Rank-based inference tools for copula regression, with property and casualty insurance applications

Insurance: Mathematics and Economics 89 (2019) 1–15

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Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime

Rank-based inference tools for copula regression, with property and casualty insurance applications ∗

Marie-Pier Côté a , Christian Genest b , , Marek Omelka c a

École d’actuariat, Université Laval, 2425, rue de l’Agriculture, Québec (Québec), Canada G1V 0A6 Department of Mathematics and Statistics, McGill University, 805, rue Sherbrooke ouest, Montréal (Québec), Canada H3A 0B9 c Department of Probability and Mathematical Statistics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic b

article

info

Article history: Received April 2018 Received in revised form March 2019 Accepted 19 August 2019 Available online 6 September 2019 Keywords: Empirical copula process Goodness-of-fit test Inversion of Kendall’s tau Maximum pseudo-likelihood Residuals Weak convergence

a b s t r a c t Rank-based procedures are commonly used for inference in copula models for continuous responses whose behavior does not depend on covariates. This paper describes how these procedures can be adapted to the broader framework in which (possibly non-linear) regression models for the marginal responses are linked by a copula that does not depend on covariates. The validity of many of these techniques can be derived from the asymptotic equivalence between the classical empirical copula process and its analog based on suitable residuals from the marginal models. Moment-based parameter estimation and copula goodness-of-fit tests are shown to remain valid under weak conditions on the marginal error term distributions, even when the residual-based empirical copula process fails to converge weakly. The performance of these procedures is evaluated through simulation in the context of two general insurance applications: micro-level multivariate insurance claims, and dependent loss triangles. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

1. Introduction Joint regression analysis of multivariate data using copulas can be traced back to the work of Song (2000) and Oakes and Ritz (2000). It can be broadly described as follows. Consider a d × 1 vector Y = (Y1 , . . . , Yd )⊤ of responses and let X = (X1 , . . . , Xq )⊤ be an associated q × 1 vector of covariates with support SX ⊆ Rq . For each j ∈ {1, . . . , d}, a (possibly non-linear) regression model is first developed for Yj as a function of X = x, yielding a conditional distribution function defined, for all y ∈ R, by Fjx (y) = Pr(Yj ≤ y | X = x). These margins are then bound through a copula to account for the residual dependence between the response variables. Initially, this dependence is often taken to be the same across all values of x. This simplifying assumption stipulates that there exists a copula C such that, for all y = (y1 , . . . , yd ) ∈ Rd and x ∈ SX , Hx (y1 , . . . , yd ) = Pr(Y1 ≤ y1 , . . . , Yd ≤ yd | X = x)

= C {F1x (y1 ), . . . , Fdx (yd )}.

(1)

This copula regression modeling approach has found applications in many fields, e.g., survival analysis (Oakes and Ritz, ∗ Corresponding author. E-mail addresses: [email protected] (M.-P. Côté), [email protected] (C. Genest), [email protected] (M. Omelka).

2000; He and Lawless, 2005; Barriga et al., 2010; Fachini et al., 2014), genetics (Li et al., 2006), and hydrology (Ben Alaya et al., 2014). In insurance contexts, elliptical copulas have been used to combine generalized linear models (GLMs) for the components of auto or multi-peril homeowners insurance claims; see, e.g., Frees and Wang (2005, 2006), Frees and Valdez (2008), Frees et al. (2010), Czado et al. (2012), and Shi et al. (2016). Frees and Sun (2010) also used a Gaussian copula regression with Gamma margins to model whole-life and term insurance demand jointly. In recent years, copula regression has also been successfully applied to account for dependence among loss triangles. For example, Shi and Frees (2011) considered Gaussian and Frank copulas as a way of coupling GLMs with log-normal and Gamma error distributions for personal and commercial auto lines. A reserving methodology for dependent loss triangles based on the Gaussian copula and development year effects in the margins was also proposed by De Jong (2012). Extensions using hierarchical copula structures were studied by Abdallah et al. (2015) and Côté et al. (2016), who showed the value of this approach for reserve estimation and risk capital evaluation in aggregated portfolios by fitting multivariate models to run-off triangle data for various business lines. While copula regression is gaining momentum in actuarial science and beyond, tools for the selection and validation of dependence structures in this context are still in their infancy. This is in contrast with the simpler case of multivariate responses whose behavior does not depend on any covariate. In the latter

https://doi.org/10.1016/j.insmatheco.2019.08.001 0167-6687/© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

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M.-P. Côté, C. Genest and M. Omelka / Insurance: Mathematics and Economics 89 (2019) 1–15

setting, rank-based inference for copula models is well established; see, e.g., Genest and Favre (2007) or Genest and Nešlehová (2012) for reviews. Central to the development of rank-based inference for copula models is the empirical copula, i.e., the sample distribution of the vectors of normalized ranks. Its asymptotic behavior can be used to establish the validity of many estimation and test procedures. When the response variables are not affected by covariates, the limit of the corresponding empirical process has been studied intensively; see, e.g., Segers (2012) or Genest et al. (2017) and references therein. When the responses depend on covariates, Gijbels et al. (2015) suggested that the empirical copula be based on the residuals from the marginal models. In the time series literature, a similar idea can be traced back at least to Chen and Fan (2005), and Rémillard (2017) showed that under broad conditions, the limit of the empirical copula based on residuals from such models is the same as if the innovations could be observed. Gijbels et al. (2015) established an analog for copula regression models with location-scale margins in the case of q = 1 covariate and d = 2 response variables. This finding was then extended to arbitrary q and d by Neumeyer et al. (2019), who also allow for time dependence. Unfortunately, the conditions under which Neumeyer et al. (2019) derived their results are too restrictive for model (1), where time dependence is irrelevant. For example, they exclude some Gamma GLMs and error distributions such as the exponential or the Weibull, which are commonly used in insurance and elsewhere (He and Lawless, 2005; Fachini et al., 2014). In this paper, we establish the validity of rank-based inference techniques for an extended class of copula regression models. Our results cover situations frequently seen in actuarial applications where the error terms of the regressions for continuous responses belong to parametric location-scale families, possibly after transforming the response variables. We prove the convergence of a weighted version of the empirical copula process under assumptions on the margins that are satisfied by the common continuous GLMs. We then show that this result is sufficient to deduce the consistency of popular parameter estimation procedures based on the inversion of Spearman’s rho and Kendall’s tau. Our result also guarantees the validity of the usual copula goodness-of-fit tests based on the Cramér–von Mises or the Kolmogorov–Smirnov statistics (Genest et al., 2009). Within model (1), our results are more broadly applicable than those of Gijbels et al. (2015) and Neumeyer et al. (2019) when the marginals of the d ≥ 2 responses can be transformed into parametric location-scale models depending on q ≥ 1 covariates. Our results also differ from those of Portier and Segers (2018), who studied the limit of an empirical copula process based on residuals deduced from a nonparametric smoothing of the margins when q = 1. As they make no location-scale assumption, their notion of residual is very different from ours. Their conditions on the underlying copula are also stricter than ours and their result holds on a fixed proper subset of [0, 1]d , whereas ours covers the entire space asymptotically. Finally, all these authors focused on the limiting behavior of the empirical copula process; only Neumeyer et al. (2019) deduced from it the validity of some rank-based inference tools. In contrast, we show the consistency of some method-of-moment estimators and goodness-of-fit tests even in cases where the empirical process fails to converge. The broad class of location-scale models considered here is described in Section 2. The empirical copula is then defined in Section 3 and under the assumptions listed in Section 4, the weak limit of a weighted version of the corresponding empirical process is established in Proposition 1. It is further shown in Proposition 2 that the limit of suitable functionals of the empirical copula based on residuals is the same as if the marginal

parameters were known. Consequences for the limiting behavior of rank-based parameter estimation and goodness-of-fit testing techniques are outlined in Section 4.3. The finite-sample behavior of these procedures is then investigated in Section 5 in two general insurance applications of copula regression: micro-level data and dependent loss triangles. All mathematical developments are relegated to an Appendix and additional simulation results are available in an Online Supplement. 2. Copula regression with location-scale error laws Consider a vector Y = (Y1 , . . . , Yd )⊤ of responses and let X = (X1 , . . . , Xq )⊤ be an associated vector of covariates with support SX ⊂ Rq . Assume that for all x ∈ SX , the distribution of Y given X = x can be written as in (1) in terms of a copula C which is functionally independent of x. Further assume that the distribution of Y given X = x is continuous for all x ∈ SX , so that C is unique. In practice, the margins F1x , . . . , Fdx and the copula C are unknown but often assumed to belong to parametric classes, viz. F1x ∈ F1,ξ1 , . . . , Fdx ∈ Fd,ξd ,

C ∈ Cθ ,

indexed by parameters ξ 1 , . . . , ξ d living in open sets Ξ1 , . . . , Ξd , respectively. While inference on the margins is standard, there are no tools for selecting the family Cθ . In most copula regression applications, researchers have either assumed that C is known (Oakes and Ritz, 2000) or that it belongs to a given class Cθ (Frees et al., 2010; Masarotto and Varin, 2012). At times, the copula has been picked from a short list (Frees and Valdez, 2008; Shi and Frees, 2011). In the absence of covariates, the choice of copula can be informed by rank-based techniques as reviewed, e.g., in Genest and Favre (2007). Côté et al. (2016) illustrate how to mimic such rank-based copula selection procedures when margins are lognormal or Gamma regressions. The key idea is to transform the responses, resulting in error terms that are independent of X. Indeed when Y given X is log-normal with parameters µ = X⊤ β and σ 2 , then ε = ln(Y ) − X⊤ β is N (0, σ 2 ) and independent of X. Likewise, if Y given X has a Gamma distribution with scale exp(X⊤ β) and shape parameter α , then ε = Y /exp(X⊤ β) has a Gamma distribution with shape α and unit scale. Here again the distribution of ε does not depend on X. Although the error term ε cannot be observed, its value corresponding to a pair (Y , X) can be approximated by plugging estimates of the unknown parameters in the relation linking ε to Y and X. The resulting quantity is called a residual by analogy with regression. When the relation between ε and Y is monotonic, as in the above examples, the copula of the vector of error terms is the same as the copula C of the response vector Y given X = x. If, in addition, the copula C does not depend on x, the vector of residuals can then be used to form an empirical copula. This approach can actually be expanded to a much broader class of distributions satisfying the location-scale assumption stated below. See Table 1 for special cases of models encompassed in this framework. Assumption I: Location-scale distribution for the error terms. For each j ∈ {1, . . . , d}, there exists a known transformation ψj which is increasing on the support of Yj and known functions µj : SX × Ξj → R and σj : SX × Ξj → R+ such that the random variable

εj =

ψj (Yj ) − µj (X; ξ j ) σj (X; ξ j )

is stochastically independent from X.

M.-P. Côté, C. Genest and M. Omelka / Insurance: Mathematics and Economics 89 (2019) 1–15

3

Table 1 Examples of distributions fulfilling the location-scale assumption. Density of Y

ε

Parameters

Exponential-Burr(α, 1, 1) ln(Y ) − ln{m(X; ξ )}

Burr XII

αβ yβ−1 m−β {1 + (y/m)β }−α−1

α, m(X; ξ ) > 0 β (X; ξ ) > 0

1/β (X; ξ ) Gamma(α, 1)

Gamma

α>0 σ (X; ξ ) > 0

σ −α yα−1 e−y/σ /Γ (α )

Y /σ (X; ξ ) N (0, 1)

Gaussian

µ(X; ξ ) ∈ R σ (X; ξ ) > 0

φ{(y − µ)/σ }

{Y − µ(X; ξ )}/σ (X; ξ ) Exponential-GB(1, 1, c , p, q) ln(Y ) − ln{m(x; ξ )}

Generalized Beta

|λ|ypλ−1 {1 − (1 − c)(y/m)λ }q−1 mpλ B(p, q){1 + c(y/m)λ }p+q

m(X; ξ ), λ(X; ξ ) > 0

1/λ(x; ξ )

c ∈ [0, 1), p, q > 0

Gompertz(α, 1)

Gompertz

αβ exp(β y + α − α eβ y )

α, β (X; ξ ) > 0

Y β (X; ξ )

µ(X; ξ ) ∈ R σ (X; ξ ) > 0

{Y − µ(X; ξ )}/σ (X; ξ )

Gumbel(0, 1)

Gumbel

σ −1 exp(−z − e−z ),

z = (y − µ)/σ

Inverse Gaussian(1, c)

Inverse Gaussian

√

λ

{ exp

2π y3

−λ(y − m) 2m2 y

2

}

m(X; ξ ) > 0, c > 0

Y /m(X; ξ )

λ(X; ξ ) = c m(X; ξ ) N (0, 1)

Log-Normal

µ(X; ξ ) ∈ R σ (X; ξ ) > 0

φ[{ln(y) − µ}/σ ]

{ln(Y ) − µ(X; ξ )}/σ (X; ξ ) Pareto(α, 1)

Pareto

ασ α y−(α+1) ,

y>σ

α, σ (X; ξ ) > 0

Y /σ (X; ξ )

m(X; ξ ) > 0

Exponential-Weibull(1, 1) ln(Y ) − ln{m(X; ξ )}

Weibull α−1

αy

−α

m

α

exp{−(y/m) }

3. Definition of the empirical copula Given a copula family, the joint likelihood involves many parameters and is typically difficult to maximize globally. For this reason, a two-step approach is often adopted. First, the marginal models are fitted, e.g., by least squares or maximum likelihood. The resulting estimates can then be used as initial values in maximizing the overall likelihood. Alternatively, some authors apply a probability integral transform based on the estimated margins to the data and regard the resulting pseudo-observations as realizations of the copula C . These approaches are not completely satisfactory in that they do not allow to test for model adequacy. In addition, the validity of the second approach has never been established. In this paper, another two-step approach is advocated, where the marginal models are first fitted and the estimated residuals are formed; in the second step, the standardized ranks of the latter are used as pseudo-observations from the copula C . Let (X1 , Y1 ), . . . , (Xn , Yn ) be mutually independent copies of a random vector (X, Y), where Y has a continuous distribution. If the parameters ξ 1 , . . . , ξ d were known, one could then compute the error terms defined, for all i ∈ {1, . . . , n} and j ∈ {1, . . . , d}, by

εij =

ψj (Yij ) − µj (Xi ; ξ j ) . σj (Xi ; ξ j )

For each j ∈ {1, . . . , d}, the error terms ε1j , . . . , εnj then form a random sample from distribution Fjε which, by Assumption I, is stochastically independent of X. The corresponding empirical distribution function is defined, for all y ∈ R, by n

1∑ ˆ Fjε (y) = 1(εij ≤ y), n

i=1

where 1 denotes the indicator function.

1/α (X; ξ )

α (X; ξ ) > 0

One could then estimate C by the oracle empirical copula defined, for all u = (u1 , . . . , ud ) ∈ [0, 1]d , by Cn (u) =

n 1∑

n

1{ˆ F1ε (εi1 ) ≤ u1 , . . . , ˆ Fdε (εid ) ≤ ud }.

i=1

This estimator is known to be consistent and asymptotically Gaussian under the following regularity condition; see, e.g., Theorem 1 in Genest et al. (2017). II: Copula smoothness. For each j ∈ {1, . . . , d}, the map u ↦ → C (j) (u) = ∂ C (u)/∂ uj exists and is continuous on the set {u ∈ [0, 1]d : 0 < uj < 1}. Unfortunately, Cn is not a feasible estimator, as it depends on the unobserved marginal error terms. As an alternative, consider the residuals defined, for each i ∈ {1, . . . , n} and j ∈ {1, . . . , d}, by

ˆ εij =

ψj (Yij ) − µj (Xi ;ˆ ξj ) , σj (Xi ;ˆ ξj )

where ˆ ξ j is a suitable estimator of ξ j . For each j ∈ {1, . . . , d}, let also ˆ Fjˆε be the empirical distribution function based on the residuals ˆ ε1j , . . . ,ˆ εnj defined, for all y ∈ R, by n

1∑ ˆ Fjˆε (y) = 1(ˆ εij ≤ y). n

i=1

Under Assumption I, a plausible rank-based nonparametric estimator of the copula C is then given, for all u = (u1 , . . . , ud ) ∈ [0, 1]d , by n

1∑ ˆ Cn (u) = 1{ˆ F1ˆε (ˆ εi1 ) ≤ u1 , . . . , ˆ Fdˆε (ˆ εid ) ≤ ud }. n

i=1

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M.-P. Côté, C. Genest and M. Omelka / Insurance: Mathematics and Economics 89 (2019) 1–15

What will be shown below is that ˆ Cn has the same asymptotic behavior as Cn under suitable regularity conditions. The consistency of ˆ Cn will then follow from the consistency of Cn , which is known to hold under Assumption II. 4. Regularity assumptions and main results The conditions under which the asymptotic behavior of ˆ Cn and rank-based inference procedures will be established are stated in Section 4.1. The main results are given in Section 4.2 and their consequences for inference are briefly described in Section 4.3. 4.1. General regularity assumptions Unlike rank-based inference in the absence of covariates, the estimation of the dependence structure in a copula regression model requires that the marginal models be specified correctly, at least to the extent that Assumption I holds. The true underlying copula can be recovered from the data only if the marginal parameters are estimated consistently. This assumption, formally stated below, is satisfied, e.g., by the maximum likelihood estimators in a GLM. III: Consistency of the √ marginal parameter estimators. For each ξ j − ξ j ) = OP (1). j ∈ {1, . . . , d}, one has n (ˆ

√

Cn − To show the weak convergence of the process ˆ Cn = n (ˆ C ), regularity conditions must be imposed on the distribution Fjε of the error and the corresponding density fjε for each j ∈ {1, . . . , d}. Two cases can be distinguished, depending on the presence or not of a shift µj in the definition of the error in Assumption I. IV: Regularity of the error distributions in the presence of a shift. For each j ∈ {1, . . . , d}, one has lim y fjε (y) = 0.

|y|→∞

(2)

1 Furthermore, the mapping u ↦ → fjε {Fj− ε (u)} is continuous, and either one of the following conditions holds:

(a) one has limu↓0 limu↑1

{1 + |Fj−ε 1 (u)|}fjε {Fj−ε 1 (u)} = 0, {1 + |Fj−ε 1 (u)|}fjε {Fj−ε 1 (u)} = 0;

∀u∈(0,1) fjε {Fj−ε 1 (u)} ≤ Lj u−ηj (1 − u)−ηj . If the distribution chosen for Yj is such that µj (X; ξ j ) = 0 almost surely in Assumption I, then the residuals are calculated as

ˆ εij = ψj (Yij )/σj (Xi ;ˆ ξ j ). In this case, Assumption IV may be replaced by Assumption IV⋆ , stated below. The proofs of the main results are written for the general case of Assumption IV, but the modifications are straightforward for the case of scale adjustment only under the corresponding parts of Assumption IV⋆ . IV⋆ : Regularity of the error distributions in the absence of a shift. For each j ∈ {1, . . . , d}, Eq. (2) is valid. Furthermore, the 1 mapping u ↦ → fjε {Fj− ε (u)} is continuous, and either one of the following conditions holds:

limu↓0 limu↑1

|Fj−ε 1 (u)|fjε {Fj−ε 1 (u)} = 0, |Fj−ε 1 (u)|fjε {Fj−ε 1 (u)} = 0;

∀u∈(0,1) |Fj−ε 1 (u)|fjε {Fj−ε 1 (u)} ≤ Lj u−ηj (1 − u)−ηj . For each j ∈ {1, . . . , d}, Assumptions IV and IV⋆ are less restrictive on Fjε than the condition used in Gijbels et al. (2015). For example, they cover all exponential distributions and, if µj (X; ξ j ) = 0, all Gamma and Weibull distributions. They also include location-scale models in which µj (X; ξ j ) ̸ = 0 and εj is Gamma or Weibull with shape parameter larger than 1/2. Finally, the location and scale functions are assumed to behave nicely, at least in a neighborhood of the true parameter value. V: Assumptions on conditional location and scale. For each j ∈ {1, . . . , d}, there exists a neighborhood U(ξ j ) of the true parameter value ξ j such that the (first order) partial derivatives of the functions µj (x; ξ ) and σj (x; ξ ) with respect to ξ are uniformly bounded as functions of both arguments on SX × U(ξ j ). Moreover, inf

x∈SX ,t∈U(ξ j )

σj (x; t) > 0.

The requirement that the scale function is strictly positive holds whenever the link function restricts the domain of the scale properly. 4.2. Main results In what follows, all results are stated in the general case where

µ1 , . . . , µd in Assumption I are non-zero almost surely. Whenever µj (X, ξ j ) ≡ 0 for some values of j ∈ {1, . . . , d}, Assump-

tion IV can be replaced at no cost by the weaker Assumption IV⋆ for these specific values of j. The following proposition, which is proved in Appendix A, is the key technical result which will be used below to validate rank-based inference tools for copula regression models under various conditions. In what follows, a ∧ b = min(a, b) and operations between vectors are meant to apply component-wise. Proposition 1. Suppose that Assumptions I–V hold and for each j ∈ {1, . . . , d} and u ∈ [0, 1], set wj (u) = 1 if Assumption IV (a) holds and wj (u) = uη¯ j (1 − u)η¯ j for some η¯ j > ηj otherwise. For all u = (u1 , . . . , ud ) ∈ [0, 1]d , further let w (u) = w1 (u1 ) ∧· · ·∧wd (ud ). Then,

√

(3)

(b) there exist ηj ∈ [0, 1) and a constant Lj ∈ (0, ∞) such that

(a) one has

(b) there exist ηj ∈ [0, 1) and a constant Lj < ∞ such that

sup | n w (u) {ˆ Cn (u) − Cn (u)}| = oP (1) u∈In

with In = I1n × · · · × Idn , where for each j ∈ {1, . . . , d}, Ijn = [0, 1] if Assumption IV (a) holds for j and, otherwise, Ijn =

[( 1 )1/(1+η¯ j ) ( 1 )1/(1+η¯ j ) ] , 1− √ . √ n

n

First consider the case where Assumption IV (a) is valid. Given the √ weak convergence of the oracle empirical copula process n (Cn −C ), Proposition 1 then leads to the following multivariate, multi-covariate extension of the main findings in Gijbels et al. (2015). Corollary 1. Suppose that Assumptions I–V hold, and that Assumption IV (a) is satisfied. Then, as n → ∞,

√ Cn (u) − Cn (u)}| = oP (1) sup | n {ˆ

(4)

u∈[0,1]d

and [0, 1]d , by ˆ Cn (u) = √ hence the process defined, for all u ∈ n {ˆ Cn (u) − C (u)} converges in the space ℓ∞ ([0, 1]d ) to a centered Gaussian process GC defined by GC (u) = BC (u) −

d ∑ j=1

C (j) (u) BC (u(j) ),

M.-P. Côté, C. Genest and M. Omelka / Insurance: Mathematics and Economics 89 (2019) 1–15

where BC is a Brownian bridge on [0, 1]d whose covariance function is given, for all u, v ∈ [0, 1]d , by E{BC (u)BC (v)} = C (u ∧ v) − C (u) C (v) and u(j) denotes the vector whose jth entry is uj and the d − 1 others are 1. Thus when Assumption IV (a) holds, one can use Corollary 1 to validate rank-based inference procedures for copula regression models whose responses satisfy the location-scale structure embodied in Assumption I. For example, when C ∈ Cθ = {C (·; θ ) : θ ∈ O}, one can deduce from Corollary 1 that some common rank-based estimators of θ are consistent. When Assumption IV (a) fails, it may happen that Eq. (4) no longer holds. Nevertheless, Proposition 1 implies that the weighted difference w ˆ Cn is still asymptotically negligible for a suitably chosen weight function w . As it turns out, this is sufficient to determine the asymptotic behavior of many interesting functionals of the empirical copula based on residuals, notably those commonly used to estimate dependence parameters. To be specific, consider a functional T defined on the space D of distribution functions on [0, 1]d , e.g., the Kendall tau functional C ↦ → T (C ) =

1 2d−1

−1

[0,1]d

J ⊆{1,...,d}

[0,1]d

C (u) dC (u) .

|C1 (u(J) ) − C2 (u(J) )|d{C1 (u) + C2 (u) + u}, (5) (J)

where the jth component of u equals uj if j ∈ J and 1 otherwise. Under this condition, which is verified not only for Kendall’s tau, but also Spearman’s rho and many other functionals of interest, one has the following result. Proposition 2. Suppose that Assumptions√I–V hold and that the functional T satisfies (5). Then, as n → ∞, n {T (ˆ Cn ) − T (Cn )} = oP (1). As an immediate consequence of this result, the asymptotic distribution of dependence measures such as Kendall’s tau or Spearman’s rho are the same whether they are computed from the residuals ˆ εij or the unobserved errors εij . The same conclusion holds for the wider class of functionals T satisfying condition (5), as stated below. Corollary 2. Suppose that√Assumptions I–V hold and that the 2 functional T satisfies √ (5). If n {T (Cn ) − T (C )}2 ⇝ N (0, σ ) as ˆ n → ∞, then also n {T (Cn ) − T (C )} ⇝ N (0, σ ). Because it does not rely on Condition (3), this result is more broadly applicable than its analog based on Corollary 1. Some of its consequences for rank-based inference in copula regression models are discussed next. 4.3. Consequences for inference Assume that C ∈ Cθ and let ˆ θ n be a rank-based, moment-like estimator of θ based on the residuals ˆ εij . Further suppose that this estimator can be expressed as a functional that satisfies condition (5), which is valid, e.g., for the inversion of Spearman’s rho or Kendall’s tau when θ ∈ R. Corollary 2 then ensures that ˆ θ n has the same limiting distribution as the analog ˜ θ n of ˆ θ n based on the unobservable errors εij , i.e.,

√

n (ˆ θn − ˜ θ n ) = oP (1).

vs. H1 : C ∈ / Cθ .

H 0 : C ∈ Cθ

(7)

Specifically, let

∫ ˆn = n CM

[0,1]d

{ˆ Cn (u) − C (u; ˆ θ n )}2 dνn (u),

and

√ n

ˆn = KS

sup |ˆ Cn (u) − C (u; ˆ θ n )|,

u∈[0,1]d

where dνn stands either for the Lebesgue measure or for dˆ Cn . Let also

∫ CMn = n

[0,1]d

{Cn (u) − C (u;˜ θ n )}2 dνn (u),

and

√ KSn =

n

sup |Cn (u) − C (u; ˜ θ n )|,

u∈[0,1]d

Suppose that there exists a constant M > 0 such that for all C1 , C2 ∈ D,

|T (C1 ) − T (C2 )| ∑ ∫ ≤M

A similar phenomenon occurs for the Cramér–von Mises and Kolmogorov–Smirnov statistics used to test the hypotheses

}

{ ∫ −1 + 2d

5

(6)

ˆ n and KS ˆ n based on the oracle estimators Cn be the analogs of CM and ˜ θn . ˜ θ n is such that the process √ Further suppose that the estimator { n {Cn (u) − C (u;˜ θ n )} : u ∈ [0, 1]d } is tight with respect to the uniform norm on the space of continuous functions on [0, 1]d . One then has the following result. Proposition 3. Suppose that Assumptions I–V hold, that Eq. (6) is valid and that there exists a neighborhood U(θ ) of the true parameter value θ such that

⏐ ⏐ ⏐∂ ⏐ C (u; a)⏐⏐ < ∞. a∈U(θ ) u∈[0,1]d ∂ a sup

sup ⏐⏐

(8)

Further suppose that hypothesis H0 specified in Eq. (7) is verified. If Assumption IV (a) holds or if max(η1 , . . . , ηd ) < 1/2, then, as n → ∞,

ˆ n − CMn = op (1). CM Moreover, if Assumption IV (a) is satisfied, then, as n → ∞,

ˆ n − KSn = op (1). KS Thus under suitable regularity conditions, the goodness-ofˆ n and KS ˆ n are asymptotically equivalent to fit test statistics CM their analogs CMn and KSn which could only be observed if the error terms were known. Therefore, the p-values for these tests can be approximated by a parametric bootstrap or the multiplier method, whose validity yet remains to be formally established. 5. Simulation studies In this section, the finite-sample behavior of rank-based inference techniques is studied for copula regression models that are typical in insurance. As is common in that field, the copulas involve a scalar parameter θ which we take as Kendall’s tau. Two estimation procedures based on residuals are compared, namely the empirical version of Kendall’s tau and the maximum pseudolikelihood (MPL) estimator (Genest et al., 1995). The power and ˆ n with dνn taken level of the goodness-of-fit test based on CM to be dˆ Cn are also examined. Note that while the consistency of the MPL estimator follows from Corollary 1 for copulas with bounded density (Neumeyer et al., 2019), the limiting behavior of this estimator in the more frequent case of unbounded copula densities is still unknown.

6

M.-P. Côté, C. Genest and M. Omelka / Insurance: Mathematics and Economics 89 (2019) 1–15

Fig. 1. Boxplots of τˆ for the two estimation methods in the trivariate micro-level application with n = 1000.

The comparisons are carried out in two common insurance settings. The first application is in the broad context of microlevel modeling of dependent risks. The second example illustrates the case of dependent loss triangles. Both simulation studies were conducted in the R Computing Environment (R Core Team, 2017) using packages copula (Hofert et al., 2015) and simsalapar (Hofert and Mächler, 2016). The latter and its accompanying paper were of great help in setting up the parallel computations and random number generation. 5.1. Micro-level modeling of dependent claim amounts There is interest in the actuarial community for models tailored to individual claimants, accidents, or policyholders. In particular, amounts paid under various insurance coverages for a given accident, claimant or insurance policy are typically dependent. As they are also partially explained by individual-specific covariates, their modeling usually requires the estimation of a dependence structure and covariate-dependent margins. Such models are developed, e.g., by Frees and Valdez (2008), Shi et al. (2016), and Côté et al. (2019). To explore the performance of the two estimation techniques, we considered a specific case inspired by Accident Benefits claim data from a Canadian insurance company (Côté et al., 2019). In this application, the responses are payments for medical benefits, income replacement benefits, and allocated expenses for a claimant involved in a car accident. As these payments are all related to the same event, they are dependent. Assume that this dependence is captured by a one-parameter copula. Covariates such as claimant’s age, gravity of accident (minor, major or catastrophic), presence of a legal counselor, and the number of different injuries were sampled from the data base. All covariates are indicator variables except X4 , the initial number of injuries, taking values in {1, . . . , 10}. To illustrate the flexibility of the method, heterogeneous margins were considered in the simulation study, viz. the log-normal, GB2, and exponential distributions. The parameter values are specified in Table E.1 in Appendix E. In the Online Supplement,

other results are shown for a similar example in the bivariate setup with Gamma margins and various shape parameters, and for log-normal and GB2 margins with scale varying with covariates. The copulas considered are exchangeable: Gumbel, Clayton, Frank, Gaussian and Student t with 4 degrees of freedom. Three levels of dependence are investigated as measured by Kendall’s tau, viz. τ ∈ {0.25, 0.50, 0.75}, for sample size n ∈ {200, 1000}. Note that these are small sample sizes in this context, as insurers usually have thousands of records available. For each of the 10,000 repetitions, the covariates X are sampled, the responses Y are generated, and the marginal parameters are estimated, thereby leading to residuals that are then used in estimating the copula parameter and for goodness-of-fit tests. Boxplots of the copula parameter estimates based on the trivariate samples are displayed in Fig. 1, for the five copula families, the three levels of dependence and n = 1000. Based on these plots, the performances of the two estimators seem satisfactory and comparable, except for the Clayton copula, where the empirical Kendall tau (iTau) has a smaller bias than the maximum pseudo-likelihood (MPL) method. Table 2 shows the ratio of the estimated mean squared errors for the two estimators in the different scenarios and for either d = 2 or 3 variables, i.e., (Y1 , Y2 ) and (Y1 , Y2 , Y3 ), respectively. Values smaller than 1 indicate that the Kendall’s tau is preferable to MPL. It is not possible to conclude that one estimator is always better than the other on the basis of the estimated mean squared error. Additional tables breaking up the mean squared error into bias and variance are presented in the Online Supplement for other sample sizes. The conclusions remain the same. Fig. 2 describes the level and power of the goodness-of-fit ˆ n involving Kendall’s tau (light) or MPL (dark). test based on CM Displayed is the rejection rate of H0 at the 5% nominal level when n = 200 pairs (Y1 , Y2 ) are observed. The test is a useful tool for model validation, as already well documented in simpler contexts (Genest et al., 2009). When dependence is moderate or strong and MPL estimation is used, the test does not hold its level for the Clayton copula, but this is not a problem with Kendall’s tau. When testing

M.-P. Côté, C. Genest and M. Omelka / Insurance: Mathematics and Economics 89 (2019) 1–15

7

Fig. 2. Percentage of rejection of the null hypothesis at the 5% level in the bivariate micro-level application when n = 200. Table 2 Relative efficiency of the MPL estimator with respect to the Kendall’s tau estimator in the micro-level application. d

2

3

n

200

C |τ

0.25

0.50

0.75

1000 0.25

0.50

0.75

Gumbel Clayton Frank Gaussian t4 Gumbel Clayton Frank Gaussian t4

1.068 1.343 1.006 1.033 0.986 1.048 1.208 1.088 0.996 0.974

1.107 1.200 1.029 1.124 1.089 1.125 1.053 1.100 1.131 1.081

1.078 0.699 0.935 1.198 1.158 0.930 0.420 0.773 1.176 1.076

1.048 1.237 1.005 1.079 1.010 1.017 1.201 1.092 1.066 0.991

1.063 1.197 1.012 1.126 1.077 1.078 1.116 1.092 1.140 1.072

1.110 0.967 0.982 1.184 1.158 1.068 0.696 0.959 1.201 1.178

relating to AP i. The loss reserving exercise amounts to predicting the future loss ratios, which can be done by fitting a GLM with the two main effects. The simulation study is performed with a model similar to, but of smaller dimension than, the one in Côté et al. (2016). ( ℓ) For business line ℓ ∈ {1, 2}, incremental loss ratios Yij have a Gamma distribution Fijℓ with shape parameter α (ℓ) and mean

µ(ijℓ) = exp(ζ (ℓ) + κi(ℓ) + λ(j ℓ) ), (ℓ)

( ℓ)

where the term κi represents the effect of AP i for line ℓ and λj is the DP effect, with the identifiability constraint

κ1(ℓ) = λ(1ℓ) = 0. (ℓ)

whether the copula family is Clayton, Kendall’s tau actually performs better both in terms of level and power, especially when the dependence is mild. However, Fig. 2 suggests that when testing whether the copula family is Gumbel, Frank or t4 , MPL estimation holds its level quite well and yields higher power than when Kendall’s tau is used. There is no overall best choice when testing for the Gaussian copula. 5.2. Insurance loss triangle data Consider dependent development triangles for property and casualty insurance claims. As in Shi and Frees (2011) or Côté et al. (2016), there are typically two non-random categorical covariates: accident period (AP) and development period (DP). If there are Na APs and at most Na DPs in the data, there are then Na (Na + 1)/2 observations, corresponding to APs and DPs i ∈ {1, . . . , Na } and j ∈ {1, . . . , Na + 1 − i}, respectively. The response is the incremental loss ratio whose numerator is the amount pertaining to, and paid in the jth period following, claims incurred in period i; the denominator is the earned premiums

( ℓ)

(ℓ)

In this setup, the errors εij = Yij /µij satisfy the locationscale assumption. The parameters of the generating mechanism, given in Table E.2 in Appendix E, are realistic as they are the estimated parameters for two lines of business and Na = 10 years in the data application of Côté et al. (2016). As in Shi and Frees (2011), it is assumed that loss ratios in corresponding cells of the two loss triangles are linked by a bivariate copula. More specifically, for i ∈ {1, . . . , Na } and j ∈ {1, . . . , Na + 1 − i}, the joint distribution of the loss ratios (1) (2) (1) (2) (Yij , Yij ) can be written as C {Fij1 (yij ), Fij2 (yij )}. The following simulation algorithm is repeated 100,000 times, for each combination of Plackett, Gumbel, Clayton, Gaussian and Frank copulas and three levels of dependence, viz. τ ∈ {0.25, ( ℓ) ( ℓ) 0.50, 0.75}. Note that as εˆ Na ,1 = εˆ 1,Na = 1 by design, they are not included in the estimation procedure. Simulation algorithm 1. Generate observations from the loss triangles using C and Fijℓ for all i ∈ {1, . . . , Na − 1}, j ∈ {1, . . . , Na − i} and ℓ ∈ {1, 2}. ( ℓ) 2. Estimate the Gamma GLM parameters µij and form the (ℓ)

residuals εˆ ij .

8

M.-P. Côté, C. Genest and M. Omelka / Insurance: Mathematics and Economics 89 (2019) 1–15

Fig. 3. Boxplots of τˆ for the two estimation methods in the loss triangle application. Table 3 Relative efficiency of the MPL estimator with respect to the Kendall’s tau estimator in the loss triangle application. C |τ

0.25

0.50

0.75

Plackett Gumbel Clayton Frank Gaussian

1.079 1.056 1.511 1.007 0.955

1.121 1.108 1.030 1.065 1.146

0.804 0.946 0.377 0.888 1.256

( ℓ)

3. Estimate the copula parameter using εˆ ij and Kendall’s tau or MPL. Fig. 3 shows the boxplots of the parameter estimates using Kendall’s tau (left) and MPL (right), for the five copula families and three levels of dependence. No estimator stands out as the preferred one in all cases. When the dependence is mild, their performance is close, as can also be seen in terms of the relative efficiencies in Table 3. When τ = 0.5 or 0.75, the bias of the Kendall’s tau estimator is systematically smaller than the one of the MPL estimator. This is especially striking in the case of the Plackett and Clayton copulas. When the dependence is strong, the Kendall’s tau estimator has a smaller MSE than the MPL estimator in all cases except for the Gaussian copula. This pattern is reversed when the dependence is mild. Table 4 breaks the MSE into bias and standard deviation. 6. Discussion Rank-based methods provide a natural set of tools for selection, inference, and validation of the dependence structure in multivariate models. This paper focused on their adaptation to the wide and flexible class of copula regression models with continuous location-scale error distributions. Classical momentbased copula parameter estimators were shown to be consistent and asymptotically normal under broader conditions than those assumed by Gijbels et al. (2015) and Neumeyer et al. (2019). The key idea is that the weak convergence of the empirical copula process studied by these authors is neither a necessary nor a sufficient condition for the consistency of all rank-based estimators and tests. One can, instead, rely on the convergence,

established in Proposition 1, of a weighted version of the empirical copula process based on suitably defined residuals from the marginal models. The conditions under which this result holds encompass a broad class of copula regression models, as documented in Table 1, including cases of direct interest for actuarial applications that were not covered by previous studies. Two major consequences of this result were detailed in Section 4. Under the conditions stated in Proposition 2, it follows that the inversion of Spearman’s rho or Kendall’s tau leads to consistent estimates of the copula parameters even if the empirical copula process fails to converge weakly. Similarly, the conditions of Proposition 3 imply that goodness-of-fit tests based on the Cramér–von-Mises statistic are not only consistent under Assumption IV (a) but also when max(η1 , . . . , ηd ) < 1/2 holds in Assumption IV (b). However, the approach used here did not allow us to extend beyond the case of Assumption IV (a) the results of Neumeyer et al. (2019) on the consistency of the Kolmogorov–Smirnov goodness-of-fit test or asymptotic normality of the maximum pseudo-likelihood method. Regarding estimation, their result is sufficient to conclude in the case of bounded copula densities, but the simulations described here suggest that this method also performs well in many cases of unbounded copula density (though possibly not in the Clayton case). Broad conditions under which this method is valid for copula regression models with continuous margins remain to be found. As these models are also frequently used in insurance and genetic applications where the margins are discontinuous, rank-based inference techniques for arbitrary response variables could also be developed along the lines of Genest et al. (2017). Acknowledgments The authors sincerely thank three anonymous referees and Prof. Johanna G. Nešlehová (McGill University) whose constructive comments and suggestions helped improve earlier drafts of this paper. The first author gratefully acknowledges support through fellowships from the Natural Sciences and Engineering Research Council of Canada (NSERC) and the James C. Hickman Scholarship Program of the Society of Actuaries. This work was also financed through research grants to the second author by the

M.-P. Côté, C. Genest and M. Omelka / Insurance: Mathematics and Economics 89 (2019) 1–15

9

Table 4 Estimated bias, standard deviation and root mean squared error of the estimators τˆ based on Kendall’s tau (iTau) or MPL in the loss triangle application.

√

τ

C | estimator

0.25

Bias

Plackett Gumbel Clayton Frank Gaussian Plackett Gumbel Clayton Frank Gaussian Plackett Gumbel Clayton Frank Gaussian

0.50

0.75

SD

MSE

iTau

MPL

iTau

MPL

iTau

MPL

−0.0112 0.0013 0.0020 −0.0105 −0.0002 −0.0217 −0.0044 −0.0098 −0.0206 −0.0007 −0.0257 −0.0068 −0.0296 −0.0300 −0.0008

−0.0122 0.0088 0.0059 −0.0065 0.0208 −0.0350 −0.0054 −0.0432 −0.0207 0.0159 −0.0520 −0.0198 −0.0867 −0.0387 −0.0026

0.1067 0.1074 0.1077 0.1050 0.1047 0.0906 0.0877 0.0882 0.0831 0.0824 0.0627 0.0515 0.0571 0.0495 0.0469

0.1025 0.1055 0.0875 0.1049 0.1050 0.0808 0.0832 0.0758 0.0803 0.0752 0.0549 0.0496 0.0586 0.0476 0.0416

0.1073 0.1074 0.1077 0.1056 0.1047 0.0932 0.0878 0.0887 0.0857 0.0824 0.0678 0.0520 0.0643 0.0578 0.0470

0.1033 0.1059 0.0877 0.1051 0.1071 0.0880 0.0834 0.0872 0.0829 0.0769 0.0756 0.0534 0.1046 0.0614 0.0417

Canada Research Chairs Program, NSERC, the Canadian Statistical Sciences Institute, and the Fonds de recherche du Québec — Nature et technologies, and to the third author via the Grant Agency of the Czech Republic (Grant GACR 19-00015S, Grantová Agentura České Republiky).

where the term OP (1) no longer depends on x and, for all u ∈ Ijn ,

Appendix A. Proof of Proposition 1

A.1. Decomposition of the process w An

For simplicity, the result will be established for

∑ q (u) = 1 C 1{ˆ εi1 ≤ ˆ F1−ˆε1 (u1 ), . . . , ˆ εid ≤ ˆ Fd−ˆε1 (ud )}, n n

√

q −˜ n (C Cn ) can be written, n

n 1 ∑[ −1 An (u) = √ 1[εi1 ≤ e1 {ˆ F1x (u1 ), Xi }, . . . , n

i=1

and

i=1

n 1∑

n

−1 εid ≤ ed {ˆ Fdx (ud ), Xi }]

Fd−ε1 (ud )}, 1{εi1 ≤ ˆ F1−ε1 (u1 ), . . . , εid ≤ ˆ

] − 1{εi1 ≤ ˆ F1−ε1 (u1 ), . . . , εid ≤ ˆ Fd−ε1 (ud )} .

i=1

which are equivalent to ˆ Cn and Cn , respectively, up to an OP (1/n) term. As the transformations ψ1 , . . . , ψd are known, one can assume without loss of generality that they are identity functions. Let Sε be the support of ε = (ε1 , . . . , εd )⊤ . For u ∈ (0, 1), x ∈ SX and j ∈ {1, . . . , d}, let −1 ˆ Fjx−1 (u) = µj (x;ˆ ξ j ) + σj (x;ˆ ξj ) ˆ Fjˆ ε (u)

and, for all y ∈ R, Fjx (y) = Fjε {ej (y, x)},

ej (y, x) =

y − µj (x; ξ j )

σj (x; ξ j )

,

⇔

εij ≤ ej {Fjx (u), Xi }.

ˆ−1

where the expectation is taken over the joint distribution of X 1 and ε while keeping ˆ Fjx−1 (uj ) and ˆ Fj− ε (uj ) fixed, i.e., nonrandom, for all j ∈ {1, . . . , d}. This notation is used for simplicity; see van der Vaart and Wellner (2007) for a justification.

¯n A.2. Treatment of w A

(A.2)

For each j ∈ {1, . . . , d}, let U(ξ j ) be a neighborhood of the true value of ξ j for which the conditions in Assumption V hold. Consider the set of functions defined on SX × Sε by

so that, for all i ∈ {1, . . . , n} and u ∈ [0, 1], −1 ˆ εij ≤ ˆ Fjˆ ε (u)

Proceeding as in Appendix B of Gijbels et al. (2015), write

w An = w A¯ n + w En , where A¯ n = An − En and √ [ −1 −1 F1x (u1 ), X}, . . . , εd ≤ ed {ˆ Fdx (ud ), X}] En (u) = n E 1[ε1 ≤ e1 {ˆ ] −1 −1 ˆ ˆ − 1{ε1 ≤ F1ε (u1 ), . . . , εd ≤ Fdε (ud )} , (A.6)

(A.1)

where

F = {(x, ϵ) ↦ → 1[ϵ1 ≤ e1 {z1 σ1 (x; t1 ) + µ1 (x; t1 ), x}, . . . ,

(A.3)

The outline of the proof is√as follows. In Appendix A.1, the q −˜ weighted empirical process w n (C Cn ) = w An is rewritten as n

w An = w A¯ n + w En , ¯ n = An − En and En is a suitable expectation defined where A ¯ n is asymptotically in (A.6). It is shown in Appendix A.2 that w A negligible uniformly in u ∈ In ; the same is done for w En in Appendix A.3. These two proofs depend on the fact, established in Appendix A.4, that whenever the conditions of Proposition 1 are satisfied for a given j ∈ {1, . . . , d}, one has

ϵd ≤ ed {zd σd (x; td ) + µd (x; td ), x}] : z1 , . . . , zd ∈ R, t1 ∈ U(ξ 1 ), . . . , td ∈ U(ξ d )}. For each f ∈ F , further let n 1 ∑ Zn (f ) = √ f (Xi , εi ), n i=1

where for each i ∈ {1, . . . , n}, εi = (εi1 , . . . , εid ). By Assump(u) (u) tion III, one can then write An (u) = Zn (fˆn ) − Zn (fn ), where fˆn(u) (x, ϵ) = 1 ϵ1 ≤ e1 {ˆ F1−ˆε1 (u1 ) σ1 (x;ˆ ξ 1 ) + µ1 (x;ˆ ξ 1 ), x}, . . . ,

[

] ϵd ≤ ed {ˆ Fd−ˆε1 (ud ) σd (x;ˆ ξ d ) + µd (x;ˆ ξ d ), x} ,

√

sup sup |w ¯ j (u) n Yjx (u)| = OP (1), u∈Ijn x∈SX

(A.5)

and w ¯ j (u) = u(η¯ j +ηj )/2 (1 − u)(η¯ j +ηj )/2 with ηj as in Assumption IV and η¯ j > ηj .

In view of (A.3), the process An = for all u ∈ [0, 1]d , as

n

˜ Cn (u) =

1 Yjx (u) = Fjx {ˆ Fjx−1 (u)} − Fjε {ˆ Fj− ε (u)}

(A.4)

(u)

and fn (x, ϵ) = 1{ϵ1 ≤ ˆ F1−ε1 (u1 ), . . . , ϵd ≤ ˆ Fd−ε1 (ud )}.

10

M.-P. Côté, C. Genest and M. Omelka / Insurance: Mathematics and Economics 89 (2019) 1–15

Proceeding as in Appendix C of Gijbels et al. (2015), one can show that the class F is P-Donsker with respect to the measure P corresponding to the joint distribution of X and ε. Using this fact, one can then mimic the arguments in Appendix B of Gijbels ¯ n , and hence w A¯ n , is asymptotically et al. (2015) to show that A negligible provided that, for all j ∈ {1, . . . , d}, sup sup |Yjx (u)| = oP (1),

(A.7)

u∈Ijn x∈SX

whose validity Appendix A.4.

follows

from

(A.4),

which

is

proved

in

A.4. Proof of Eq. (A.4)

√

The proof relies on a decomposition of the process n Yjx as a sum of a uniform quantile process in the classical sense (known to be uniformly bounded in probability) and an analog involving covariates, which is also bounded in probability when properly weighted. To be specific, let Gjn be the empirical distribution function of the (unobservable) uniform random sample Fjε (ε1j√ ), . . . , Fjε (εnj ). 1 For each j ∈ {1, . . . , d} and u ∈ (0, 1), let Qjn (u) = n {G− jn (u) − u}, so that

√

The process En defined in (A.6) involves an expectation over X and ε. Upon integration over the distribution of the latter, one finds, for all u ∈ [0, 1]d ,

√

En (u) =

−1 −1 n EX [C [F1X {ˆ F1X (u1 )}, . . . , FdX {ˆ FdX (ud )}]

Applying the mean value theorem, one deduces that, for all u ∈ [0, 1], d ∑

Fjx−1 (u)} − u. Given that from Theorem B(ii) where Yjx (u) = Fjx {ˆ in Csörgő et al. (1993), the uniform quantile process Qjn is uniformly bounded in probability, it suffices to show that

√

sup sup |w ¯ j (u) n Yjx (u)| = OP (1). It must thus be shown that for every δ ∈ (0, 1), there exist Kδ ∈ (0, ∞) and Nδ ∈ N such that, for all n ≥ Nδ , the event Kδ

{

Bn = ∀u∈Ijn ∀x∈SX : |Fjx {ˆ Fjx−1 (u)} − u| ≤

√

EX {C (j) (uX ) n YjX (uj )},

√

}

w ¯ j (u) n

has probability larger than 1 − δ . To this end, first introduce

j=1

where ux = (u1x , . . . , udx ) and, for each j ∈ {1, . . . , d}, YjX (u) is defined in (A.5) and ujx is comprised between Fjx {ˆ Fjx−1 (uj )} and 1 Fjε {ˆ Fj− ε (uj )}. To show that w En is asymptotically negligible uniformly in u ∈ In , it suffices to work summand by summand. If Assumption IV (a) holds for some j ∈ {1, . . . , d}, then wj ≡ 1 and one can mimic the proof of Theorem 4 in Gijbels et al. (2015). If instead Assumption IV (b) holds for that j, let pj (u) = w (u)/w ¯ j (uj ) for all u ∈ In , and write

√

w(u) EX {C (uX ) n YjX (uj )} = √ pj (u)C (j) (u) EX {w ¯ j (uj ) n YjX (uj )} √ + pj (u)EX [{C (j) (uX ) − C (j) (u)} w ¯ j (uj ) n YjX (uj )]. (j)

n Yjx (u) − Qjn (u),

u∈Ijn x∈SX

− C [F1ε {ˆ F1−ε1 (u1 )}, . . . , Fdε {ˆ Fd−ε1 (ud )}]].

En (u) =

√ n Yjx (u) =

A.3. Treatment of w En

In view of Assumptions II, III and Eq. (A.4), the second term is oP (1) uniformly in u ∈ In . To show that the same holds for the first term, it suffices to check that, uniformly in uj ∈ Ijn ,

√

EX {w ¯ j (uj ) n YjX (uj )} = oP (1)

⏐ ⏐ µ (x;ˆ ⏐ j ξ j ) − µj (x; ξ j ) ⏐ ⏐ and σj (x; ξ j ) x∈SX ⏐ σ (x;ˆ ⏐ ⏐ j ξ j ) − σj (x; ξ j ) ⏐ Sjn = sup ⏐ ⏐, σj (x; ξ j ) x∈SX Mjn = sup ⏐

and note that Assumptions III and V guarantee that both

√

√

Mjn = OP (1/ n)

and

Sjn = OP (1/ n).

(A.9)

Thus, one can assume that n is large enough to ensure that {Sjn < 1} occurs with probability at least 1 − δ . Conditionally on this event, one then has, for all u ∈ Ijn and x ∈ SX , −1 ˆ Fjˆ ε (u)

σj (x;ˆ ξ j ) µj (x;ˆ ξ j ) − µj (x; ξ j ) + σj (x; ξ j ) σj (x; ξ j ) ≤ˆ F −1 (u)[1 + Sjn sign{ˆ F −1 (u)}] + Mjn jˆ ε

jˆ ε

and hence, considering the definition of Fjx given in Eq. (A.1),

[

because |C (j) (u)| ≤ 1 and pj (u) is bounded and vanishes at uj = 0 or 1. To this end, first note that

√ ¯ j (uj ) En (u(j) ) w ¯ j (uj )EX { n YjX (uj )} = w

]

−1 ˆ−ε 1 (u)}] + Mjn . sup Fjx {ˆ Fjx−1 (u)} ≤ Fjε ˆ Fjˆ ε (u)[1 + Sjn sign{Fjˆ

x∈SX

Now for arbitrary u ∈ Ijn and K ∈ (0, ∞), consider the random variable

√ [ F −1 [u + K /{w ¯ j (u) n}] − Mjn ]

Ujn (u, K ) = ˆ Fjˆε

=w ¯ j (uj ) An (u(j) ) − w ¯ j (uj ) A¯ n (u(j) ),

jε

−1 1 + Sjn sign{ˆ Fjˆ ε (u)}

where u(j) denotes the vector whose jth entry is uj and whose d − 1 other components all equal 1. With probability 1, one has, for all u ∈ [0, 1]d ,

1 −1 with the convention that Fj− ε (y) = Fjε {max(0, y ∧ 1)} for any y ∈ R. Now observe that, for all x ∈ SX ,

n 1 ∑ −1 An (u(j) ) = √ 1{ˆ εij ≤ ˆ Fjˆ ε (uj )} n

u ≤ Ujn (u, K )

i=1

n 1 ∑

−√

n

⇒

Fjx {ˆ Fjx−1 (u)} − u ≤

K

√ . w ¯ j (u) n

Completely analogously, one has 1 1{εij ≤ ˆ Fj− ε (uj )} = 0.

(A.8)

i=1

Therefore, w ¯ j (uj ) An (u(j) ) = oP (1) and hence the conclusion holds provided that w ¯ j (uj )A¯ n (u(j) ) is oP (1). This follows from the same ¯ n (u(j) ) as a argument as in Appendix A.2 because one can view A ¯ n involving only uj ∈ Ijn . one-dimensional analog of the process A

[

]

−1 ˆ−ε 1 (u)}] − Mjn . Fjx−1 (u)} ≥ Fjε ˆ Fjˆ inf Fjx {ˆ ε (u)[1 − Sjn sign{Fjˆ

x∈SX

For arbitrary u ∈ Ijn and K ∈ (0, ∞), consider the random variable

√ [ F −1 [u − K /{w ¯ j (u) n}] + Mjn ] jε . −1 1 − Sjn sign{ˆ Fjˆ ε (u)}

Ljn (u, K ) = ˆ Fjˆε

M.-P. Côté, C. Genest and M. Omelka / Insurance: Mathematics and Economics 89 (2019) 1–15

Then, for all x ∈ SX , u ≥ Ljn (u, K )

and Fjx {ˆ Fjx−1 (u)} − u ≥ −

⇒

K

√

O∗n2 = Mjn + sup On2 (u)[1 + Sjn sign{zjn (u, K )}] = OP (1/ n).

√ .

w ¯ j (u) n

u∈Ijn

∗

Therefore, letting P denote outer probability, one has

Thus one has, for all u ∈ Ijn ,

P (Bn ) ≥ P (Bn ∩ {Sjn < 1})

] {Ljn (u, K ) ≤ u ≤ Ujn (u, K )} ∩ {Sjn < 1}

{Ljn (u, K ) ≤ u} ⊇ {L¯ jn (u, K ) + On ≤ u} { = En1 (u, K ) ≤ −On +

] {Ljn (u, K ) ≤ u} ∩ {Sjn < 1}

¯¯ j (u) = uηj (1 − u)ηj and where w

∗

∗

∗

[⋂

≥ P∗

[⋂

≥P

u∈Ijn

u∈Ijn

+P

∗

[⋂

11

En1 (u, K ) = L¯ jn (u, K ) − u +

] {u ≤ Ujn (u, K )} ∩ {Sjn < 1} − 1,

K

K

} √

¯¯ j (u) n 2w

,

√ .

¯¯ j (u) n 2w

√

u∈Ijn

and hence it suffices to show that given δ ∈ (0, 1), there exist Kδ ∈ (0, ∞) and Nδ ∈ N such that, for all n ≥ Nδ , each of the two outer probabilities on the right-hand side of the above chain is greater than 1 − δ/2. In what follows, the first term is considered; the second term can be treated in a similar way. First observe that proceeding as −1 Fjˆ in Appendix A.2 with ˆ ε (u) replaced by y, one has, uniformly in y ∈ R,

√

n [ˆ Fjˆε (y) − ˆ Fjε (y) − E{ˆ Fjˆε (y) − ˆ Fjε (y)}] = oP (1),

(A.10)

provided that the following simpler version of (A.7) holds, i.e., sup sup |Fjε [ej {yσj (x;ˆ ξ j ) + µj (x;ˆ ξ j ), x}] − Fjε (y)| = oP (1),

Then, conditionally on the event {−On + K /(2 n) ≥ 0}∩{Sjn < 1}, one has P∗

[⋂

] { √ } {Ljn (u, K ) ≤ u} ≥ P ∗ sup En1 (u, K ) ≤ −On + K /(2 n) u∈Ijn

u∈Ijn

{

} ≥ P ∗ sup En1 (u, K ) ≤ 0 . u∈Ijn

√

Now note that supu∈Ijn |On1 (u)| = OP (1/ n). Thus by Lemma D1 in Appendix D, the last probability can be made arbitrarily large for all sufficiently large n by taking K large enough. √ Finally, the same holds also for Pr[{Sjn < 1} ∩ {On ≤ K /(2 n)}], which completes the proof of Proposition 1. □ ∗

y∈R x∈SX

where ej is defined in Eq. (A.2). However, this is an immediate consequence of the continuity of Fjε and (A.9). Next, setting On (y) = ˆ Fjε (y) − E{ˆ Fjε (y)}, one can deduce from (A.10) that, for all y ∈ R,

Appendix B. Proof of Proposition 2

ˆ Fjˆε (y) = E{ˆ Fjˆε (y)} + On (y) [ { σj (X;ˆ ξ j ) − µj (X; ξ j ) }] ξ j ) µj (X;ˆ + + On (y) = EX Fjε y σj (X; ξ j ) σj (X; ξ j )

M

J ⊆{1,...,d}

(A.11) write

{

1 Ljn (u, K ) = ˆ Fjˆε Fj− u− ε

K

√ w ¯ j (u) n

}

−1 Sjn sign{ˆ Fjˆ ε (u)} −1 1 − Sjn sign{ˆ Fjˆ ε (u)}

√

,

On2 (u) =

Mjn −1 1 − Sjn sign{ˆ Fjˆ ε (u)}

.

Letting K

{

1 y = zjn (u, K ) = Fj− u− ε

√

}

w ¯ j (u) n

{1 + On1 (u)} + On2 (u),

in approximation (A.11), and upon taking the supremum over x ∈ SX , one finds Ljn (u, K ) ≤ L¯ jn (u, K ) + On ,

√

where On = supu∈Ijn On {zjn (u, K )} = OP (1/ n) by Theorem A of Csörgő et al. (1993) and

[

{

1 L¯ jn (u, K ) = Fjε Fj− u− ε

K

√

w ¯ j (u) n

}

{1 + O∗n1 (u)} + O∗n2

]

with ∗

[

On1 (u) = sign

1 Fj− ε

{

u−

K

√

∫ [0,1]d

] {1 + On1 (u)} + On2 (u) ,

}]

√ w ¯ j (u) n × {1 + On1 (u)} × [1 + Sjn sign{zjn (u, K )}] − 1

[0,1]d

As the right-hand side is a finite sum of terms, it suffices to show that n

where On1 (u) and On2 (u) are OP (1/ n) terms given by On1 (u) =

1 √

| n {T (ˆ Cn ) − T (Cn )}| ∑ √ ∫ ≤ n |ˆ Cn (u(J) ) − Cn (u(J) )| d{ˆ Cn (u) + Cn (u) + u}.

√ up to a term of order oP (1/ n) whatever the value of y. Further

[

Given that Cn and ˆ Cn are in D, it follows from (5) that

|ˆ Cn (u(J) ) − Cn (u(J) )| dνn (u) = oP (1)

(B.1)

for each subset J ⊂ {1, . . . , d} and νn either equal to ˆ Cn , Cn or Lebesgue’s measure. In what follows, we consider only the first two cases, and leave the third one to the reader, as it is analogous and in fact more straightforward. To show (B.1) for a specific measure νn , we proceed by induction on the cardinality of the set of indices J. The result clearly holds when |J | = 1, i.e., assume that a single element of u(J) , say uj , is different from 1. Proceeding as in (A.8), one then finds that ˆ Cn (u(J) ) − Cn (u(J) ) = 0 with probability 1, and hence (B.1) holds. Now suppose that (B.1) holds for each J such that |J | = d − 1. We will show that it then holds also for |J | = d; the remaining cases when 1 < |J | < d − 1 can be handled analogously. To this end, write

√

∫ n [0,1]d

√

=

|ˆ Cn (u) − Cn (u)| dνn (u)

∫

|ˆ Cn (u) − Cn (u)| dνn (u) ∫ √ + n |ˆ Cn (u) − Cn (u)| dνn (u), n

In

(B.2)

[0,1]d \In

where In = I1n × · · · × Idn Proposition 1.

⊂ [0, 1]d was introduced in

12

M.-P. Côté, C. Genest and M. Omelka / Insurance: Mathematics and Economics 89 (2019) 1–15

It will be shown that the two terms on the right-hand side of (B.2) are oP (1). To deal with the first summand, note that, by Proposition 1,

√

∫

|ˆ Cn (u) − Cn (u)| dνn (u)

n

and analogously for Cn (u). One then has

|ˆ Cn (u) − Cn (u)| ≤ |ˆ Cn (1, u2 , . . . , ud ) − Cn (1, u2 , . . . , ud )| n n ⏐ ⏐1 ∑ ⏐ ⏐1 ∑ ⏐ ⏐ ⏐ ⏐ 1{ˆ F1ˆε (ˆ εi1 ) > u1 }⏐ + ⏐ 1{ˆ F1ε (εi1 ) > u1 }⏐ +⏐ n

In

1 √

∫ =

w (u) ∫

In

= oP (1) In

w(u)

n

Thus one gets

dνn (u).

√

1

w(u)

In

dνn (u) ≤

⌈n∆jn ⌉ d ∑ 1 ∑ j=1

n

1

i=⌊n∆jn ⌋

(i/n)η¯ j (1 − i/n)η¯ j

[0,1]d \In

≤

∫ d 1 ∑ ∑ √ n

(k)

(1) Ljn

and

(k)

= oP (1) + o(1) = oP (1),

|ˆ Cn (u) − Cn (u)| dνn (u) = oP (1).

(B.3)

For k = 0, one has uj < ∆jn and so one can simply bound, for

Appendix C. Proof of Proposition 3

√

sup |ˆ Cn (u) − C (u; ˆ θ n )| = sup |Cn (u) − C (u;˜ θ n )| + oP (1/ n), u∈[0,1]d

ˆ n − KSn = op (1). which implies that as n → ∞, KS Turning to the Cramér–von Mises statistic, one can write ˆn = CM

n

1{ˆ Fjˆε (ˆ εij ) ≤ uj } +

i=1

n 1∑

n

1{ˆ Fjε (εij ) ≤ uj }

∫ Ajkn =

n

(0)

|ˆ Cn (u) − Cn (u)| dνn (u)

√ ≤

∆jn

n

∫ (2u + 2/n)dνn(j) (u) =

|A11n | =

√

n (∆jn )2 + o(1)

[0,1]d \In

0

=n

1/2

n

−1/(1+ηj )

∫ + o(1) = o(1),

(j)

ˆ Cn (u) = ˆ Cn (1, u2 , . . . , ud ) n 1∑ − 1{ˆ F1ˆε (ˆ εi1 ) > u1 , ˆ F2ˆε (ˆ εi2 ) ≤ u2 , . . . , ˆ Fdˆε (ˆ εid ) ≤ ud } i=1

+

(B.4)

where νn stands for the measure induced by the function ˆ Cn (u(j) ) (j) or Cn (u ). Now consider k = 1 and suppose, to simplify notation, that j = 1. Write

n

ajn (u)akn (u) dνn (u).

Term A11n . First write

Ljn

∫

[0,1]d

It will be shown below that A22n = CMn + oP (1) and that the remaining terms are asymptotically negligible.

This further implies that n

Ajkn ,

where

i=1

≤ 2 uj + . √

3 3 ∑ ∑ j=1 k=1

2

∫

√

a3n (u) = oP (1/ n).

Thus, one has

(0)

u in Ljn ,

|ˆ Cn (u) − Cn (u)| ≤

(B.5)

where the induction assumption was used in the last equality. A similar argument can be used for any other j ∈ {1, . . . , d}. The induction is completed by combining (B.4) and (B.5) to get (B.3). □

u∈[0,1]d

Ljn

n 1∑

{2(1 − u) + 2/n} dνn(1) (u)

√

Thus, for a given j ∈ {1, . . . , d}, it is sufficient to show that, for k ∈ {0, 1}, n

∆1n

a1n (u) = oP (1/ n),

= {u ∈ [0, 1] : uj > ∆jn }.

√

1

n

First assume that Assumption IV (a) holds and consider the Kolmogorov–Smirnov statistic. Combined with (6) and (8), Corollary 1 yields that, uniformly in u ∈ [0, 1]d ,

d

∫

∫

ˆ Cn (u) − C (u; ˆ θ n ) = a1n (u) + a2n (u) + a3n (u).

where (0)

|ˆ Cn (1, u2 , . . . , ud ) − Cn (1, u2 , . . . , ud )| dνn (u)

For all u ∈ [0, 1]d , let a1n (u) = ˆ Cn (u) − Cn (u), a2n (u) = Cn (u) − C (u; ˜ θ n ), and a3n (u) = C (u;˜ θ n ) − C (u;ˆ θ n ), so that

|ˆ Cn (u) − Cn (u)| dνn (u),

Ljn = {u ∈ [0, 1]d : uj < ∆jn }

(1)

+

Ljn

j=1 k=0

∫

n

√

|ˆ Cn (u) − Cn (u)| dνn (u)

n

√ ≤

,

is uniformly bounded in n, and hence the first term on the right-hand side of (B.2) is oP (1). To deal with the second summand, note that when u ∈ [0, 1]d \ In , there is at least one component of u (say uj ) such that either uj < ∆jn or uj > ∆jn . So, one has

∫

|ˆ Cn (u) − Cn (u)| dνn (u)

(1)

L1n

L1n

where for each j ∈ {1, . . . , d}, Ijn = [∆jn , ∆jn ]∫. As one can take η¯ j ∈ (0, 1) for all j ∈ {1, . . . , d}, it follows that In {w(u)}−1 dνn (u)

√

∫ n

Furthermore, given that each marginal distribution induced by the measure νn is uniform on the set {1/n, 2/n, . . . , 1}, one has

∫

i=1

2 ≤ |ˆ Cn (1, u2 , . . . , ud ) − Cn (1, u2 , . . . , ud )| + 2(1 − u1 ) + .

n w (u)|ˆ Cn (u) − Cn (u)| dνn (u) 1

n

i=1

{ˆ Cn (u) − Cn (u)}2 dνn (u)

{ˆ Cn (u) − Cn (u)}2 dνn (u).

(C.1)

In

Given that max(η1 , . . . , ηd ) < 1/2 by assumption, one can choose η¯ 1 > η1 , . . ., η¯ d > ηd such that max(η¯ 1 , . . . , η¯ d ) < 1/2. Thus by Proposition 1,

∫

{ˆ Cn (u) − Cn (u)}2 dνn (u) ≤ oP (1) In

∫

1 In

w

2 (u)

dνn (u) = oP (1),

where the last equality is obtained similarly as in the proof of Proposition 2.

M.-P. Côté, C. Genest and M. Omelka / Insurance: Mathematics and Economics 89 (2019) 1–15

To deal with the other summand on the right-hand side of (C.1), one can also proceed as in the proof of Proposition 2. The difference is that (B.4) becomes

∫ n

(0)

2 |ˆ Cn (u) − Cn (u)| dνn (u) ≤ n

∫

∆jn

4 (u + 1/n) dν 2

(j) n (u)

0

Ljn

= O(n) n−3/{2(1+ηj )} = O(n1−3/{2(1+ηj )} ) = o(1), (j)

where νn stands for the measure induced by the function ˆ Cn (u(j) ) or Cn (u(j) ). An argument similar to (B.5) can be used to handle

∫ n

(1) Ljn

Proof. To simplify notation, write

{

∫ A22n − CMn = n

[0,1]d

u∈Ijn

For brevity, we will only consider An = supu∈Ijn ∩(0,1/2] En1 (u, K ); similar arguments can be used for supu∈Ijn ∩[1/2,1) En1 (u, K ) and infu∈Ijn En2 (u, K ). ∗ For arbitrary but fixed δ > √ 0, let M > 0 and Nδ be√such that for all n ≥ Nδ∗ , Pr(On > M / n) < δ/2, and also M / n ≤ 1/2. Further let

{ unK = max u −

K

and anu = √ On ≤ M / n,

En1 (u, K ) + u −

√ ,

u

}

K

{ = u − min

w ¯ j (u) n 2

1 Fj− ε (unK ).

{Cn (u) − Cn (u;˜ θ n )}2 d{ˆ Cn (u) − Cn (u)}.

√

}

On = max sup |On1 (u)|, |On2 | = OP (1/ n).

2 |ˆ Cn (u) − Cn (u)| dνn (u).

Term A22n . If νn is the Lebesgue measure, then one gets exactly A22n = CMn . Thus, it remains to consider dνn = dˆ Cn . Then, one has

13

K

√ ,

sup |ˆ Cn (u) − Cn (u)| = oP (1)

}

w ¯ j (u) n 2

Note that for all n ≥ Nδ∗ , provided that

√ ≤ Fjε {anu + |anu On1 (u)| + On2 }

¯¯ j (u) n 2w

= unK + fjε (y∗ju ){|anu On1 (u)| + On2 } ≤ unK + fjε (y∗ju )(1 + |anu |){|On1 (u)| + |On2 |}, √ ≤ unK + fjε (y∗ju )(1 + |anu |)2M / n,

Now, Proposition 1 implies that

u

(D.1)

u∈[0,1]d

where the equality stems from the mean value theorem with

and so the asymptotic negligibility of the difference A22n − CMn can be proved analogously as Proposition 4 of Genest et al. (2013).

y∗ju = anu + λ|anu On1 (u)| + λOn2

Term A33n . With the help of (6) and (8), one has

∫ |A33n | =

[0,1]d

{C (u;˜ θ n ) − C (u;ˆ θ n )}2 dνn (u)

for some λ ∈ (0, 1) that depends on u. Using the definition of unK , we can then rewrite the inequality (D.1) as K

{

En1 (u, K ) ≤ − min

√ ,

u}

K

+ √ ¯¯ j (u) n w ¯ j (u) n 2 2w √ + fjε (y∗ju )(1 + |anu |)2M / n.

≤ O(1) n ∥˜ θn − ˆ θ n ∥2 = oP (1). Term Ajkn for j ̸ = k. Using the results √ for A11n , A22n and A33n and Hölder’s inequality, one gets |Ajkn |≤ Ajjn Akkn = oP (1), and the proof is complete. □

u = u1+ηj −ηj ≥ (∆jn )1+ηj u−ηj = n−(1+ηj )/{2(1+η¯ j )} u−ηj ,

Appendix D. Auxiliary result

where we have used the fact that (∆jn )1+ηj = n−(1+ηj )/{2(1+η¯ j )} . †

Thus, for n ≥ NK = {2K (1 − u)−ηj }2(1+η¯ j )/(η¯ j −ηj ) ,

Lemma D1. Suppose that Assumption IV (b) holds for some j ∈ ¯¯ j (u) = uηj (1 − u)ηj . {1, . . . , d} and ηj > 0. For u ∈ (0, 1), let w Let also On1 (u), On2 be arbitrary√sequences of random elements such √ that supu∈Ijn |On1 (u)| = OP (1/ n) and On2 = OP (1/ n). For any fixed K > 0, consider the variables En1 (u, K ) = Fjε

+

[

1 Fj− ε

K

{ u−

K

¯¯ j (u), and also By design w ¯ j (u) < w

}

] {1 + On1 (u)} + On2 − u

√

w ¯ j (u) n

− min

K

{

√ ,

u}

w ¯ j (u) n 2

+

K

K

√ ≤− √ . ¯¯ j (u) n ¯¯ j (u) n 2w 2w

To conclude, it is thus sufficient to show that, for all n sufficiently large, sup

u∈Ijn ∩(0,1/2]

fjε (y∗ju )(1 + |anu |) ≤ K /(2M),

(D.2)

as this would imply that

√

√

¯¯ j (u) n}. fjε (y∗ju )(1 + |anu |)2M / n ≤ K /{2w

√

¯¯ j (u) n 2w

Two cases must be distinguished depending on the support of Fjε .

and

[

{

1 En2 (u, K ) = Fjε Fj− u+ ε

−

K

K

} √

w ¯ j (u) n

] {1 + On1 (u)} + On2 − u

√ .

¯¯ j (u) n 2w

Then for every δ > 0, there exist Kδ ∈ (0, ∞) and Nδ ∈ N such that, for all n ≥ Nδ ,

{

}

P ∗ sup En1 (u, Kδ ) ≤ 0 > 1 − δ u∈Ijn

1 ∗ Case I: Fj− ε √(u) → −∞ as u ↓ 0. Assume n ≥ Nδ and On ≤ M / n. Then for any u ∈ Ijn and λ ∈ (0, 1), we have λ max{|On1 (u)|, |On2 |} ≤ 1/2 and hence

|y∗ju | ≥ |anu |{1 − λ|On1 (u)|} − λ|On2 | ≥ |anu |/2 − 1/2. Consequently, 2 |y∗ju | + 1 ≥ |anu | and it follows that fjε (y∗ju )(1 + |anu |) = fjε (y∗ju ) (1 + |y∗ju |)

1 + |anu | 1 + |y∗ju |

≤ 2 fjε (y∗ju ) (1 + |y∗ju |).

and

(D.3)

P ∗ inf En2 (u, Kδ ) ≥ 0 > 1 − δ,

Next pick an arbitrary fixed ϵ ∈ (0, 1/2). By Assumption IV (b), there exists Lϵ ∈ (0, ∞) such that, for all u ∈ (0, 1 − ϵ],

where P ∗ denotes outer probability.

1 −1 fjε {Fj− ε (u)}{1 + |Fjε (u)|} ≤ Lϵ .

{

}

u∈Ijn

(D.4)

14

M.-P. Côté, C. Genest and M. Omelka / Insurance: Mathematics and Economics 89 (2019) 1–15

Let also Nδ,ϵ ≥ Nδ∗ be sufficiently large so that for any n ≥ Nδ,ϵ and any λ ∈ (0, 1), one also has

√

√

Table E.1 Marginal specifications of the pseudo-data generating mechanism in the bivariate and trivariate micro-level simulation studies.

1 −1 −1 Fj− ε (1/2) + λ|Fjε (1/2)|M / n + λM / n < Fjε (1 − ϵ ).

Y1

Then, combining (D.3) and (D.4), one can conclude that sup

u∈Ijn ∩(0,1/2]

fjε (y∗ju )(1 + |anu |) ≤

sup

u∈Ijn ∩(0,1/2]

2 fjε (y∗ju ) (1 + |y∗ju |) ≤ 2Lϵ .

So (D.2) holds provided that K is chosen so that K /(2M) ≥ 2Lϵ . Thus the desired result holds when Kδ = 4MLϵ and Nδ = † max(Nδ,ϵ , NK ). 1 Case 2: supu∈(0,1/2] |Fj− ε (u)| ≤ c for some c ∈ (0, ∞). In this case, one has

fjε (y∗ju )(1 + |anu |) ≤ (1 + c) fjε {anu + λOn3 (u)},

(D.5) Nδ∗

where On3 (u) = |anu On1 (u)| + On2 . Now assume n ≥ and √ † On ≤ M / n. We will show that there exists Nδ > 0 such that, † for all n ≥ Nδ , 1 Fj− ε (u/4)

≤ anu + λOn3 (u) ≤

1 Fj− ε (3/4).

(D.6)

Indeed, first note that the upper bound holds as soon as †

1 −1 2 n ≥ Nδ,1 = M 2 (1 + c)2 /{Fj− ε (3/4) − Fjε (1/2)} .

To establish the lower bound, observe that the mean value theorem implies that −1 1 Fj− ε (u/2) − Fjε (u/4) =

1

u

−

1 Fj− ε (u/4)

≥

1 supv∈[u/4,u/2] fjε {Fj− ε (v ) }

(D.7) where the last line is justified by the fact that u ≤ 1/2 and ηj ∈ (0, 1), which implies that (1/4)1+ηj (1 − u/4)ηj ≥ 7/128. From the definition of ∆jn , the rightmost term in (D.7) is bounded above by 128 Lj

7n(ηj −η¯ j )/(2{1+η¯ j )} 128Lj

√

n

M(1 + c)

≥

√

n

,

Table E.2 Marginal parameters in the loss triangle simulation study, from Côté et al. (2016).

ℓ

1

2

ℓ

α (ℓ)

10.700

24.046

ζ (ℓ)

κ2(ℓ)

−0.750

0.053

λ(2ℓ)

κ

−0.729

−0.156

κ

−0.651

0.239

κ

−0.741

0.137

κ

−0.574

0.120

κ

−0.574

0.003

κ

−0.658

−0.160

κ

−1.157

0.169

(ℓ) 3 (ℓ) 4 (ℓ) 5 (ℓ) 6 (ℓ) 7 (ℓ) 8 (ℓ) 9

λ(3ℓ) λ(4ℓ) λ(5ℓ) λ(6ℓ) λ(7ℓ) λ(8ℓ)

λ(9ℓ)

1

−3.628

2

−3.501

2.061

0.815

2.061

0.817

2.018

0.849

1.818

0.717

1.297

0.283

0.773

−0.115

−0.493

−1.001

−0.429

−1.375

√

1 −1 Fj− ε (u/2) − Fjε (u/4) ≥ M(1 + c)/ n.

Combining this result with the fact that

√ √ |On3 (u)| ≤ (1 + |anu |)M / n ≤ M(1 + c)/ n †

for each u ∈ Ijn ∩ (0, 1/2], one can conclude that for any n ≥ Nδ = † † max(Nδ,1 , Nδ,2 ),

√

1 −1 anu + λOn3 (u) ≥ Fj− ε (u/2) − M(1 + c)/ n ≥ Fjε (u/4),

which establishes the desired lower bound in (D.6) and the exis† tence of Nδ . Finally, it follows from Assumption IV (b) that sup

v∈[u/4,3/4]

1 fjε {Fj− ε (v )} ≤

4Lj uηj (1 − u)ηj

.

Together with (D.5), this implies that, for all u ∈ Ijn ∩ (0, 1/2], fjε (y∗ju )(1 + |anu |) ≤ (1 + c)

4Lj uηj (1 − u)ηj

,

See Tables E.1 and E.2. Appendix F. Supplementary data Supplementary material related to this article can be found online at https://doi.org/10.1016/j.insmatheco.2019.08.001. References

where the last inequality holds for all sufficiently large n, say † n ≥ Nδ,2 . For any such n, one thus has

fjε {anu + λOn3 (u)} ≤

Y3 Exponential 7.750 0.370 1.086 −0.025 0.090 −0.221 −0.253 – – –

Appendix E. Parameters used in the simulation study

≥ 7u1+ηj /(128 Lj ) ≥ 7(∆jn )ηj +1 /(128 Lj ),

≥

p q

GB2 13.770 0.786 2.264 −0.031 0.114 −0.395 −0.122 0.345 7.434 64.311

which implies that (D.2) is satisfied provided that K /(2M) ≥ (1 + c)8Lj . Thus the desired result holds when Kδ = 16(1 + c)MLj † and Nδ = max(Nδ∗ , Nδ ). □

u/4

≥ (u/4)1+ηj (1 − u/4)ηj /Lj

7n−(ηj +1)/{2(1+η¯ j )}

σ

Y2

Log-Normal 7.267 0.947 3.433 0.062 0.185 −0.056 0.041 0.802 – –

1 4 fjε {Fj− ε (vu )}

for some vu ∈ [u/4, u/2]. Therefore, by Assumption IV (b), 1 Fj− ε (u/2)

Model β0 (Intercept) β1 (Major accident) β2 (Catastrophic accident) β3 (Presence of legal counselor) β4 (Initial number of injuries) β5 (Age < 22) β6 (Age > 55)

Abdallah, A., Boucher, J.-P., Cossette, H., 2015. Modeling dependence between loss triangles with hierarchical Archimedean copulas. ASTIN Bull. 45, 577–599. Barriga, G.D., Louzada-Neto, F., Ortega, E.M., Cancho, V.G., 2010. A bivariate regression model for matched paired survival data: Local influence and residual analysis. Stat. Methods Appl. 19, 477–495. Ben Alaya, M.A., Chebana, F., Ouarda, T., 2014. Probabilistic Gaussian copula regression model for multisite and multivariable downscaling. J. Clim. 27, 3331–3347. Chen, X., Fan, J., 2005. Estimation and model selection of semiparametric copula-based multivariate dynamic models under copula misspecification. J. Econometrics 135, 125–154. Côté, M.-P., Genest, C., Abdallah, A., 2016. Rank-based methods for modeling dependence between loss triangles. Eur. Actuar. J. 6, 377–408. Côté, M.-P., Genest, C., Stephens, D.A., 2019. A Bayesian approach to modeling multivariate micro-level insurance claims in the presence of unsettled claims (submitted for publication). Csörgő, M., Horváth, L., Shao, Q.-M., 1993. Convergence of integrals of uniform empirical and quantile processes. Stochastic Process. Appl. 45, 283–294. Czado, C., Kastenmeier, R., Brechmann, E.C., Min, A., 2012. A mixed copula model for insurance claims and claim sizes. Scand. Actuar. J. 2012 (4), 278–305. De Jong, P., 2012. Modeling dependence between loss triangles. N. Am. Actuar. J. 16, 74–86. Fachini, J.B., Ortega, E.M., Cordeiro, G.M., 2014. A bivariate regression model with cure fraction. J. Stat. Comput. Simul. 84, 1580–1595.

M.-P. Côté, C. Genest and M. Omelka / Insurance: Mathematics and Economics 89 (2019) 1–15 Frees, E.W., Meyers, G., Cummings, A.D., 2010. Dependent multi-peril ratemaking models. ASTIN Bull. 40, 699–726. Frees, E.W., Sun, Y., 2010. Household life insurance demand. N. Am. Actuar. J. 14, 338–354. Frees, E.W., Valdez, E.A., 2008. Hierarchical insurance claims modeling. J. Amer. Statist. Assoc. 103, 1457–1469. Frees, E.W., Wang, P., 2005. Credibility using copulas. N. Am. Actuar. J. 9, 31–48. Frees, E.W., Wang, P., 2006. Copula credibility for aggregate loss models. Insurance Math. Econom. 38, 360–373. Genest, C., Favre, A.-C., 2007. Everything you always wanted to know about copula modeling but were afraid to ask. J. Hydrol. Eng. 12, 347–368. Genest, C., Ghoudi, K., Rivest, L.-P., 1995. A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82, 543–552. Genest, C., Huang, W., Dufour, J.-M., 2013. A regularized goodness-of-fit test for copulas. J. SFdS 154, 64–77. Genest, C., Nešlehová, J., 2012. Copulas and copula models. In: El-Shaarawi, A.H., Piegorsch, W.W. (Eds.), Encyclopedia of Environmetrics, second ed. Wiley, Chichester, pp. 541–553. Genest, C., Nešlehová, J.G., Rémillard, B., 2017. Asymptotic behavior of the empirical multilinear copula process under broad conditions. J. Multivariate Anal. 159, 82–110. Genest, C., Rémillard, B., Beaudoin, D., 2009. Omnibus goodness-of-fit tests for copulas: A review and a power study. Insurance Math. Econom. 44, 199–213. Gijbels, I., Omelka, M., Veraverbeke, N., 2015. Estimation of a copula when a covariate affects only marginal distributions. Scand. J. Stat. 42, 1109–1126. He, W., Lawless, J.F., 2005. Bivariate location–scale models for regression analysis, with applications to lifetime data. J. R. Stat. Soc. Ser. B 67, 63–78. Hofert, M., Kojadinovic, I., Mächler, M., Yan, J., 2015. copula: Multivariate Dependence with Copulas. R package version 0.999-14.

15

Hofert, M., Mächler, M., 2016. Parallel and other simulations in R made easy: An end-to-end study. J. Stat. Softw. 69, 1–44. Li, M., Boehnke, M., Abecasis, G.R., Song, P.X.-K., 2006. Quantitative trait linkage analysis using Gaussian copulas. Genetics 173, 2317–2327. Masarotto, G., Varin, C., 2012. GaussIan copula marginal regression. Electron. J. Statist. 6, 1517–1549. Neumeyer, N., Omelka, M., Hudecová, Š., 2019. A copula approach for dependence modeling in multivariate nonparametric time series. J. Multivariate Anal. 171, 139–162. Oakes, D., Ritz, J., 2000. Regression in a bivariate copula model. Biometrika 87, 345–352. Portier, F., Segers, J., 2018. On the weak convergence of the empirical conditional copula under a simplifying assumption. J. Multivariate Anal. 166, 160–181. R Core Team, 2017. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. Rémillard, B., 2017. Goodness-of-fit tests for copulas of multivariate time series. Econometrics 5 (1), article number 13. Segers, J., 2012. Weak convergence of empirical copula processes under nonrestrictive smoothness assumptions. Bernoulli 18, 764–782. Shi, P., Feng, X., Boucher, J.-P., 2016. Multilevel modeling of insurance claims using copulas. Ann. Appl. Stat. 10, 834–863. Shi, P., Frees, E.W., 2011. Dependent loss reserving using copulas. ASTIN Bull. 41, 449–486. Song, P.X.-K., 2000. Multivariate dispersion models generated from Gaussian copula. Scand. J. Stat. 27, 305–320. van der Vaart, A.W., Wellner, J.A., 2007. Empirical processes indexed by estimated functions. In: Cator, E.A., Jongbloed, G., Kraaikamp, C., Lopuhaä, H.P., Wellner, J.A. (Eds.), IMS Lecture Notes Monograph Series 2007: Asymptotics: Particles, Processes and Inverse Problems. In: Institute of Mathematical Statistics, vol. 55, pp. 234–252.