On semiparametric estimation of ruin probabilities in the classical risk model

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On semiparametric estimation of ruin probabilities in the classical risk model

Esterina Masiello

To cite this version:

Esterina Masiello. On semiparametric estimation of ruin probabilities in the classical risk model. Scandinavian Actuarial Journal, Taylor & Francis (Routledge), 2012, 2014 (4), pp.283-308.

�10.1080/03461238.2012.690247�. �hal-00266449�

Nonparametric Statistical Analysis of Ruin Probability

Esterina Masiello

Dipartimento di Statistica, Probabilit`a e Statistiche Applicate P.le Aldo Moro 5, 00185 Rome Italy

e-mail: esterina.masiello@uniroma1.it

Abstract

The ruin probability of an insurance company is a central topic in risk theory. In this paper, the classical Poisson risk model is considered and a nonparametric estimator of the ruin probability is provided. Strong consistency and asymptotic normality of the estimator are estabilished. Bootstrap confidence bands are also studied. Further, a simulation example is presented in order to investigate the finite sample properties of the proposed estimator.

Key words Poisson risk model, ruin probability, nonparametric estimation, asymptotics.

1. Introduction and preliminaries

We consider the classical Poisson risk model, where the inter-arrival times T1, T2, . . .form a sequence of independent random variables (r.v.’s) with common exponential distribution function (d.f.) A(t) = P(T ≤ t) = 1 −e^−λt and finite meanµT = 1/λ. The claim sizesX1, X2, . . .are positive independent and identically distributed (i.i.d.) r.v.’s, with common d.f.F(x) =P(X ≤x), finite meanµand finite varianceσ². Moreover, the two sequences {T_i}and{X_i}are supposed to be independent. We further assume that the insurance company has an initial surplusx > 0and that premiums arrive at a known steady rate, say,c >0 per unit time.

LetΨ(x)denote the probability of ruin starting with initial reservex. Exact calculation and approximation of ruin probabilities have been a great source of inspiration and technique development in actuarial mathematics since the seminal paper by Lundberg (1903). The problem is that, apart from some special cases, a general expression for Ψ does not exist. We will construct a nonparametric estimator Ψ_n(x) of Ψ(x) based on a sample T₁, . . . , T_n of inter-arrival times and a sampleX1, . . . , Xnof corresponding claims. The d.f. of inter-arrival times and claims are both supposed unknown. More specifically, the inter-arrival time distribution is assumed to be exponential with unknown parameterλ. As far as the claim distribution is concerned, no specific parametric model is assumed.

Such a statistical problem has been considered by different authors. As remarked by Embrechts et al (1997), literature has mostly been concerned with asymptotic expansions; few results focus on statistical estimation ofΨ. Clearly, one could work out a parametric estimation procedure or use nonparametric

methods. As remarked in Pitts (1994), the latter approach is particularly interesting in view of its applicability. Also Frees (1986a) and Croux and Veraverbeke (1990) have proposed a nonparametric estimator forΨ. The estimator by Croux and Veraverbeke (1990) is a linear combination of U-statistics; they generalize techniques in Frees (1986b) whose estimator is based on the sample reuse concept.

For a nonparametric approach to the estimation ofΨsee also Hipp (1989a, 1989b).

Our approach is closely connected to that developed in Pitts (1994), but the technique used to prove asymptotic results for the estimator is different from ours since that author takes a functional view of the stochastic model, following the tradition of Gill (1989), Gr¨ubel (1989) and Gr¨ubel and Pitts (1993).

Also the ideas in Frees (1986a), as well as the ideas in Harel et al. (1995), on nonparametric estimation of the renewal function are basic for our approach.

2. The nonparametric estimator

In practical applications, the parameterλof the inter-arrival times distribution and the claim size d.f.F will not be given but have to be estimated with available data.

Let us consider a random sample ofninter-arrival timesT1, . . . , Tnwith the corresponding claimsX₁, . . . , X_n. Suppose that the assumptions of the Cramer- Lundberg model are satisfied. If the model under consideration is stable, that is λµ/c <1, it is possible to express the ruin probabilityΨas a compound geometric tail probability by mean of the so-called Pollaczeck-Khinchine formula (see, e.g.

Feller (1971), Grandell (1990) or Embrechts and Veraverbeke (1982)):

Ψ(x) =

∞

X

k=1

(1−ρ)ρ^k(1−F_I^∗k(x)) x≥0 (1)

whereρ = λµ/c,F_I(x) = _µ¹Rx

0 y dF(y)is the integrated tail distribution and F_I^∗k denotes thek-fold convolution of the d.f.FI. Fork = 1,2, . . .,F_I^∗k(x) = P(Y1+. . .+Yk ≤ x), whereY1, . . . , Yk are independent r.v.’s with common density functionf_I(x) = (1/λ)(1−F(x)),x≥0.

Remark 1. F_I(x) is also called equilibrium distribution of the d.f. F(x). In insurance, it can be interpreted as the d.f. of the amount of the drop in the surplus, or equivalently the difference between the initial surplus and the surplus at the time it first falls below the initial surplus. In other words, defined S(y)as a random sum of claim amounts,F_I is the d.f. ofS(y)−cy, given thatS(y)−cy >0and S(t)−ct≤0,0≤t < y.

The Pollaczeck-Khinchine formula is not entirely satisfying as a tool for computing Ψ(x) because of the infinite sum of convolution powers but it will be our key tool for estimating the ruin probability. From now on, for the sake of simplicity and without loss of generality, the rate of premium income per unit time will be taken to be equal to 1(c= 1); otherwise , it is enough to refer to the transformed claimsX˜i=Xi/c.

Our approach is the following: we first estimateE(T) =µ_T by its maximum likelihood estimatorTn= ¹_nPn

i=1Ti, andE(X) =µbyXn= _n¹Pn

i=1Xi. The next step consists in estimatingρby the ratioρˆ=Xn/Tn. Finally, we replace the k-fold convolution ofFI in Eq. (1) by the following estimator (Frees, 1986b):

F_n^∗k(x) = n

k −1

X

c

I_(Yi

1+...+Y_ik≤x) (2) where{i1, i2, . . . , ik}is a subset of sizekof{1,2, . . . , n}andP

cis the sum over all ⁿ_k

different combinations of{i1, i2, . . . , ik}.

We have easily estimated the quantities at the right hand side of Eq. (1) by the corresponding sampling counterparts. It seems therefore natural to estimateΨ(x) by

Ψn(x) =

∞

X

k=1

(1−ρ) ˆˆ ρ^kF^∗k_n (x) whereF^∗k_n (x) = 1−F_n^∗k(x).

Some of the properties of this estimator are shortly outlined in Conti and Masiello (2006a) and Conti and Masiello (2006b). In this paper, we obtain the asymptotic distribution of the estimator (Section 2.1) and study bootstrap confidence bands (Section 4). Moreover, in Section 5, a short simulation exemple is provided in order to illustrate the finite sample behavior of the estimator. An estimator for thep-th quantile of the ruin probability function is also proposed in Section 3, together with its main asymptotic properties.

Remark 2.The estimatorΨn(x)is a functional ofFn(x)andρ, i.e.ˆ Ψn=Φ(Fn,ρ).ˆ As a possible approach to the study of statistical properties of the proposed estimator, we could try to extend methods in Pitts (1994), who has studied the problem in the special case of a known value of ρ. In order to derive strong consistency and asymptotic normality results for the “output” estimator Ψn, it would suffice to establish analytical properties of continuity and differentiability for the functionalΦand combine them with the statistical properties of the “input”

estimatorsF_nandρ. We will take a slightly different approach following ideas inˆ Harel et al. (1995) in order to prove weak convergence of the stochastic process in the Skorokhod topology, but this will be discussed in the sequel.

Remark 3.Based on the estimator in (2), Frees (1986b) introduces a nonparametric estimator of the renewal functionH(t) =P∞

k=1F^∗k(t)given by Hn(t) =

m

X

k=1

F_n^∗k(t)

wherem = m(n)is a positive integer depending on n, such thatm ≤ nand m ↑ ∞ as n ↑ ∞. We left in the definition of our estimatorΨn(x)an infinite sum. On this subject, observe that the results which we are going to obtain are essentially asymptotic results applied when the sample sizentends to infinity.

2.1. Basic asymptotic results

In the above section, we have provided a point estimate forΨ(x). In order to study its behavior, we now need to obtain the asymptotic distribution of the estimator Ψn(x).

The key results of the present section are Theorem 2 and Theorem 3, where the strong consistency and the asymptotic normality of the estimator are estabilished.

To formulate our main theorems, we need some preliminary results. The asymptotic distribution of ρˆis obtained first, since it will play a central role in all subsequent developments.

Theorem 1Under the assumptions of the Cram´er-Lundberg model and ifρ <1, asngoes to infinity, the following results hold:

– ρˆ^a.s.→ ρ – √

n( ˆρ−ρ)tends in law to a normal distribution with mean zero and variance σ²_ρ= _µ¹2

T

σ²+_µ^µ4² T

σ²_T. Proof We may first write:

ˆ

ρ=f(Xn, Tn) =Xn

T_n

Take a first order Taylor expansion off(Xn, Tn)at the point(µ, µT)to obtain:

f(X_n, T_n) =f(µ, µ_T) + ∂f

∂Xn

_(µ,µ

T)

(X_n−µ) + ∂f

∂Tn

_(µ,µ

T)

(T_n−µ_T) +Rem whereRemdenotes the remainder term. It is easy to get the following relationship:

√n( ˆρ−ρ) = 1

µT

√

n(X_n−µ)− µ µ²_T

√n(T_n−µ_T) +√

n Rem (3) The asymptotic behavior of the estimatorsX_n andT_nis well-known. In fact, they converge almost surely to the corresponding “true values” as the sample size ntends to infinity. Furthermore,√

n(Xn−µ)and√

n(Tn−µT)have normal asymptotic distributions with mean zero and variancesσ²andσ²_T, respectively. As a simple application of the delta method, we obtain

√n( ˆρ−ρ)→^d N(0, σ²_ρ) whereσ²_ρ= _µ¹2

T

σ²+_µ^µ4² T

σ²_T.2

Remark 4.The mean and the variance ofρˆcan be obtained by direct calculations.

Taking into account thatρˆ= Xn/Tn and thatXn andTn are independent, we can easily obtain:

E X_n

Tn

=nE(X_n)E 1

Pn i=1Ti

(4)

SinceT_iis exponential with density:

fT(t) =

0 ift <0 λe^−λtift≥0 thenV =Pn

i=1T_iis Erlang distributed with density:

f_V(t) =

(0 ift <0

λⁿtⁿ⁻¹

(n−1)!e^−λtift≥0 Hence, we obtain

E 1

Pn i=1Ti

=E 1

V

= Z ∞

0

1 t

λⁿtⁿ⁻¹ (n−1)!e^−λtdt

= λ

(n−1)!Γ(n−1)

= λ

n−1 With obvious replacements, equation (4) becomes:

E Xn

Tn

= n

n−1 µ µ_T

When the sample sizengoes to infinity, we obtain:

E Xn

Tn

→ µ µ_T

so that the estimatorρˆis asymptotically unbiased. Let us consider the variance:

V ar Xn

Tn

=E

"

X²_n T²_n

#

−

E Xn

Tn

²

=E[X²_n]E

"

1 T²_n

#

−

E Xn

Tn

²

(5)

whereE[X²_n] =V ar(X_n) + [E(X_n)]²=σ²/n+µ²and

E 1

T²_n

!

=E

"

1

1 n

Pn i=1Ti

2

#

=n²E 1

V²

=n² Z ∞

0

1 t²

λⁿtⁿ⁻¹ (n−1)!e^−λtdt

=n² λ²

(n−1)!Γ(n−2)

= n²

(n−2)(n−1) 1 µ²_T

Once more, with obvious replacements in equation (5), we obtain the following exact result:

V ar Xn

Tn

= n

(n−2)(n−1) σ²

µ²_T + n² (n−2)(n−1)²

µ² µ²_T

and, asymptotically,

n→∞lim nV ar( ˆρ) = σ² µ²_T + µ²

µ²_T

We still need some auxiliary results. Let us define

Ψe_n(x) =

∞

X

k=1

(1−ρ) ˆˆ ρ^kF^∗k(x) We can rewrite our “empirical” process as

√n(Ψn(x)−Ψ(x)) =√

n(Ψn(x)−Ψen(x)) +√

n(Ψen(x)−Ψ(x))

=U_n(x) +V_n(x) where we have definedU_n(x) =√

n(Ψ_n(x)−Ψe_n(x))andV_n(x) =√

n(Ψe_n(x)− Ψ(x))and we study each of the two components separately. We begin by a couple of lemmas describing the behavior ofUn(x)andVn(x), respectively.

Lemma 1Let us define Un(x) = √

n(Ψn(x) − Ψen(x)). With the above assumptions, for eachx >0, asn→ ∞, we obtain the following result:

n→∞lim E[Un(x)Un(y)] = (1−ρ)²

∞

X

k=1

∞

X

j=1

ρ^k+jk j Cov(F^∗k−1(x−Y1), F^∗j−1(y−Y1))

Proof First of all, we may expressU_n(x)in the form:

U_n(x) = √

n(Ψ_n(x)−Ψe_n(x))

= √ n

∞

X

k=1

(1−ρ) ˆˆρ^k(F^∗k_n (x)−F^∗k(x))

= √ n

∞

X

k=1

(1−ρ)ρ^k(F^∗k_n (x)−F^∗k(x))

+

∞

X

k=1

n^1/4[(1−ρ) ˆˆρ^k−(1−ρ)ρ^k]n^1/4(F^∗k_n (x)−F^∗k(x))

Since the quantities n^1/4[(1 − ρ) ˆˆρ^k − (1 − ρ)ρ^k] and n^1/4

sup

x

F^∗k_n (x)−F^∗k(x)

both converge in probability to zero when

n goes to infinity, the asymptotic distribution of U_n(x)coincides with that of

√nP∞

k=1(1−ρ)ρ^k(F^∗k_n (x)−F^∗k(x)). The following relations hold:

E[Un(x)Un(y)] = (1−ρ)²

∞

X

k=1

∞

X

j=1

ρ^k+jE[√

n(F^∗k_n (x)−F^∗k(x))

√n(F^∗j_n(y)−F^∗j(y))]

Let us consider the expected value on the right hand side of this last expression, to obtain

E[√

n(F^∗k_n (x)−F^∗k(x))√

n(F^∗j_n(y)−F^∗j(y))] =n{E[F^∗k_n (x)F^∗j_n(y)]

−F^∗j(y)F^∗k(x)−F^∗k(x)F^∗j(y) +F^∗k(x)F^∗j(y)}

=nE[F^∗k_n (x)F^∗j_n(y)]−nF^∗j(y)F^∗k(x)

sinceF^∗k_n (x)is an unbiased estimator ofF^∗k(x). As pointed out by Frees (1986b), this estimator is aU-statistic and thus it is easy to establish that for eachk≥1and for eachx≥0,F_n^∗k(x)^a.s.→ F^∗k(x)asn→ ∞. Moreover, it is uniformly almost surely consistent, that is, for everyk≥1, as n goes to infinity

sup

x

|F_n^∗k(x)−F^∗k(x)|^a.s.→ 0 We thus obtain:

E[F^∗k_n (x)F^∗j_n(y)] =E

"

n k

−1

X

c

I_(Yi

1+...+Y_ik>x)

n j

−1

X

c

I_(Yi

1+...+Y_ij>y)

#

=E



 n

k −1n

j −1

X

c_k∈Cn,k

X

c_j∈Cn,j

I(Y_i1+...+Y_ik>x)

I_(Yi

1+...+Y_ij>y)

i

whereC_n,k is the set of the combinations ofnelements of classk andC_n,j is the set of the combinations ofnelements of classj;c_kandc_jare two subsets of {1, . . . , n}that havel≤min(k, j)elements in common. Supposek > j, then it is possible to writec_j = (c_j,k, c_j,k), wherec_j,kis the set of the elements ofc_jwhich are inc_k. In general,c_j,kis composed bylelements(l= 0,1, . . . , j). Hence, we have

E[F^∗k_n (x)F^∗j_n(y)] = n

k −1n

j −1

X

c_k∈Cn,k

j

X

l=0

X

c_j,k∈Ck,l

X

c_j,k∈Cn−k,j−l

E [I(Y_i1+...+Y_il+Y_il+1+...+Y_ik>x)I(Y_i1+...+Y_il+Y_hl+1+...+Y_hj>y)](6) wherec_j,kis a combination without replacement oflelements ofc_jandc_j,kis a combination without replacement ofj−l elements of{1, . . . , n}\c_j; moreover,

n−kis not smaller thanj−l. Using the independence ofY_i’s, we easily have the following relationship:

E[I_(Yi

1+...+Y_il+Y_il+1+...+Y_ik>x)I_(Yi

1+...+Y_il+Y_hl+1+...+Y_hj>y)]

=E[E[I_(Yi

1+...+Y_il+Y_il+1+...+Y_ik>x)I_(Yi

1+...+Y_il+Y_hl+1+...+Y_hj>y)|Y_i1, . . . , Y_il]]

=E[F^∗k−l(x−Y_i1−. . .−Y_il)F^∗j−l(y−Y_i1−. . .−Y_il)]

from which it follows that (6) takes the form n

k −1n

j −1

X

c_k∈Cn,k

j

X

l=0

X

c_j,k∈Ck,l

X

c_j,k∈Cn−k,j−l

E[F^∗k−l(x−Yi₁−. . .−Yi_l) F^∗j−l(y−Yi₁−. . .−Yi_l)]

Finally, with obvious replacements, we obtain the following expression:

E[Un(x)Un(y)] = (1−ρ)²

∞

X

k=1

∞

X

j=1

ρ^k+j

n n

k −1n

j −1

X

c_k∈Cn,k

j

X

l=0

X

c_j,k∈C_k,l

X

c_j,k∈Cn−k,j−l

E[F^∗k−l(x−Yi₁−. . .−Yi_l)

F^∗j−l(y−Yi₁−. . .−Yi_l)]−nF^∗j(y)F^∗k(x)

Now, letngo to infinity to obtain

n→∞lim E[U_n(x)U_n(y)] = (1−ρ)²

∞

X

k=1

∞

X

j=1

ρ^k+jk j

Cov(F^∗k−1(x−Y₁), F^∗j−1(y−Y₁)) (7) 2 Lemma 2Let us defineVn(x) =√

n(Ψen(x)−Ψ(x)). Under the same assumptions as in Lemma 1, for eachx >0, asn→ ∞, we obtain

n→∞lim E[Vn(x)Vn(y)] = σ²

µ²_T + µ² µ⁴_Tσ_T²

^∞ X

k=1

∞

X

j=1

f_k⁰(ρ)f_j⁰(ρ)F^∗k(x)F^∗j(y) wheref_l⁰(ρ)denotes the first derivative of the functionfl(ρ) = (1−ρ)ρ^l. Proof We have first

V_n(x) =√

n(Ψe_n(x)−Ψ(x))

=√ n

∞

X

k=1

[(1−ρ) ˆˆρ^k−(1−ρ)ρ^k]F^∗k(x)

=√ n

∞

X

k=1

[fk( ˆρ)−fk(ρ)]F^∗k(x)

wheref_k(t) = (1−t)t^k. Taking a first order Taylor expansion off_k( ˆρ)at the pointρ, we obtain the following relationship:

V_n(x) =

∞

X

k=1

[√

n( ˆρ−ρ)f_k⁰(ρ) +√

nRem_k]F^∗k(x)

withsup_x√ n|P∞

k=1RemkF^∗k(x)|→^p 0whenn→ ∞. Hence,Vn(x)possesses the same asymptotic distribution as√

n( ˆρ−ρ)P∞

k=1f_k⁰(ρ)F^∗k(x).

Let us identify the covariance:

E[V_n(x)V_n(y)] =E √

n( ˆρ−ρ)

∞

X

k=1

f_k⁰(ρ)F^∗k(x) √

n( ˆρ−ρ)

∞

X

j=1

f_j⁰(ρ)F^∗j(y)

Taking into account relation (3), we have:

E[V_n(x)V_n(y)] =E √

n 1 µT

(X_n−µ)−√ n µ

µ²_T(T_n−µ_T) ²

∞

X

k=1

f_k⁰(ρ)F^∗k(x) ∞

X

j=1

f_j⁰(ρ)F^∗j(y)

=

∞

X

k=1

∞

X

j=1

f_k⁰(ρ)f_j⁰(ρ)F^∗k(x)F^∗j(y) 1

µ²_TE[√

n(X_n−µ)]²− 2µ µ³_T

E[√

n(Xn−µ)]E[√

n(Tn−µT)] + µ² µ⁴_TE[√

n(Tn−µT)]²

Lettingngo to infinity and taking into account thatE(√

n(X_n−µ)) = 0and E(√

n(T_n−µ_T)) = 0, we obtain:

n→∞lim E[Vn(x)Vn(y)] = σ²

µ²_T + µ² µ⁴_Tσ²_T

^∞ X

k=1

∞

X

j=1

f_k⁰(ρ)f_j⁰(ρ)F^∗k(x)F^∗j(y)(8) 2 We are now in a position to establish the main results of the present paper in the following two theorems.

Theorem 2Under the same assumptions as in the previous lemmas, sup

x

|Ψn(x)−Ψ(x)|^a.s.→ 0 as n tends to infinity.

Proof First observe that sup

x

|Ψn(x)−Ψ(x)|= sup

x

∞

X

k=1

(1−ρ)ρ^k(F^∗k_n (x)−F^∗k(x))

+( ˆρ−ρ)

∞

X

k=1

f_k⁰(ρ)F^∗k(x)

≤sup

x

∞

X

k=1

(1−ρ)ρ^k(F^∗k_n (x)−F^∗k(x)) + sup

x

( ˆρ−ρ)

∞

X

k=1

f_k⁰(ρ)F^∗k(x)

Consider that

∞

X

k=1

(1−ρ)ρ^k(F^∗k_n (x)−F^∗k(x)) =

L

X

k=1

(1−ρ)ρ^k(F^∗k_n (x)−F^∗k(x))

+

∞

X

k=L+1

(1−ρ)ρ^k(F^∗k_n (x)−F^∗k(x)) Now, take into account that for every > 0, there exists L = L such that P∞

k=L+1(1−ρ)ρ^k < and write sup

x

∞

X

k=L+1

(1−ρ)ρ^k(F^∗k_n (x)−F^∗k(x))

≤

∞

X

k=L+1

(1−ρ)ρ^k sup

x

(F^∗k_n (x)−F^∗k(x))

≤

∞

X

k=L+1

(1−ρ)ρ^k

≤

On the basis of this last relationships, also taking into account that|ˆρ−ρ|^a.s.→ 0and thatsup_x

F^∗k_n (x)−F^∗k(x)

a.s.→ 0, the statement of the theorem easily follows.

2

Theorem 3Let D denote the set of right-continuous functions with left-hand limits, endowed with the Skorokhod topology. For eachx > 0,asn → ∞, the sequence of stochastic processesZn(x) =√

n(Ψn(x)−Ψ(x))converges weakly in the spaceDto a Gaussian processZ with mean function zero and covariance kernel given by (17).

Proof For the sake of clarity, the proof is split into three steps. In the first step we show that the finite-dimensional distributions of Zn(·)converge in law to a multivariate normal distribution while, in the second step, using arguments in

Harel et al. (1995), we state the weak convergence of the process. In the last step, calculations for the identification of the covariance kernel of the process are showed.

Step1 (Convergence of the finite-dimensional distributions). We just consider one-dimensional distributions, since the same reasoning applies also to the multidimensional case. We need to show that, if t is a continuity point of the distribution ofZ,

n→∞lim P(Zn ≤t) =P(Z ≤t) Just recall that

Z_n (x) =U_n(x) +V_n(x)

= (1−ρ)√ n

∞

X

k=1

ρ^k(F^∗k_n (x)−F^∗k(x)) +√

n( ˆρ−ρ)

∞

X

k=1

f_k⁰(ρ)F^∗k(x)(9)

From Lemma 2 and Theorem 1, Vn(x) = √

n( ˆρ−ρ)P∞

k=1f_k⁰(ρ)F^∗k(x) converges in law to a normal distribution with mean zero and varianceσ²_V obtained by (8) simply takingx=y. Then, we can write

P(Vn(x)≤t) =P(N(0, σ²_V)≤t) =Φ(t/σV) where, with obvious notation,Φ(x) =P(N(0,1)≤x).

Let us now study the termU_n(x). To simplify the notation, we will consider from now on, throughout this step,F^∗k(x)instead of the tail F^∗k(x), since the same arguments apply. For everyL≥1, we have that

Uen(x) = (1−ρ)

L

X

k=1

ρ^k√

n(F_n^∗k(x)−F^∗k(x))

+(1−ρ)

∞

X

k=L+1

ρ^k√

n(F_n^∗k(x)−F^∗k(x))

Once more to simplify the notation, let us takea_k = (1−ρ)ρ^k√

n(F_n^∗k(x)− F^∗k(x)). For fixedx, the event(Uen(x)≤t)can be written as

(Uen(x)≤t) = ^L

X

k=1

ak ≤t−

∞

X

k=L+1

ak

For every >0andL≥1, we will have (Uen(x)≤t) =

( ^L X

k=1

ak ≤t−

∞

X

k=L+1

ak

\

∞

X

k=L+1

ak

≤ )

[ ( ^L

X

k=1

ak ≤t−

∞

X

k=L+1

ak

\

∞

X

k=L+1

ak

>

)

The following chain of inequalities holds true:

P(Ue_n(x)≤t) =P ^∞

X

k=1

a_k ≤t

≤P ^L

X

k=1

a_k ≤t−

∞

X

k=L+1

a_k,

∞

X

k=L+1

a_k

≤

+P ^L

X

k=1

a_k≤t−

∞

X

k=L+1

a_k,

∞

X

k=L+1

a_k

>

≤P ^L

X

k=1

a_k ≤t−

∞

X

k=L+1

a_k,−≤

∞

X

k=L+1

a_k≤

+P

∞

X

k=L+1

ak

>

≤P ^L

X

k=1

ak ≤t+,−≤

∞

X

k=L+1

ak≤

+P

∞

X

k=L+1

ak

>

≤P ^L

X

k=1

ak ≤t+

+P

∞

X

k=L+1

ak

>

(10)

Let us consider the first term in the summation in the expression (10); we have



 a₁ . . . aL



=





(1−ρ)ρ√

n(F_n^∗1(x)−F^∗1(x)) . . .

(1−ρ)ρ^L√

n(F_n^∗L(x)−F^∗L(x))





that isLlinear combinations of U-statistics. The joint distribution is multinormal as an application of Hoeffding theorem (Serfling (1980)). For every L ≥ 1,

√nPL

k=1akconverges in law, asngoes to infinity, to a normal distribution with mean zero and variance

σ²_L= (1−ρ)²

L

X

k=1 L

X

j=1

ρ^k+jk jCov(F^∗k−1(x−Y1), F^∗j−1(x−Y1)) (11)

obtained from (7) by lettingy=x. We can finally state

n→∞lim P ^L

X

k=1

ak≤t+

=P(N(0, σ_L²)≤t+) =Φ((t+)/σL)

As far as the second term in expression (10) is concerned, by applying Chebyshev inequality we obtain

P

∞

X

k=L+1

ak

>

=P

∞

X

k=L+1

(1−ρ)ρ^k√

n(F_n^∗k(x)−F^∗k(x))

>

≤ 1 ²E

" ^∞ X

k=L+1

(1−ρ)ρ^k√

n(F_n^∗k(x)−F^∗k(x)) 2#

LettingT_n^k(x) =√

n(F_n^∗k(x)−F^∗k(x)), the last expression becomes

P

∞

X

k=L+1

a_k

>

≤ 1

²(1−ρ)²

∞

X

k=L+1

∞

X

j=L+1

ρ^k+jE[T_n^k(x)T_n^j(x)]

= 1

²(1−ρ)²

∞

X

k=L+1

∞

X

j=L+1

ρ^k+jσ_n^kj

where, following the same lines as in the proof of Lemma 1, we easily obtain:

σ^kj_n =n n

k −1n

j −1

X

c_k∈Cn,k

j

X

l=0

X

c_j,k∈Ck,l

X

c_j,k∈Cn−k,j−l

E[F^∗k−l(x−Y_i1−. . .−Y_il)F^∗j−l(x−Y_i1−. . .−Y_il)]

−nF^∗j(x)F^∗k(x) Letngo to infinity to obtain:

n→∞limP

∞

X

k=L+1

ak

>

≤ 1

²(1−ρ)²

∞

X

k=L+1

∞

X

j=L+1

ρ^k+jσ^kj

Hence

n→∞lim P ^∞

X

k=1

a_k≤t

≤Φ((t+)/σ_L) + 1

²(1−ρ)²

∞

X

k=L+1

∞

X

j=L+1

ρ^k+jσ^kj

Letting nowLtend to infinity andtend to zero in such a way that the quantity

1

²(1−ρ)²P∞ k=L+1

P∞

j=L+1ρ^k+jσ^kjtends to zero, we finally obtain

n→∞lim P

∞

X

k=1

ak ≤t

!

≤Φ(t/σ_∞) (12)

whereσ_∞is obtained from (11) by lettingLtend to infinity.

To prove the reverse inequality, let us consider the following relationships P

∞

X

k=1

ak≤t

=P ^L

X

k=1

ak≤t−

∞

X

k=L+1

ak

≥P ^L

X

k=1

ak≤t−

∞

X

k=L+1

ak,

∞

X

k=L+1

ak

≤

≥P ^L

X

k=1

ak≤t−,

∞

X

k=L+1

ak

≤

≥P ^L

X

k=1

ak≤t−

+P

∞

X

k=L+1

ak

≤

−1

=P ^L

X

k=1

ak≤t−

−P

∞

X

k=L+1

ak

>

Note that the last two relationships are easily obtained on the basis of the following simple consideration. LetAandBbe two sets; it is well known thatP(A∪B) = P(A) +P(B)−P(A∩B), from which we obtain

P(A∩B) =P(A) +P(B)−P(A∪B)

≥P(A) +P(B)−1 andP(A) +P(B)−1≤P(A∩B)≤min(P(A), P(B)).

As before, it is now easy to prove that

n→∞lim P

∞

X

k=1

ak ≤t

!

≥Φ(t/σ_∞) (13)

and from (12) and (13) the relationship

n→∞lim P

∞

X

k=1

ak ≤t

!

=Φ(t/σ_∞)

The same technique applies, with small changes, to prove the convergence of all finite-dimensional distributions ofZ_n(·), and this concludes Step 1.

Step2 (Weak convergence of the process). We want to state weak convergence of the process in the spaceDof right-continuous functions with left-hand limits, endowed with the Skorokhod topology. To this purpose, we follow ideas in Harel et al (1995); the same reasoning applied by these authors to the empirical renewal process may in fact be applied in this specific context.

The starting point is the well known fact that the empirical process√

n(Fn−F) converges in distribution toB^◦◦Fin the spaceD. Here,B^◦denotes the Brownian bridge and“◦⁰⁰denotes composition of functions. We want to deduce from this the corresponding result for√

n(Ψn−Ψ).

The main problem is thatZ_nis a nonlinear function of the empirical process.

The solution to such a problem is found in the following lines, which provide calculations for a linearization of the stochastic processZn(x) = √

n(Ψn(x)− Ψ(x)). Once more, let us consider the following decomposition of the process Z_n(x):

Zn(x) =Un(x) +Vn(x)

=√

n(1−ρ)ˆ

∞

X

k=1

ˆ

ρ^k(F^∗k_n (x)−F^∗k(x))

+√ n

∞

X

k=1

[(1−ρ) ˆˆ ρ^k−(1−ρ)ρ^k]F^∗k(x)

We just take in consideration the first term in the summation, since the proof of weak convergence of the second term is trivial. Moreover, to simplify the notation, let us considerF^∗kinstead ofF^∗kand, therefore,

Ue_n=√ n

∞

X

k=1

(1−ρ) ˆˆρ^k(F_n^∗k−F^∗k)

=√

n(F_n−F)∗

∞

X

k=1

(1−ρ) ˆˆρ^k(F_n^∗k−1+F_n^∗k−2∗F +· · ·+F_n∗F^∗k−2+F^∗k−1)

=√

n(Fn−F)∗

∞

X

k=1

(1−ρ) ˆˆρ^kF_n^∗k∗

∞

X

k=1

(1−ρ) ˆˆρ^kF^∗k

=√

n(Fn−F)∗Ψn∗Ψen (14)

=√

n(Fn−F)∗Ψen∗Ψen+√

n(Fn−F)∗(Ψn−Ψen)∗Ψen

Observe that, by (14),

√n(Ψ_n−Ψe_n) =Ue_n=√

n(F_n−F)∗Ψ_n∗Ψe_n We easily obtain:

Uen =√

n(Fn−F)∗Ψen∗Ψen+√

n(Fn−F)∗Ψn∗Ψen∗(Fn−F)∗Ψen

=√

n(Fn−F)∗Ψen∗Ψen+√

n(Fn−F)∗(Fn−F)∗Ψn∗Ψen∗Ψen

LetG=Ψe_n∗Ψe_n. We finally obtain:

Ue_n=√

n(F_n−F)∗G+√

n(F_n−F)^∗2∗Ψ_n∗G

We have decomposed the process into two terms: a first term which is linear in the empirical process and a second term which can be shown to be negligible as n → ∞, simply following the same lines as in Harel et al. (1995). By Theorem 3.2 in Harel et al (1995), we can definitely state weak convergence of the process.

We do not provide here a detailed proof since it proceeds in the same way as the proofs of Theorem 3.2 and Theorem 4.1 in Harel et al (1995).

Step3 (Identification of the covariance kernel). To complete the proof of the theorem, we just need to evaluate the covariance kernel of the stochastic process.

It is awkward and we need some other preliminary notations to make it readable.

Observe that

E[Zn(x)Zn(y)] =E[(Un(x) +Vn(x))(Un(y) +Vn(y))]

=E[Un(x)Un(y)] +E[Un(x)Vn(y)]

+E[Vn(x)Un(y)] +E[Vn(x)Vn(y)] (15) and consider each of the four components separately. We have already computed the first and the last term on the right hand side of (15) (see Lemma 1 and Lemma 2); let us computeE[Un(x)Vn(y)]

E[Un(x)Vn(y)] =E

(1−ρ)

∞

X

k=1

ρ^k√

n(F^∗k_n (x)−F^∗k(x))

√ n

1

µ_T(Xn−µ)− µ

µ²_T(Tn−µT) ^∞

X

j=1

f_j⁰(ρ)F^∗j(y)

= 1

µ_T(1−ρ)

∞

X

k=1

ρ^k

∞

X

j=1

f_j⁰(ρ)F^∗j(y)E[√

n(F^∗k_n (x)−F^∗k(x))

√n(Xn−µ)]− µ

µ²_T(1−ρ)

∞

X

k=1

ρ^k

∞

X

j=1

f_j⁰(ρ)F^∗j(y)

×E[√

n(F^∗k_n (x)−F^∗k(x))√

n(Tn−µT)] (16) Taking into account thatE[√

n(F^∗k_n (x)−F^∗k(x))√

n(Tn −µT)] = 0, we obtain that (16) is equal to

(1−ρ) µT

∞

X

k=1

∞

X

j=1

ρ^kf_j⁰(ρ)F^∗j(y)E[√

n(F^∗k_n (x)−F^∗k(x))√

n(Xn−µ)]

The computation of this last expression is not difficult. First of all, we have E[√

n(F^∗k_n (x)−F^∗k(x))√

n(Xn−µ)] =nE[F^∗k_n (x)Xn−F^∗k_n (x)µ

−F^∗k(x)Xn+F^∗k(x)µ]

=nE[F^∗k_n (x)Xn]−nµF^∗k(x) Furthermore

E[F^∗k_n (x)X_n] =E



 n

k −1

X

c_k∈Cn,k

I_(Yi

1+...+Y_ik>x)

1 n

n

X

j=1

X_j





= 1 n

1

n k

E





n

X

j=1

X_j X

c_k∈Cn,k

I_(Yi

1+...+Y_ik>x)





SinceCn,k=C_n,k^j SC_n,k^j , whereC_n,k^j is the set of all ⁿ⁻¹_k−1

combinations of nelements of classkcontainingjandC_n,k^j is the set of all ⁿ⁻¹_k

combinations ofnelements of classkthat does not containj, we obtain:

E[F^∗k(x)Xn] = 1 n

1

n k

E







n

X

j=1

Xj





 X

c_k∈C_n,k^j

I_(Yi

1+...+Y_ik−1+Yj>x)

+ X

c_k∈C_n,k^j

I(Y_i1+...+Y_ik−1+Y_ik>x)













= 1 n

1

n k

n

X

j=1





 X

c_k∈C_n,k^j

E[XjI_(Yi

1+...+Y_ik−1+Yj>x)]

+ X

c_k∈C_n,k^j

E[XjI(Y_i1+...+Y_ik−1+Y_ik>x)]







= 1 n

1

n k

n

X

j=1

n−1 k−1

E[X_jF^∗k−1(x−Y_j)] + n−1

k

µF^∗k(x)

We can definitely state that

E[Un(x)Vn(y)] = (1−ρ) µT

∞

X

k=1

∞

X

j=1

ρ^kf_j⁰(ρ)F^∗j(y) ( 1

n k

n

X

h=1

n−1 k−1

E[XhF^∗k−1(x−Yh)]

+ n−1

k

µF^∗k(x)

−nµF^∗k(x)

By adding all components together in (15), we obtain the following expression:

E[Z_n(x)Z_n(y)] = (1−ρ)²

∞

X

k=1

∞

X

j=1

ρ^k+j

n n

k −1n

j −1

X

c_k∈Cn,k

j

X

l=0

X

c_j,k∈Ck,l

X

c_j,k∈Cn−k,j−l

E[F^∗k−l(x−Yi₁−. . .−Yi_l)

F^∗j−l(y−Yi₁−. . .−Yi_l)]−nF^∗j(y)F^∗k(x)

+

∞

X

k=1

∞

X

j=1

f_k⁰(ρ)f_j⁰(ρ)F^∗k(x)F^∗j(y)

× 1

µ²_TE[√

n(X_n−µ)]²+ µ² µ⁴_TE[√

n(T_n−µ_T)]²

+2

∞

X

k=1

∞

X

j=1

(1−ρ)ρ^k 1

µ_Tf_j⁰(ρ)F^∗j(y) ( 1

n k

n

X

h=1

n−1 k−1

E[XhF^∗k−1(x−Yh)] + n−1

k

µF^∗k(x)

−nµF^∗k(x)

Now, letngo to infinity to finally obtain the following asymptotic expression:

E[Z(x)Z(y)] = (1−ρ)²

∞

X

k=1

∞

X

j=1

ρ^k+jk j Cov(F^∗k−1(x−Y1), F^∗j−1(y−Y1))

+ σ²

µ²_T + µ² µ⁴_Tσ²_T

^∞ X

k=1

∞

X

j=1

f_k⁰(ρ)f_j⁰(ρ)F^∗k(x)F^∗j(y)

+2 1 µT

(1−ρ)

∞

X

k=1

∞

X

j=1

ρ^kf_j⁰(ρ)F^∗j(y)

×Cov(X1, F^∗k−1(x−Y₁)) (17) 2 Remark 5.The estimator for the probability of ruin described in Section 2 has been constructed by replacing the truek-fold convolution of the claim size d.f.Fby the estimator introduced by Frees (1986b). Instead of doing this, one could estimate Fby the empirical d.f.F˜_n^∗1(x) =F_n^∗1(x)and then define recursively estimates of F^∗k(x)by the relationship

F˜_n^∗k(x) =

Z F˜_n^∗k−1(x−u)dF˜_n^∗1(u)

that is considering the k-fold convolution of the empirical d.f.. As observed by Frees, although F˜_n^∗k(x) is a biased estimate of F^∗k(x), it has the possible advantage of being the nonparametric maximum likelihood estimator.

Furthermore, it is a V-statistic and thus it is closely related to the U-statistic F_n^∗k(x). Remind that U-statistics and V-statistics have the same asymptotic properties.

3. Quantile estimation

In this section, we propose an estimator for thep-th quantile of the ruin probability function. For0≤p <1, define thep-th quantileξpofΨ(·)as

ξ_p=Ψ⁻¹(p) = inf{x≥0 :Ψ(x)≥p}

A natural estimateξb_pofξ_pcan be defined as

ξb_p=Ψb_n⁻¹(p) = inf{x≥0 :Ψ_n(x)≥p}

The main asymptotic properties of the proposed estimator are presented in the following two theorems.

Theorem 4Assume thatξp,0≤p < 1, is uniquely defined, i.e. for every >0, Ψ(ξp+)< Ψ(ξp) =p < Ψ(ξp−). Then, asn→ ∞

ξbp a.s.→ ξp

Proof Ψ(x)is a decreasing function and the equationΨ(x) =ppossesses a single root, namelyξ_p. From the uniqueness ofξ_p, for every >0

δ= min{p−Ψ(ξp+), Ψ(ξp−)−p}>0 We note that

P r{Ψn(ξp−)−p >0}=P r{Ψ(ξp−)−Ψn(ξp−)< Ψ(ξp−)−p}

≥P r{Ψ(ξ_p−)−Ψ_n(ξ_p−)< δ}

≥P r{|Ψn(ξp−)−Ψ(ξp−)|< δ} (18) Taking into account thatΨn is a consistent estimator ofΨ, for everyδ >0, (18) converges to 1 asn→ ∞. Therefore

P r{Ψn(ξp−)−p >0} →1 (n→ ∞) In the same way we can show that

P r{p−Ψ_n(ξ_p+)<0} →0 (n→ ∞) Then, asn→ ∞

P r{Ψn(ξp+)< p < Ψn(ξp−)} →1 ∀ >0 which implies that

P r{ξp+ >ξbp> ξp−} →1 asn→ ∞,∀ >0.2