P.-O Goffard

S Rao Jammalamadaka

S G Meintanis

Goodness-of-fit procedures for compound distributions with an application to Insurance

Keywords: 62G10, 62G20 Compound distributions, goodness-of-fit tests, Katz family, Laplace transform

Goodness-of-fit procedures are introduced for testing the validity of compound models. New tests that utilize the Laplace transform as well as classical tests based on the distribution function are investigated. A major area of application of compound laws is in insurance, to model total claims resulting from specific claim frequencies and individual claim sizes. Monte Carlo simulations are used to compare the different test procedures under a variety of specifications for these two components of total claims. A detailed application to an insurance data set is presented.

Introduction

Consider the random sum of random variables,

X = N k=1 U k , (1)
where N is a count random variable with probability mass function p N and the U k 's form a sequence of independently and identically distributed (iid) non-negative continuous random variables having a distribution function (DF) F U , and independent of N .

Such "compound " random variables defined in [START_REF] Allison | On a data-dependent choice of the tuning parameter appearing in certain goodness-of-fit tests[END_REF] come up in many practical applications and has different interpretations depending on the context. Our focus here is to consider this compound variable X as the total amount of claim associated with a nonlife insurance portfolio over a fixed time period. The random variable N represents the claim frequency while the U k 's represent the individual claim sizes. The final objective of practitioners is to identify the components of this quantity defined in [START_REF] Allison | On a data-dependent choice of the tuning parameter appearing in certain goodness-of-fit tests[END_REF] i.e. identify the distribution of the claim frequency N and that of the individual claim sizes U . We will work under the assumption that the correspondence between X and the DF's for N and U k 's within model [START_REF] Allison | On a data-dependent choice of the tuning parameter appearing in certain goodness-of-fit tests[END_REF] as being unique and demonstrate that it works in many practical situations, although there may be few exceptions in theory. We note that incomplete data situations such as these where one has only observations on the total claims X, arise in practice when an insurance company keeps track of only aggregated data by the month, the quarter, or the year. The methodology can also be useful to a reinsurance company, which has only access to partial information on X, and would like to better understand the underlying risk and improve the rate-making. Model ( 1) is also useful in the banking industry which has only access to data on annualized operational risk. As pointed out in Chaudhury [START_REF] Chaudhury | A review of the key issues in operational risk capital modeling[END_REF], the data available to assess operational risk are typically incomplete as banks often report aggregate losses, often discarding small losses. Such loss of information also occurs when data is merged say after the acquisition of another banking operation.

Assume we observe aggregated claim sizes X 1 , . . . , X n , and we wish to assess the conformity of a given compound model in the composite situation whereby the DF's involved depend on unknown parameters. Specifically we write p N := p N (•; ϑ N ) and F U := F U (•; ϑ U ), for the component DF's with the parameter vector ϑ = (ϑ N , ϑ U ) treated as unknown. If F X denotes the DF of the compound r.v. X, we wish to test the composite null hypothesis

H 0 : The DF of X in (1.1) is F X 0 ≡ F X 0 (•; ϑ), for some ϑ ∈ Θ, (2)
where Θ denotes an appropriate parameter space.

Two types of nonparametric goodness-of-fit (GOF) tests are considered. The first type is based on a dissimilarity measure between the population DF and the empirical DF of the available data; see e.g. D'Agostino and Stephens [START_REF] D'agostino | Goodness-of-Fit Techniques[END_REF], or Thas [START_REF] Thas | Comparing Distributions[END_REF] for reviews on the subject of DF-based GOF tests. Since the random variable X typically has a point mass at 0 corresponding to N = 0, the standard Kolmogorov-Smirnov (KS) and Cramèr-von Mises (CvM) GOF tests need some corresponding modifications. Although these procedures are originally meant to handle continuous data, extensions have been proposed to assess the adequacy for discrete, grouped, or mixed data. See for instance Schmid [START_REF] Schmid | On the Kolmogorov and Smirnov limit theorems for discontinuous distribution functions[END_REF], Walsh [START_REF] Walsh | Bounded probability properties of Kolmogorov-Smirnov and similar statistics for discrete data[END_REF], Noether [START_REF] Noether | Note on the Kolmogorov statistic in the discrete case[END_REF], Slakter [START_REF] Slakter | A comparison of the Pearson chi-square and Kolmogorov goodnessof-fit tests with respect to validity[END_REF], Conover [START_REF] Conover | A Kolmogorov goodness-of-fit test for discontinuous distributions[END_REF], Gleser [START_REF] Gleser | Exact power of goodness-of-fit tests of Kolmogorov type for discontinuous distributions[END_REF], and Dimitrova et al. [START_REF] Dimitrova | Computing the Kolmogorov-Smirnov distribution when the underlying cdf is purely discrete, mixed or continuous[END_REF] for the KS GOF test. Regarding the CvM criterion, the reader is referred to the works of Choulakian et al. [START_REF] Choulakian | Cramér-von Mises statistics for discrete distributions[END_REF], Henze [START_REF] Henze | Empirical-distribution-function goodness-of-fit tests for discrete models[END_REF], Spinelli and Stephens [START_REF] Spinelli | Cramér-von Mises tests of fit for the Poisson distribution[END_REF], Spinelli [START_REF] Spinelli | Testing fit for the grouped exponential distribution[END_REF] and Lockhart et al. [START_REF] Lockhart | Cramér-von Mises statistics for discrete distributions with unknown parameters[END_REF]. We propose modified estimators of the KS and CvM test statistics that take care of the point mass at 0 and also address the lack of closed form expression for the DF of X.

A second group of procedures we employ here, measure the model discrepancy in terms of the distance between the population Laplace transform (LT) and the empirical LT.

Statistical tests involving this approach work directly with transform-based statistics, thus avoiding LT inversion which is often complex and costly. These methods are quite convenient in cases where the DF is complicated while the LT is readily available. Such methods are relatively new but since their introduction, they have been used in various estimation and testing problems; see for instance Henze [START_REF] Henze | A new flexible class of omnibus tests for exponentiality[END_REF], Henze and Klar [START_REF] Henze | Goodness-of-fit tests for the inverse Gaussian distribution based on the empirical laplace transform[END_REF], Henze and Meintanis [START_REF] Henze | Tests of fit for exponentiality based on the empirical Laplace transform[END_REF], Meintanis and Iliopoulos [START_REF] Meintanis | Tests of fit for the Rayleigh distribution based on the empirical Laplace transform[END_REF], Besbeas and Morgan [START_REF] Besbeas | Efficient and robust estimation for the one-sided stable distribution of index 1/2[END_REF], Ghosh and Beran [START_REF] Ghosh | On estimating the cumulant generating function of linear processes[END_REF], Milošević and Obradović [START_REF] Milošević | New class of exponentiality tests based on Uempirical Laplace transform[END_REF], and Allison et al. [START_REF] Allison | An 'apples to apples' comparison of various tests for exponentiality[END_REF].

The test statistics involved in these procedures lead to non-standard asymptotic distributions for which finding critical values requires sophisticated numerical methods. Moreover, due to the fact that the parameters of the null distribution have to be estimated a priori, the tests are not distribution free. We overcome these difficulties using a parametric bootstrap approach, which has gained popularity in approximating the null distribution in goodness-of-fit testing.

The paper is organized as follows. Section 2 provides a brief background on compound distributions and reviews moment-based estimation of the parameters. Section 3 details the goodness of fit testing procedures tailored to the distribution of aggregated claim sizes. Section 4 reports the results of a Monte-Carlo simulation study conducted to compare of the GOF tests in terms of power. Section 5 presents an application of our GOF procedures to a real data set from the insurance industry. We conclude with discussion in Section 6. Asymptotic results are contained in the Appendix.

Preliminaries

Compound Distribution

Recall that X = N k=1 U i , where N is a counting rv with probability mass function p N and the U k 's are iid non-negative continuous random variables with DF F U , and independent of N . Given the fact that N can take the value zero with probability p N (0), the DF of X is given by F X (x) = p N (0) + 1 -p N (0) F X|N >0 (x), x ≥ 0,

where F X|N >0 denotes the DF of X provided that N > 0. Note that the conditional probability distribution of X|N > 0 is continuous. The Laplace transform (LT) of X, defined as L X (t) := E(e -tX ), may be expressed as

L X (t) = G N L U (t) , t ≥ 0, (4)
where G N (t) := E(t N ) denotes the probability generating function of N and L U is the LT of the claim size distribution.

We let the distribution of the claim sizes be quite general (apart from its parametric form), but model the claim frequency via a counting distribution from the broad class viz. Katz family. Recall that the distribution of N belongs to the Katz family [START_REF] Katz | Unified treatment of a broad class of discrete probability distributions[END_REF], written N ∼ KF(a, b), if the probability mass function satisfies the recursive equation

p N (k) = a + b k p N (k -1), for k ≥ 1. ( 5
)
A characterization is given in Sundt and Jewell [START_REF] Sundt | Further results on recursive evaluation of compound distributions[END_REF]. Prominent members of this family (5) include 1. The binomial N ∼ Bin(α, p) (α ∈ N, 0 < p < 1) which satisfies [START_REF] Charpentier | Computational actuarial science with R[END_REF] with a = -p/(1 -p) and b = (α + 1)p/(1 -p).

2. The Poisson N ∼ Pois(λ) (λ > 0) which satisfies [START_REF] Charpentier | Computational actuarial science with R[END_REF] with a = 0 and b = λ.

3. The negative binomial N ∼ Neg-Bin(α, p) (α > 0, 0 < p < 1) with probability mass function

p N (k) = Γ(α + k) Γ(α)Γ(k + 1) p α (1 -p) k , for k ≥ 0, (6)
which satisfies [START_REF] Charpentier | Computational actuarial science with R[END_REF] with a = 1 -p and b = (α -1)(1 -p).

These discrete distributions are commonly used to model claim frequencies and this choice seems quite general and justified.

Throughout this paper, we will denote by n 0 ≤ n the number of zeros and by X + 1 , . . . , X + n-n 0 the non-zero values within the sample X 1 , . . . , X n .

Moments based estimation for aggregate claims

The Method of Moments estimator is obtained by matching the empirical moments with the theoretical moments of the parametric model. If N ∼ KF(a, b) then the moments of X may be expressed in terms of the moments of U via the recursive relations 1. (geometric-exponential): Assume that the claim sizes follow an exponential distribution, U ∼ Exp(θ), and that the claim frequency is geometric N ∼ Neg-Bin(1, p). Thus we have that a = 1 -p, b = 0, in the Katz family parametrization in [START_REF] Charpentier | Computational actuarial science with R[END_REF]. Substituting in [START_REF] Choulakian | Cramér-von Mises statistics for discrete distributions[END_REF] and solving yields the parameters estimates

(1 -a)E X k+1 = k i=0 k i k + 1 i + 1 a + b E U i+1 E X k-i , for k ≥ 0, (7)
θ = m 2 -X2 2 X and p = θ θ + X . (8)
2. (Poisson-exponential): Assume a Poisson frequency for N ∼ Pois(λ), with claim size following an exponential distribution U ∼ Exp(θ), with density

f U (x) = 1 θ e -x/θ , for x ≥ 0. ( 9
)
The Poisson distribution with parameter λ corresponds to a = 0, b = λ in the Katz family parametrization in [START_REF] Charpentier | Computational actuarial science with R[END_REF], while for the exponential distribution with parameter θ we have, E(U ) = θ, and E (U 2 ) = 2θ 2 . Substituting in [START_REF] Choulakian | Cramér-von Mises statistics for discrete distributions[END_REF] and solving yields the parameter estimates

θ = m 2 2 X and λ = 2 X2 m 2 . ( 10
)
3. (Poisson-gamma): Assume that the claim sizes follow a gamma distribution, U ∼ gamma(r, θ), with density

f U (x) = e -x/θ x r-1 θ r Γ(r) , for x ≥ 0. ( 11
)
Let the claim frequency be Poisson distributed N ∼ Pois(λ). We have that

a = 0, b = λ, E(U ) = rθ, E (U 2 ) = r(r + 1)θ 2
, and E (U 3 ) = r(r + 1)(r + 2)θ 3 . Substituting in [START_REF] Choulakian | Cramér-von Mises statistics for discrete distributions[END_REF] and solving yields the parameters estimates

r = 2m 2 2 -m 3 X m 3 X -m 2 2 , θ = m 2 X( r + 1)
, and λ = X

r θ . ( 12
)
Remark 2.1. The estimates of the parameters in Examples ( 8) and (12) may turn out to be negative due to the lack of fit of the model. The partial Method of Moments presented below often resolves this difficulty.

The "partial Method of Moments" idea is as follows: whenever the data consists of one or more X i that take the value zero i.e. n 0 > 0, consider adding to the system of equations ( 7), an additional estimation equation corresponding to the probability of this event. If N ∼ Bin(α, p) or N ∼ Neg-Bin(α, p), it is given by

p N (0) = p α , (13)
and when N ∼ Pois(λ), it is

p N (0) = e -λ . (14)
This probability that N = 0 is estimated by p N (0) = n 0 /n. The parameters of the claim sizes distribution follow from the other MME equations. The resulting estimates are referred to as partial Method of Moments Estimators (partial -MMEs) in the remainder.

The following example provides the expressions of the partial -MMEs in the geometricexponential, Poisson-gamma and Poisson-inverse Gaussian cases.

Example 2.

1. Assume that the claim frequency is geometric N ∼ Neg-Bin(1, p), then b = 0 and a = 1 -p, and p is estimated via

p = n 0 n ,
and if the claim sizes are exponentially distributed U ∼ Exp(θ), then θ is estimated via

θ = p X 1 -p .
Hence, the partial-MME cannot be negative in the geometric-exponential case.

2. Assume that the claim frequency is Poisson distributed N ∼ Pois(λ), then a = 0 and b = λ, and λ is estimated via

λ = -log n 0 n ,
and if the claim sizes are gamma distributed U ∼ Gamma(r, θ), then using [START_REF] Choulakian | Cramér-von Mises statistics for discrete distributions[END_REF], the Gamma parameters are estimated via

r = X2 λm 2 -X2 and θ = X λ r . ( 15
)
These estimators do not involve the third order moment anymore, and if λ is in a reasonable range (λ

> X2 m 2 )
, their values will be non-negative. 3. Assume that the claim frequency is Poisson distributed N ∼ Pois(λ), then a = 0 and b = λ, and λ is estimated via

λ = -log n 0 n .
Let the claim sizes be inverse-Gaussian distributed U ∼ IG(µ, ϕ), with density

f U (x) = 1 2πx 3 ϕ 1/2 exp - (x -µ) 2 2µ 2 ϕx , x > 0. ( 16
)
Substituting in [START_REF] Choulakian | Cramér-von Mises statistics for discrete distributions[END_REF] and solving yields the parameter estimates

µ = X λ and ϕ = λm 2 -X2 X . (17)
3 Goodness-of-fit tests for aggregate claims

Tests based on the distribution function

As already mentioned, a DF-based GOF test compares the population DF F X 0 (x; ϑ) = P 0 (X ≤ x ϑ), x ∈ R, to its empirical counterpart, the empirical DF, defined by

F X n (x) = 1 n n i=1 I(X i ≤ x). (18)
Note that the empirical DF may be rewritten as

F X n (x) = n 0 n + n -n 0 n F X|N >0 n (x),
where F

X|N >0 n

denotes the empirical DF of X given that N > 0, which can be estimated via

F X|N >0 n (x) = 1 n -n 0 n-n 0 i=1 I(X + i ≤ x). (19)
We estimate the population DF as F X 0 (x) := F X 0 (x; ϑ) where ϑ = ϑ(X 1 , . . . X n ) is some asymptotically efficient estimator. As we assumed that the distribution of N belongs to the Katz family, the population DF may be approximated via the so-called Panjer algorithm, see [START_REF] Panjer | Recursive evaluation of a family of compound distributions[END_REF] for more details.

Kolmogorov-Smirnov test for compound distributions

The Kolmogorov-Smirnov GOF test employs the distance

KS F X 0 , F X n = √ n sup x∈(0,∞) |F X 0 (x) -F X n (x)| := √ n sup x∈(0,∞) D KS n (x). ( 20
)
Denote by X + 1:n-n 0 , . . . , X + n-n 0 :n-n 0 , the order statistics associated to the sample X + 1 , . . . , X + n-n 0 , and define the intervals I i = [X + i:n-n 0 , X + i+1:n-n 0 ) for i = 0, . . . , n -n 0 , with the convention X 0:n-n 0 = 0 and X n-n 0 +1:n-n 0 = ∞ . Then for x ∈ I i ,

D KS n (x) = n 0 + i n -F X 0 (x) , i = 0, . . . , n -n 0 , so that sup x∈I i D KS n (x) = max n 0 + i n -F X 0 X + i:n-n 0 , F X 0 X + i+1:n-n 0 - n 0 + i n .
Considering successively the intervals I i , we estimate the KS distance [START_REF] Henze | Goodness-of-fit tests for the inverse Gaussian distribution based on the empirical laplace transform[END_REF] by

KS n = max (D -, D + ) ,
where

D -= max 0≤i≤n-n 0 n 0 + i n -F X 0 X + i:n-n 0 , and D + = max 0≤i≤n-n 0 F X 0 X + i+1:n-n 0 - n 0 + i n .

Cramér-von Mises test for compound distributions

The Cramér-von Mises GOF test uses the criterion

CM F X 0 , F X n = n +∞ 0 F X 0 (x) -F X n (x) 2 dF X 0 (x). ( 21
)
Given the mixed nature of the distribution of X, the probability measure follows from differentiation in (3) with

dF X 0 (x) = p N 0 (0)δ 0 (x) + 1 -p N (0) dF X|N >0 0 (x), x ≥ 0, (22)
where δ 0 (x) denotes the Dirac measure at 0. Reinserting [START_REF] Janssen | Global power function of goodness-of-fit tests[END_REF] into the integral (21) yields

CM F X 0 , F X n = n p N (0) p N (0) - n 0 n 2 + 1 -p N (0) +∞ 0 F X 0 (x) -F X n (x) 2 dF X|N >0 0 (x) . ( 23
)
Because the non-negative data points X + 1 , . . . , X + n-n 0 are assumed to be distributed as F X|N >0 0 under the null hypothesis, we can write the integral in [START_REF] Katz | Unified treatment of a broad class of discrete probability distributions[END_REF] as

p N (0) - n 0 n 2 n n-n 0 i=1 F X|N >0 (X + i ) - n -n 0 n + +∞ 0 [1 -p N (0)]F X|N >0 0 (x) - n -n 0 n F X|N >0 n (x) 2 dF X|N >0 (x). ( 24
)
By expanding the square and applying the change of variable u = F X|N >0 (x), the integral (24) may be rewritten as

[1 -p N (0)] 2 3 -[1 -p N (0)] n -n 0 n 1 n -n 0 n-n 0 i=0 [1 -F X|N >0 0 (X + i ) 2 ] + n -n 0 n n -n 0 + n-n 0 i=0 (1 -2i)F X|N >0 0 (X + i:n ) . (25)
Finally combining [START_REF] Katz | Unified treatment of a broad class of discrete probability distributions[END_REF], [START_REF] Lockhart | Cramér-von Mises statistics for discrete distributions with unknown parameters[END_REF] and [START_REF] Meintanis | Tests of fit for the Rayleigh distribution based on the empirical Laplace transform[END_REF] allows to estimate the CvM statistics as

CM n = n p N 0 (0) p N 0 (0) - n 0 n 2 + 1 -p N 0 (0) p N 0 (0) - n 0 n 2 n i=1 F X|N >0 0 (X + i:n ) - n -n 0 n + [1 -p N 0 (0)] 2 3 -[1 -p N 0 (0)] n -n 0 n 1 n -n 0 n-n 0 i=0 [1 -F X|N >0 0 (X + i ) 2 ] + n -n 0 n n -n 0 + n-n 0 i=0 (1 -2i) F X|N >0 0 (X + i:n ) , (26)
where

F X|N >0 0 (x) := F X|N >0 0
(x; ϑ) and p N 0 (0) := p N 0 (0; ϑ N ) is the parametric estimator of the probability that N = 0 under the null hypothesis.

Tests based on the Laplace transform

LT-based GOF tests are based on a distance between the LT L X 0 (t; ϑ) := E 0 (e -tX |ϑ), t > 0, (we often write L X 0 (t) for simplicity) corresponding to the null hypothesis, and its empirical counterpart, the empirical LT, given by

L X n (t) = 1 n n i=1 e -tX i .
Typically, such a test statistic is expressed as an integrated distance between L X n (t) and L X 0 (t) involving a weight function w(t) > 0, t ≥ 0. The main motivation of the LT approach lies in tractability of the LT L X 0 (•), given by (4). Two approaches are described in Sections 3.2.1 and 3.2.2.

L 2 dissimilarity measure

An obvious choice is to consider the discrepancy between the theoretical and empirical Laplace transform

SE n (t) = L X n (t) -L X 0 (t), (27)
and integrate SE 2 n (•) against the weight function w(t) as

S n,w = n ∞ 0 SE 2 n (t)w(t)dt, (28)
where L X 0 (t) = L X 0 (t; ϑ). Choosing the exponential weight function w(t) = e -βt , β > 0, allows us to write the test statistic in [START_REF] Panjer | Recursive evaluation of a family of compound distributions[END_REF] as

S n,β = 1 n n i,j=1 1 X i + X j + β -2 n i=1 ∞ 0 L X 0 (t)e -(X i +β)t dt + n ∞ 0 L X 0 (t) 2 e -βt dt. ( 29
)
Depending on the hypothesized LT, numerical integration may be required for the evaluation. A classical work-around in LT-based GOF testing to avoid numerical integration is to define a dissimilarity measure relying on a differential equation which we discuss next. The issue of the choice of the weight parameter β is postponed to Section 4.

Dissimilarity measure based on a differential equation

If under the null hypothesis, N ∼ KF(a, b) then the LT of X satisfies a differential equation. Start by noting that

dG N (t) = a + b 1 -at G N (t), (30)
where df (t) denotes the first derivative of the function f with respect to t. Differentiating with respect to t on both sides of (4) yields

dL X (t) = dL U (t)dG N L U (t) , t ≥ 0, (31)
and reinserting [START_REF] Schmid | On the Kolmogorov and Smirnov limit theorems for discontinuous distribution functions[END_REF] into [START_REF] Slakter | A comparison of the Pearson chi-square and Kolmogorov goodnessof-fit tests with respect to validity[END_REF] leads to the following differential equation

dL X (t) 1 -aL U (t) -(a + b)L X (t)dL U (t) = 0. ( 32
)
Equation [START_REF] Spinelli | Testing fit for the grouped exponential distribution[END_REF] motivates us to define a dissimilarity measure as

DE n (t) = dL X n (t) 1 -a L U 0 (t) d L U 0 (t) -( a + b)L X n (t), t ≥ 0, (33)
where

dL X n (t) = - 1 n n i=1 X i e -tX i , L U 0 (t) = L U 0 (t; ϑ U ) and d L U 0 (t) = dL U 0 (t; ϑ U ).
The corresponding test statistic (analogous to [START_REF] Panjer | Recursive evaluation of a family of compound distributions[END_REF]) is defined by

T n,w = n ∞ 0 DE 2 n (t)w(t)dt, (34)
with rejection for large values of T n,w . Letting w(t) = e -βt , β > 0 and by straightforward computations we have from [START_REF] Stute | Bootstrap based goodness-of-fit-tests[END_REF],

T n,β = 1 n n i,j=1 X i X j K (2) β (X i + X j ) + 1 n n i,j=1 X i K (1) β (X i + X j ) + ( a + b) 2 n n i,j=1 K (0) β (X i + X j ), (35)
where

K (k) β (x) = ∞ 0 1 -a L U 0 (t) d L U 0 (t) k e -(x+β)t w(t)dt, for k = 0, 1, 2. ( 36
)
The exponential weight function e -βt , β > 0, allows us to derive tractable formulas when the claim sizes distribution is gamma or inverse Gaussian as shown in the following example.

Example 3. First note that K (0)

β (x) = (x + β) -1 . 1.
Let U be gamma distributed Gamma(r, θ) with LT given by L U 0 (t) = (1 + θt) -r . We have that

K (1) β (x) = a (x + β + θ) rθ(x + β) 2 - e (x+β)/θ θ r r(x + β) r+2 Γ u r + 2;
x + β θ and K

(2)

β (x) = e (x+β)/θ θ 2r r 2 (x + β) 2r+3 Γ u 2r + 3; x + β θ -2a e (x+β)/θ θ r r 2 (x + β) r+3 Γ u r + 3; x + β θ + a 2 (x 2 + 2θx + 2θ 2 ) (rθ) 2 (x + β) 3 ,
where Γ u (r; x) = +∞ x y r-1 e -y dy denotes the upper incomplete gamma function.

2. Let U be inverse Gaussian distributed IG(µ, ϕ) with LT given by L U 0 (t) = exp . We have that

K (1) β (x) = a √ ϕe (x+β) 2µ 2 ϕ (x + β) 3/2    e - (x+β) 2µ 2 ϕ √ x + β µ √ ϕ + 1 √ 2 erfc √ 2 x + β µ 2 ϕ    + e c 2 µ 2 (x + β)ϕ µ 2 ϕ x + β c √ 2 e -c 2 + 1 √ 2 erfc (c) + 2µ 2 √ ϕ (x + β) 3/2 e -c 2 + µ 2 (x + β) 2 √ 2 erfc(c)
and

K (2) β (x) = 2 3 e d 2 (x + β) 3 ϕ(x + β) 1 √ 2 erfc(d) + 3 ϕ(x + β) 2 e -d 2 + 3ϕ(x + β) 4 d √ 2 e -d 2 + 1 √ 2 erfc(d) + ϕ(x + β) 2
d 2 + 4 2 e -d 2 -2ae c 2 1 √ 2 erfc(c) + 3 ϕxe -c 2 + 3ϕx c √ 2 e -c 2 + 1 √ 2 erfc(c) + [ϕ(x + β)] 3/2 c 2 + 4 2 e -c 2 + a 2 1 µ 2 (x + β) + 2ϕ (x + β) 2 , where erfc(x) = 2 √ π +∞ x
e -t 2 dt denotes the complementary error function.

Asymptotic results including the limit distribution of the LT-based test statistic S n,w under the null hypothesis are given in the Appendix. This distribution, as well as the limit distributions corresponding to the other LTor DFbased tests considered here, is extremely complicated. Therefore, in the next section we resort to resampling techniques in order to obtain critical values and actually carry out the tests.

Simulation study

This section presents the result of a Monte-Carlo experiment designed to assess the power of the GOF procedures. In the first subsection, we investigate the impact of the choice of the parameter β in the weight function on the performance of the LT based GOF procedures. In the second subsection, the DF and LT based GOF tests are compared in terms of power. Parametric bootstrap resampling is used to approximate the distribution of the test statistic under the null hypothesis. This type of resampling has been set on a firm theoretical basis, see e.g., Stute et al. [START_REF] Stute | Bootstrap based goodness-of-fit-tests[END_REF], Henze [START_REF] Henze | Empirical-distribution-function goodness-of-fit tests for discrete models[END_REF], and Genest and Rémillard [START_REF] Genest | Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models[END_REF], and is typically called upon when the asymptotic null distribution of any given test is too complicated to apply in practice. For the sake of completeness, the parametric bootstrap principle is recalled hereafter. Say we wish to assess the fit of an iid sample X 1 , . . . , X n of aggregated claim data to a parametric model characterized by its DF F X 0 (x, ϑ). The parameter of the model is inferred as ϑ = ϑ(X 1 , . . . , X n ) and the test statistic TS ∈ {CM n , KS n , S n,w , T n,w } is computed. Bootstrap samples are then drawn from F X 0 (x, ϑ). We compute the test statistic for each one of the samples and the critical values follow from quantile estimation. The steps of the parametric bootstrap routine are given in Algorithm 1 where B ∈ N denotes the number of bootstrap samples and α ∈ (0, 1) is the confidence level of the GOF test.

Algorithm 1 Parametric bootstrap for goodness-of-fit test

1: compute ϑ := ϑ(X 1 , . . . , X n ) 2: compute TS. 3: for k = 1 → B do 4: simulate X * k,1 , . . . , X * k,n from F X 0 (x, ϑ) 5: compute ϑ * k := ϑ(X * k,1 , . . . , X * k,n ) 6: compute TS * k 7: end for 8: compute T S * α := Quantile(TS * 1 , . . . , TS * B ; α) 9: if TS > T S * α then reject H 0 10: else accept H 0 11: end if
In the sequel, we study the probability of rejection of a sample generated by a fixed model F X when the model tested is F X 0 (x, ϑ). It requires to generate M ∈ N samples X k,1 , . . . , X k,n , k = 1, . . . , M drawn from F X and apply Algorithm 1. The warp-speed strategy suggested by Giacomini et al. [START_REF] Giacomini | A warp-speed method for conducting Monte Carlo experiments involving bootstrap[END_REF] allows us to reduce the running time required for our experiment. The idea is to generate only one bootstrap sample from F X 0 for each Monte Carlo sample simulated from F X . The parametric bootstrap routine augmented by the warp-speed strategy is provided in Algorithm 2.

Algorithm 2 Rejection probability via parametric bootstrap and warp-speed method

1: for k = 1 → M do 2: simulate X k,1 , . . . , X k,n from F X 3: compute ϑ k := ϑ(X k,1 , . . . , X k,n ) 4: compute TS k 5: simulate X * k,1 , . . . , X * k,n from F X 0 (x, ϑ) 6: compute ϑ * k := ϑ(X * k,1 , . . . , X * k,n ) 7:
compute TS *

SE 2 n (t) → n 0 n -p N 0 (0) 2 , as t → +∞.
Choosing small values of β leads to capturing a difference in the atom of probability at 0, while choosing a large β allows to detect variations in the right tail. Note also that opting for the partial-MME method will make the SE n (t) distance tend toward 0 for large values of t.

For further scrutiny on the role of the weight function and the weight parameter β, we consider the discrepancy SE(t) in [START_REF] Noether | Note on the Kolmogorov statistic in the discrete case[END_REF] and take the Taylor series expansion of the exponential function figuring in the empirical Laplace transform therein. This leads to

SE 2 n (t) = ∞ k,ℓ=1 (-t) k+ℓ k!ℓ! µ k -E 0 (X k ) µ ℓ -E 0 (X ℓ ) , (37)
where

µ k = n -1 n k=1 X k i , k ≥ 1,
denote the empirical moments of the sample X 1 , . . . , X n and E 0 (X k ) := E 0 (X k ; ϑ) is the estimate of the corresponding expectation of the aggregated claim size under H 0 obtained by replacing ϑ by ϑ. Integrating (37) term-by-term against the exponential weight function w(t) = e -βt yields

S n,β = n ∞ k,ℓ=1 k + ℓ k (-1) k+ℓ β k+ℓ+1 µ k -E 0 (X k ) µ ℓ -E 0 (X ℓ ) . (38)
Thus the weight function tunes how the difference between the empirical and theoretical moments enter the test statistic S n,w . Namely, lowering the value of β allows one to take into account higher order moments. This analysis holds too for the DE n (t) distance with

DE 2 n (t) = +∞ k,ℓ=1 (-t) k+ℓ k!ℓ! 1 n n i=1 Q k+1 (X i ) 1 n n i=1 Q ℓ+1 (X i ) , (39)
where

(Q k ) k≥1 is a sequence of polynomials satisfying E 0 [Q k (X)] = 0, for k ≥ 1. The polynomial Q k (x)
is of order k in x and its coefficients may be expressed in terms of the parameters of the model specified under H 0 . For instance, if a compound Poissonexponential Pois(λ) -exp(θ) is assumed under H 0 then we have that

Q 1 (x) = x -λ θ, Q 2 (x) = x 2 -( λ + 2) θx, Q 3 (x) = x 3 -θ( λ + 4)x 2 + 2 θ 2 x, . . . (40)
Integrating ( 39) term-by-term against the exponential weight function w(t) = e -βt yields

T n,β = n ∞ k,ℓ=1 k + ℓ k (-1) k+ℓ β k+ℓ+1 1 n n i=1 Q k+1 (X i ) 1 n n i=1 Q ℓ+1 (X i ) . (41)
The value of β is calibrated to select the moments that will influence the test decision. We note also that with moment estimation, E 0 (X k ) = µ k , k = 1, 2, so that the corresponding terms in [START_REF] Van Der Vaart | Weak Convergence and Empirical Processes[END_REF] and (41) vanish when choosing the Method of Moments estimator.

A Monte Carlo experiment is further conducted to gain insight on how to choose β to optimize the performance of the Laplace transform GOF procedures. We test the adequacy of a compound Poisson-exponential model Pois(λ) -Exp(θ) to data coming from a Poisson-gamma model Pois(λ = 1) -gamma(r, θ = 1). The probability of rejection is computed when varying the value the shape parameter r ∈ {0.5, 0.75, 1, 2, 4} for both of the Laplace transform based procedures as well as the two available inference methods (MME and partial-MME). We set the sample size to n = 100 and use Algorithm 2 with M = 10, 000 Monte Carlo runs. Figure 1 displays the level (when r = 1) of the test for β ranging from 10 -13 to 10 3 . The probability of rejection (expected to be around 5%) is too high when using the T n,β (t) statistic and too low when using the S n,β when β < 10 -7 . The sampling error on the parameter estimates might explain this fact as the variance is higher for large integrated distance. Figure 2 displays the power (when r ̸ = 1) of the tests. The power of the test always decreases with β which reflects that the distance between the Laplace transforms vanishes as t approaches 0. Two behaviors may be observed on Figure 2 depending on the value of r. When r < 1, the rejection probability increases before reaching a maximum and decreasing. When r > 1, the power admits a plateau before decreasing. It is common in the goodness-of-fit testing literature to opt for the values of β that fare well in the majority of the cases. In this connection we note that there q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.00 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.00 0.25 0.50 0.75 1.00 -10 -5 0 ln(β) ln [START_REF] Dimitrova | Computing the Kolmogorov-Smirnov distribution when the underlying cdf is purely discrete, mixed or continuous[END_REF] probability of rejection (a) r = 0.5 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.0 [START_REF] Dimitrova | Computing the Kolmogorov-Smirnov distribution when the underlying cdf is purely discrete, mixed or continuous[END_REF] probability of rejection (b) r = 0.75 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 -10 -5 0 ln(β) ln [START_REF] Dimitrova | Computing the Kolmogorov-Smirnov distribution when the underlying cdf is purely discrete, mixed or continuous[END_REF] probability of rejection (c) r = 2 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.00 exist data-driven selection methods to determine proper values of the parameter β in LT-based goodness-of-fit tests, such as those proposed recently by Allison and Santana [START_REF] Allison | On a data-dependent choice of the tuning parameter appearing in certain goodness-of-fit tests[END_REF] and Tenreiro [START_REF] Tenreiro | On the automatic selection of the tuning parameter appearing in certain families of goodness-of-fit tests[END_REF]. However, given the computing time associated with the test statistic we decided not to implement such search methods here, and employ in the comparative study below the values β = 10 -3 , 10 -2 when using the S n,β and β = 0.1, 1 when using the S n,β , which performed well in our preliminary study.

0.1 0.2 0.3 0.4 -10 -5 0 ln(β) ln

Comparison of the GOF procedures

In this subsection, we compare the GOF procedures in terms of probability of rejection when the input samples differ from the model stated under the null hypothesis.

Test 1. In this first test, we generate samples from a Poisson-Weibull model Pois(λ = 1)-Weibull(r, θ = 1) and test with our GOF method the adequacy of a Poisson-exponential, Poisson-gamma and Poisson-inverse Gaussian model. The Weibull distribution Weibull(r, θ) admits a probability density function given by

f U (x) = r θ x θ r-1 exp - x θ r , for x > 0. ( 42
)
We set the sample size to 100 and use the partial-MME to infer the parameters in the model specified in H 0 . Figure 3 displays the powers computed via our parametric bootstrap routine with 10, 000 Monte Carlo runs changing the shape parameter r in the claim sizes distribution. The Poisson-Weibull coincides with the Poisson-exponential and Poissonq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.00 0.25 0.50 0.75 1.00 0.5 1.0 1.5 2.0 shape parameter probability of rejection (a) H 0 : Poisson-exponential q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.0 0.1 0.2 0.3 0.4 0.5 1.0 1.5 2.0 shape parameter probability of rejection (b) H 0 : Poisson-gamma q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.00 gamma models when the shape parameter is set to 1 so the power tends toward 5% on Figures 3(a) and 3(b) as r gets closer to 1. The GOF procedures do well when testing for a Poisson-exponential with very high power as r gets farther from 1, see Figure 3(a). The results are a bit disappointing when testing for a Poisson-gamma distribution, the DF based procedures achieve greater power in this case, see Figure 3(b). The procedures associated to the S n,β distance outperform greatly the other methods when testing for a Poisson-inverse Gaussian model.

Test 2. In this second test, samples are generated from zero-modified Poisson-exponential zmpois(λ = 5, p 0 ) -exp(θ = 1) and mixed Poisson-exponential mpois(p, λ 1 = 1, λ 2 = 5)-exp(θ = 1) and we assess the adequacy of a Poisson-exponential model. The probability mass function of the zero-modified Poisson distribution zmpois(λ, p 0 ) is given by

p N (k) = p 0 , for k = 0, 1-p 0 1-e -λ λ k e -λ k! , for k ≥ 1, (43)
and the probability mass function of the mixed Poisson distribution mpois(p, λ 1 , λ 2 ) is given by

p N (k) = p λ k 1 e -λ 1 k! + (1 -p) λ k 2 e -λ 2 k! , for k ≥ 0. ( 44
)
The Poisson-exponential under H 0 is fitted using the MME based on samples of size 100. Figure 4 displays the probability of rejection are computed via our parametric bootstrap routine with 10, 000 Monte Carlo runs letting the parameter p vary in the alternative claim frequency distributions. The GOF procedures all detect reasonably well the modification q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.25 0.50 0.75 1.00 0.25 0.50 0.75 p 0 probability of rejection (a) Samples drawn from a zmpois(p, λ = 5)exp(θ = 1) model q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.1 Other cases have been studied, the simulation data may be found in the online supplements [START_REF] Goffard | Online accompaniment for "Goodness-of-fit tests for compound distributions with applications in insurance[END_REF]. The main conclusion is that none of the procedures stands out in all and every case. This conclusion is corroborated by analytic methods which lead to the conclusion that any given goodness-of-fit test has nontrivial power only towards a given direction away from the null hypothesis; see Janssen [START_REF] Janssen | Global power function of goodness-of-fit tests[END_REF], and Escanciano [START_REF] Escanciano | On the lack of power of omnibus specification tests[END_REF]. Therefore we suggest, in a practical situation, to apply all the procedures to see if they lead to the same conclusion.

An application to insurance data

We illustrate our inference and goodness-of-fit procedures on an actuarial data set called itamtplcost accessible from the R package CASdatasets (see also the book of Charpentier [START_REF] Charpentier | Computational actuarial science with R[END_REF]). This dataset contains losses (in excess of 500, 000 euros) of an Italian Motor-TPL company since 1997. It comprises two variables Date and UltimateCost, and 457 observations. Table 1 We start by looking at the individual claim data before applying our methods to the monthly aggregated data, in the hope that they lead to similar inference and conclusions.

Figure 5 displays the exponential and Pareto Quantile-Quantile plots. A linear relationq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqqq qqqq q q q q q q q q q q 8 10 12 ship is observed between the lower order quantiles on Figure 5(a) and between the higher order quantiles on Figure 5(b). This, in turn, suggests the use of a splicing model with an exponential-type distribution to model the small claims and a Pareto-type distribution to fit the larger losses. The claim sizes distribution tested in the sequel are the exponential, gamma and inverse Gaussian distributions. Due to the heavy tail of the data, these distributions are not likely to result in a good fit. Hence, we decided to conduct the analysis over the whole data set and then on the smaller claims only. The small claims are defined on the basis of a threshold. The cut-off point between small and large claims is the upper order statistic that minimizes the asymptotic mean squared error of the Hill estimator, as it is a standard procedure in extreme value theory; see for instance the case study in the book by Beirlant et al. [START_REF] Beirlant | Statistics of Extremes: Theory and Applications[END_REF]Chapter 6]. Figure 6 displays the mean-excess plot and the Hill plot of the loss data. The threshold is set at 1, 766, 751 euros (correq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q 1.0e+06 (a) Mean excess plot q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 2 of variance when considering the small claims solely. The quartiles are stable while the mean in the small claims subset decreases to get closer to the median. Table 3 reports the Method of Moments estimators of the parameters of the exponential, gamma and inverse Gaussian distribution. Figure 7 displays the histograms of the claim sizes, on which are superposed the densities of the exponential, gamma and inverse Gaussian distributions with the parameters provided in Table 3. Table 4 reports the value of the AIC as well as the outcome of the KS and CvM GOF test for the exponential, gamma and inverse Gaussian distributions. On the basis of these results it seems that the gamma is the single distribution that provides a better fit for both of the datasets. We note how the rejection when using the KS GOF test is a close call when testing the gamma and inverse Gaussian model for the small claims. In order to apply our method, the original data needs to be processed so as to consider claim amounts aggregated monthly. For each time period, we collect the claim frequency as well as the sums of the incurred claims. Table 5 provides an overview of the processed data.

Conclusion and perspectives

Several goodness-of-fit tests for compound distributions were investigated, both classical as well as tests based on the Laplace transform. In either case the test criteria were tailored to specific versions of the null hypothesis that are popular in applications. The message drawn from a detailed Monte Carlo study is that all criteria respect the nominal level of the test and at the same time have reasonable power against some interesting alternatives, with the Laplace transform based test having a certain edge in terms of power. The real-data application shows the potential of the suggested methods for practitioners in order to also identify the components of an aggregate claim probability distribution, namely the claim frequency and the claim size distribution, when the only available data are the aggregated losses.

There are clearly several possible directions in which the current results can be extended. For instance, one can explore alternate estimation methods for the parameters, go outside the Katz family for counting models, or consider situations where multivariate data is available on the compound variable X. 3) made in order to prove consistency, as well as those pertaining to the limit null distribution, (A.4)-(A.6), are standard in the context of testing goodness-of-fit based on the empirical LT; see for instance Henze [START_REF] Henze | A new flexible class of omnibus tests for exponentiality[END_REF], Henze and Klar [START_REF] Henze | Goodness-of-fit tests for the inverse Gaussian distribution based on the empirical laplace transform[END_REF], and Henze and Meintanis [START_REF] Henze | Tests of fit for exponentiality based on the empirical Laplace transform[END_REF].

provided in De Pril [ 29 ,

 29 Equation 3]. Solving the system[START_REF] Choulakian | Cramér-von Mises statistics for discrete distributions[END_REF] for ϑ N = (a, b) and ϑ U yields the Method of Moments Estimators (MMEs).Denote by X = n -1 n i=1 X i and m k = n -1 n i=1 X i -X k the sample mean and the sample centered moments of order k ≥ 2, respectively. The following examples provide expressions for the MMEs in the geometric-exponential, Poisson-exponential, and Poissongamma cases: Example 1.

k 8 : 4 . 1

 841 end for 9: compute TS * α := Quantile(TS * 1 , . . . , TS * M ; α) 10: return The probability of rejection M -1 M k=1 I TS k >TS * α Investigation of the impact of the weight function The goal of this subsection is to investigate the role of the weight parameter β in LT-based procedures and prompt the discussion for the choice of good values of this parameter. It is well known from Tauberian theorems, see for instance Feller [12, Chapter XIII.5] that the tail behavior of a probability distribution concentrated on the positive half-line is reflected by the behavior of the Laplace transform at 0 and vice-versa. It is especially true in our context, that for SE n (t) we have

Figure 1 :

 1 Figure 1: Level of the LT based GOF test depending on the distance between Laplace transform and inference techniques used: (dotted) S n,β and MME ; (dashed) S n,β and partial-MME; (solid) T n,β and MME ; (dotdash) T n,β and partial-MME.

  ) ln[START_REF] Dimitrova | Computing the Kolmogorov-Smirnov distribution when the underlying cdf is purely discrete, mixed or continuous[END_REF] probability of rejection(d) r = 4

Figure 2 :

 2 Figure 2: Power of the LT based GOF test for the different combinations of distances and inference techniques: (dotted) S n,β and MME ; (dashed) S n,β and partial-MME; (solid) T n,β and MME ; (dotdash) T n,β and partial-MME.

  (c) H 0 : Poisson-inverse Gaussian

Figure 3 :

 3 Figure 3: Power of the various GOF tests in Test 1: (dotted) Cramér-von Mises ; (dash) Kolmogorov-Smirnov; (solid) S n,β , β = 10 -3 ; (dotdash) S n,β , β = 10 -2 ;(two dash) T n,β , β = 0.1 ; (long dash) T n,β , β = 1.

Figure 4 :

 4 Figure 4: Power of the various GOF tests in Test 2: (dotted) Cramer-von Mises ; (dash) Kolmogorov-Smirnov; (solid) S n,β , β = 10 -3 ; (dotdash) S n,β , β = 10 -2 ; (two dash) T n,β , β = 0.1 ; (long dash) T n,β , β = 1.

  Empirical Quantiles of the log loss (b) Q-Q plot to assess the adequacy to the Pareto distribution.

Figure 5 :

 5 Figure 5: Quantile-Quantile Plots.

Figure 6 :

 6 Figure 6: Mean-Excess plot and Hill plot.

  Histogram of the claim sizes.

  Histogram of the small claim sizes.

Figure 7 :

 7 Figure 7: Histograms of the data along with the density of the exponential (solid), gamma (dotted) and inverse Gaussian (dashed) distributions.

0 Z 2 0 KRemark 7 . 1 .

 02071 Now the limit distribution of the test statistic follows from Theorem 2 and the Continuous Mapping theorem. Specifically we have S n,w = (t)w(t)dt := Z w where Z(t) is the process defined in Theorem 2. The distribution of Z w is the same as that of ∞ j=1 λ j N 2 j , where λ 1 , λ 2 , ..., are the eigenvalues corresponding to the integral operatorAg(s) = ∞ (s, t)g(t)w(t)dt,i.e. the solutions of the equation Ag(s) = λg(s), and where N j , j ≥ 1, are iid standard normal variates. The assumptions (A.1)-(A.

Table 1 :

 1 shows the first 5 observations of the dataset itamtplcost. The first five observations from the dataset itamtplcost.

	Date	Claim size
	08/01/1997	726,986.95
	02/03/1997	1,222,682.37
	18/03/1997	428,543.10
	07/04/1997	258,786.06
	11/04/1997	637,117.61

Table 2 .

 2 We note the swift decrease

		Overall	Small claims
	No of observations	457	408
	E(U)	1, 015, 352.15	837, 255.78
	St.Dev(U)	680, 742.00	371, 646.54
	Skewness(U)	2.55	0.22
	Kurtosis(U)	12.65	-0.35
	Minimum	2, 160.73	2, 160.73
	Maximum	6, 639, 499.57	1, 764, 900.46
	Q1	627, 718.53	595, 464.36
	median	844, 010.92	790, 992.29
	Q3	1, 224, 316.09	1, 066, 904.00

Table 2 :

 2 Descriptive statistics of the claim data.

Table 3 :

 3 Estimated parameters for the exponential and gamma distributions.

	Model	parameters	Overall	Small claims
	exponential	scale	1, 015, 352.15	837, 255.78
	gamma	shape	2.23	5.09
		scale	455, 404.21	164, 564.55
	inverse Gaussian mean	1, 015, 352.15	837, 255.78
		dispersion	4.42E-07	2.35E-07

Table 4 :

 4 Measure of the adequacy of the exponential and the gamma distributions.

  Table 6 reports the estimated parameters of

	Time period	claim frequency	total claim size
	01/1997	1	726, 986.95
	02/1997	0	0.00
	03/1997	2	1, 651, 225.47
	04/1997	2	895, 903.67
	05/1997	0	0.00

Table 5 :

 5 Monthly aggregated data. the Poisson and geometric distributions accompanied by the Akaike information criterion and the χ 2 distance. The Poisson is better suited than the geometric distribution in view of the lower χ 2 distance and AIC values.

	Data	model	parameters	χ 2 distance	AIC
	Overall	Poisson	λ = 2.38	15.17	752.79
		geometric	p = 0.70	45.42	790.27
	Small claims	Poisson	λ = 1.93	17.14	705.48
		geometric	p = 0.66	32.67	724.57

Table 6 :

 6 Inference and measure of adequacy of the Poisson and geometric distribution over the claim frequency data.Table7gives the partial-MMEs for the Poisson-exponential, Poisson-gamma, Poissoninverse Gaussian and geometric-exponential compound models. We note that these values

	model		Overall	Small claims
	Poisson-exponential	λ =	2.42	2.52
		θ =	999, 750.14	630, 059.684
	Poisson-gamma	λ =	1.86	1.52
		r =	1.89	5.03
		θ =	687, 167.20	207, 806.41
	Poisson-inverse Gaussian λ =	1.86	1.37
		µ = 1, 301, 919.02	1, 058, 724.32
		ϕ =	4.05E-07	1.90E-07
	geometric-exponential	p =	0.84	0.78
		θ =	447, 546.23	445, 020.10

Table 7 :

 7 Inference and measure of adequacy of the Poisson and geometric distribution over the claim frequency data. are very different from the values estimated via the individual claim sizes and frequency data given in Tables3 and 6. Tables8, 9and 10 provides a summary of the GOF procedures applied on the data. The critical values are computed using Algorithm 1 with 10, 000 bootstrap loops. The Poisson-gamma and Poisson-Inverse Gaussian models can-

	β = 10 -3	β = 10 -2

Table 8 :

 8 Summary of the Laplace transform based GOF procedures using the S n,β statistic. not be discarded according to all the methods. The exponential claim sizes is discarded for all the methods except when using the S n,β test statistic based on the Laplace transform, see Table8.

	β = 10 -3	β = 10 -2

Table 9 :

 9 Summary of the Laplace transform based GOF procedures using the T n,β statistic.

	KS	CvM

Table 10 :

 10 Summary of the DF based GOF procedures.

Acknowledgements

The authors thank an anonymous referee for helpful comments and suggestions that helped improve the original manuscript. This work was initiated while Pierre-Olivier Goffard and Simos Meintanis were visiting the department of Statistics and Applied Prob-

ability at UC Santa Barbara. The authors are grateful for the warm welcome they received there. Pierre-Olivier Goffard's work is conducted within the Research Chair DIALog under the aegis of the Risk Foundation, an initiative by CNP Assurances.

Appendix: Consistency and limit null distribution

In this section, we discuss the consistency and limiting distribution of the LT-based test statistics under the null hypothesis H 0 . We focus our attention on the test criterion S n,w defined in [START_REF] Panjer | Recursive evaluation of a family of compound distributions[END_REF], but note that similar results may be obtained for the test statistic T n,w . We begin with the consistency of the test based on S n,w under the following assumptions:

(A.1) The estimator satisfies ϑ → ϑ, a.s., as n → ∞, for some ϑ ∈ Θ, with ϑ ≡ ϑ 0 when the null hypothesis H 0 is true, with ϑ 0 being the true value.

(A.

2) The LT L X 0 (•; ϑ) is continuous in ϑ.

(A.

3) The weight function satisfies, (i) w(t) > 0, ∀t > 0, except for a set of measure zero, (ii)

a.s., as n → ∞.

Proof. : Clearly the strong consistency of the empirical Laplace transform and the continuity of L 0 (•; ϑ) imply that SE n → L X (t) -L X 0 (t; ϑ), a.s., as n → ∞. Then since SE 2 n (t) ≤ 4, the result follows by Lebesgue's dominated convergence theorem.

The right-hand side of (45) is positive unless L X (t) = L X 0 (t; ϑ), for all t > 0. However, by the uniqueness of the LT, the last identity holds true only under the null hypothesis H 0 , in which case ϑ ≡ ϑ 0 , thus implying the strong consistency of the test that rejects H 0 for large values of S n,w .

We continue with the limit distribution of the test statistic S n,w under the null hypothesis H 0 . For simplicity we assume that ϑ is a scalar parameter. To this end assume that (A.4) The estimator ϑ := ϑ n satisfies the Bahadur representation

where ℓ(•; •) are such that E(ℓ(X; ϑ 0 )) = 0 and E(ℓ 2 (X; ϑ 0 )) < ∞.

22

(A.5) The LT L X 0 (t; ϑ) is twice differentiable with respect to ϑ with a continuous second derivative in the neighborhood of the true value ϑ 0 .

(A.6) The weight function is such that

for all ϑ * in a neighborhood of ϑ 0 .

Theorem 2. Under assumptions (A.1) to (A.6) we have under H 0 ,

as n → ∞, where Z(t) is the zero-mean Gaussian process with covariance kernel K(s, t;

The covariance kernel is specified by

Proof. : Along the proof we will write Z

(1)

n if the two random processes (Z

n (t) = ε n (t), and the remainder ε n (t) is such that it has no effect on the limit null distribution of the test statistic S n,w .

With this understanding using assumption (A.5) and the second part of (A.6), a twoterm Taylor expansion yields

In turn using assumption (A.4) and the first part of (A.6) in (46) leads to

where

The result now follows by applying the Central Limit Theorem in Hilbert spaces, (see e.g. van der Vaart and Wellner [START_REF] Van Der Vaart | Weak Convergence and Empirical Processes[END_REF], p. 50) on the process Z * * n (t) given in (47).