreme value index γ. As in (4.2.8), using the Turnbull estimator rather than the classical empirical distribution we obtain an estimator of γ > 0 in the case of incomplete data:

(4.3.45)

We then compute these statistics at the positions , k = 1, …, n − 1. Such plots will assist in choosing an appropriate threshold t and estimates of the extreme value index γ to validate the tail component in the splicing.

If the upper bounds are put to , that is, if one uses the right censoring framework, then, under the random right censoring assumption of independence between the real cumulative payment at closure of the claim and the censoring variable C which is observed in case the claim is right censored, estimators of γ > 0 have been proposed in Beirlant et al. [96, 101], Einmahl et al. [315], and Worms and Worms [796]. Using the likelihood approach, Beirlant et al. [101] proposed the estimator

(4.3.46)

with (i = 1, …, n) and the proportion of non‐censored data in the top kZ‐data. Einmahl et al. [315] derived asymptotic results, while Beirlant et al. [96] proposed a bias‐reduced version. Worms and Worms [796] derived a tail index estimator which is derived through the estimation of the mean excess function of the log‐data, comparable with the estimator derived in (4.3.45):

(4.3.47)

where the Kaplan–Meier estimator can be written as

with Δ_{i, n} equal to 1 if the ith smallest observation Z_{i, n} is non‐censored, and 0 otherwise.

4.4.1 Pareto‐type Modelling

When modelling time dependence or incorporating any other covariate information in an independent data setting with Pareto‐type distributed responses Z_i, the exceedances are defined through Y_i = Z_i/t for some appropriate threshold t. Note that in many circumstances the threshold should then also be modelled along x = x_i, i = 1, …, n. As before, we assume that as

(4.4.48)

where A_i, γ_i > 0. Regression can be modelled through the scale parameter A and/or the extreme value index γ.

Changes in γ can be modelled in a parametric way using likelihood techniques. Suppose, for instance, that regression modelling of γ > 0 using an exponential link function appears appropriate in a given case study:

The log‐likelihood function is then given by

leading to the likelihood equations

Beirlant and Goegebeur [99] propose to inspect the goodness‐of‐fit of such a regression model under constant scale parameter A on the basis of a Pareto QQ‐plot using

which are indeed approximately Pareto distributed with tail index 1, when the regression model is appropriate.

The case where γ does not depend on i, while A does depend on i, was formalized in Einmahl et al. [316] assuming that there exists a tail function 1 − F and a continuous, positive function A defined on [0, 1] such that

(4.4.49)

uniformly for all and all i = 1, …, n with . A is then called the skedasis function, which characterizes the trend in the extremes through the changes in the scale parameter A. Under (4.4.49), Einmahl et al. [316] showed that the Hill estimator H_{k, n} is still a consistent estimator for γ. Assuming equidistant covariates x_i = i/n, i = 1, …, n, as in (4.2.19),

where denotes the number of Z values larger than the threshold Z_{n−k, n} with covariate value x in a neighbourhood of x_i. The contribution of the observations to is governed by a symmetric density kernel function K on [−1, 1] and K_h(x) = K(x/h)/h, so that K gives more weight to the observations with covariates closer to x_i. We hence obtain

Finally, estimators of small tail probabilities and large quantiles follow directly from (4.4.49), (4.2.12) and (4.2.13):

4.4.2 Generalized Pareto Modelling

Let Y₁, …, Y_n be independent GPD random variables and let x_i denote the covariate information vector, that is,

where γ(x), σ(x), μ(x) denote admissible functions of x, whether of parametric nature using three vectors of regression coefficients β_j (j = 1, 2, 3) of length p with , and , or of non‐parametric nature. Again this model is used as an approximation of the conditional distribution of excesses Y (x) = Z − μ(x) over a high threshold μ(x) given that there is an exceedance. The choice of an appropriate threshold μ(x) is of course even more difficult than in the non‐regression setting since the threshold can depend on the covariates in order to take the relative extremity of the observations into account.

• When parametric functions , and have been chosen, the estimators of β_j (j = 1, 2, 3) can be obtained by maximizing the log‐likelihood function
  
  where N_μ denotes the number of excesses over the threshold function μ(x).
• Alternatively, non‐parametric regression techniques are available to estimate the parameter functions γ(x), σ(x). Consider independent random variables Z₁, …, Z_n and associated covariate information x₁, …, x_n. Suppose we focus on estimating the tail of the distribution of Z at x^*. Fix a high local threshold μ(x^*) and compute the exceedances Y_i = Z_j − μ(x^*), provided Z_j > μ(x^*), . Here j is the index of the ith exceedance in the original sample, and denotes the number of exceedances over the threshold μ_x^*. Then re‐index the covariates in an appropriate way such that x_i denotes the covariate associated with exceedance Y_i. Using local polynomial maximum likelihood estimation, one approximates γ(x) and σ(x) by polynomials, centered at x^*. Let h denote a bandwidth parameter and consider a univariate covariate x. Assuming γ, respectively σ, being m₁, respectively m₂, times differentiable one has for |x_i − x^*| ≤ h,
  

  where

  

  The coefficients of these approximations can be estimated by local maximum likelihood fits of the GPD, with the contribution of each observation to the log‐likelihood being governed by a kernel function K. The local polynomial maximum likelihood estimator (β₁, β₂) = () is then the maximizer of the kernel weighted log‐likelihood function

  

  with respect to , where g(y;μ, σ) = (1/σ)(1 + (γ/σ)y)^−1−1/γ is the density of the generalized Pareto distribution.

  A more recent approach is using penalized log‐likelihood optimization based on spline functions. Let the covariates x be one‐dimensional within an interval [a, b]. The goal is to fit reasonably smooth functions h_γ and h_σ with γ(x) = h_γ(x) and σ(x) = h_σ(x) to the observations (Y_i, x_i), i = 1, …, N_μ. The penalized log‐likelihood is then given by

  

  The introduction of the penalty terms is a standard technique to avoid over‐fitting when one is interested in fitting smooth functions (see Hastie and Tibshirani [428] or Green and Silverman [408]). Next • stands for γ or σ. Intuitively the penalty functions measure the roughness of twice‐differentiable curves and the smoothing parameters λ_• are chosen to regulate the smoothness of the estimates ĥ_• : larger values of these parameters lead to smoother fitted curves.

  Let a = s₀ < s₁ < … < s_m < s_m+1 = b denote the ordered and distinct values among . A function h defined on [a, b] is a cubic spline with the above knots if the following conditions are satisfied:

  • on each interval [s_i, s_i+1], h is a cubic polynomial
  • at each knot s_i, h and its first and second derivatives are continuous.

  A cubic spline is a natural cubic spline if in addition to the two latter conditions it satisfies the natural boundary condition that the second and third derivatives of h at a and b are zero. It follows from Green and Silverman [408] that for a natural cubic spline h with knots s₁, …, s_m one has

  

  where h_• = (h_•(s₁), …, h_•(s_m)), and K is a symmetric m × m matrix of rank m − 2 only depending on the knots s₁, …, s_m. Hence

  

In order to assess the validity of a chosen regression model one can generalize the exponential QQ‐plot of generalized residuals defined before in the non‐regression case:

with

Finally, given regression estimators for (γ(x), σ(x)) using an appropriate threshold function μ(x), extreme quantile estimators are given by

where can, for instance, be taken to be equal to the Nadaraya–Watson estimator

For more details and other non‐parametric methods, refer to Davison and Ramesh [252], Hall and Tajvidi [420], Chavez‐Demoulin and Davison [200], Daouia et al. [249] [248], Gardes and Girard [368, 369], Gardes and Stupfler [370], Goegebeur, Guillou and Osmann [393], and Stupfler [713], as well as Chavez‐Demoulin et al. [201] for other non‐parametric extreme value regression methods and applications.

4.4.3 Regression Extremes with Censored Data

In Section 4.3 we discussed the problem when estimating the distribution of the final payments based on censored data using the Kaplan–Meier estimator of the distribution of the payment data. Here we propose to consider regression modelling of the final payments given the development time at the closure of a claim. Note, however, that both the final payments and development periods are right censored, both variables being censored (or not censored) at the same time. We again use the notation Z_i (i = 1, …, n) for the observed cumulative payment at the end of the study from that section, and similarly nDY_{e, i} for the observed number of development years at the end of 2010. Again Δ_{i, n} denotes the indicator of non‐censoring corresponding to the ith smallest observed value payment Z_{i, n}. Akritas and Van Keilegom [11] proposed the following non‐parametric estimator of the conditional distribution of X given a specific value of nDY assuming that X and the censoring variable C (see Section 4.3) are conditionally independent given nDY:

with weights

Denoting the weight W corresponding to the ith smallest Z value Z_{i, n} with W_{i, n} we then arrive at the following Hill‐type estimator of the conditional extreme value given nDY = d, generalizing the unconditional Worms and Worms estimator defined in (4.3.47):

Pareto QQ‐plots adapted for censoring per chosen d value can then be defined as

In order to derive a full model for the complete payments X as a function of nDY = d a local version of the splicing algorithm from Section 4.3.2 can be developed considering random right censoring on X with the kernel weights W_i = W_i(d;h) as introduced above. The EM algorithm can then be applied using a kernel weighted log‐likelihood, comparable with the approach from Section 4.4.1. For instance, given the complete version of the data, the complete likelihood function is then given by

The corresponding complete data weighted log‐likelihood function then equals

4.5.1 The Multivariate POT Approach

From (3.6.25) and (3.6.26) one observes the importance of estimating the stable tail dependence function l defined in Chapter 3. The estimation of the tail dependence can be performed non‐parametrically or using parametric models. We refer to Kiriliouk et al. [488] for fitting parametric multivariate generalized Pareto models using censored likelihood methods.

A non‐parametric estimator of an STDF is given by

(4.5.50)

with

where R_{i, j} denotes the rank of X_{i, j} among X_{1, j}, …, X_{n, j}:

The estimator is a direct empirical version of definition (3.6.22) of l with u = n/k.

A slightly different version is given by

(4.5.51)

where denotes the ith smallest observation of component j. Bias‐reduced versions of these estimators were proposed in Fougères et al. [357] and Beirlant et al. [98]. In the bivariate case, or then act as estimators of the extremal coefficient θ.

An estimator of the extremal dependence coefficient χ can be constructed on the basis of an estimator of χ(u) for u → 1 using the estimator of C(u, u)

(4.5.52)

where (j = 1, 2; i = 1, …, n) with denoting the empirical distribution function of the jth marginal and X_{i, j} the ith observation of the jth component. Hence

As an application note that an estimator of the parameter τ in the logistic dependence model can be obtained from χ(u) → 2 − 2^τ as u → 1, from which

Setting

the Hill estimator based on (i = 1, …, n) leads to an estimator of the coefficient of tail dependence η. Of course bias reduction techniques can be applied here too.

4.5.2 Multivariate Mixtures of Erlangs

Lee and Lin [533] defined a d‐variate Erlang mixture where each mixture component is the joint distribution of d independent Erlang distributions with a common scale parameter 1/λ > 0. The dependence structure is then captured by the combination of the positive integer shape parameters of the Erlangs in each dimension. We denote the positive integer shape parameters of the jointly independent Erlang distributions in a mixture component by the vector r = (r₁, …, r_d) and the set of all shape vectors with non‐zero weight by ℛ. The density of a d‐variate Erlang mixture evaluated in x > 0 can then be written as

(4.5.53)

Lee and Lin [533] showed that, given any density f(x), the d‐variate Erlang mixture

with mixing weights

satisfies . The weights α_r of the components in the mixture are defined by integrating the density over the corresponding d‐dimensional rectangle of the grid formed by the shape parameters multiplied with the common scale. When the value of λ increases, this grid becomes more refined and the sequence of Erlang mixtures converges to the underlying distribution function.

Verbelen et al. [757] provided a flexible fitting procedure for multivariate mixed Erlangs (MMEs), which iteratively uses the EM algorithm, by introducing a computationally efficient initialization and adjustment strategy for the shape parameter vectors. Randomly censored and fixed truncated data can also be dealt with.

The bivariate distribution function of the fitted bivariate GPD is given by

In order to guarantee that the marginal distributions have support on one has to impose the constraints and , which then lead to the parameter values (γ₁ = 0.57, σ₁ = 3.57) and (γ₂ = 0.65, σ₂ = 6.05).