Limiting behaviour of a geometric-type estimator for tail indices
Introduction
Let Z1,Z2,… be independent, non-negative random variables with common distribution function (d.f.) F satisfying where r is a regularly varying function at infinity and R a positive constant. Denoting by F−1 the left continuous inverse of F, i.e., F−1(s)≔inf{x:F(x)≥s}, (1) is equivalent to where L̃ is a slowly varying function at zero (see, e.g. Schultze and Steinebach, 1996 and references therein).
We shall be concerned here with the estimation of the tail coefficient R in (1) or, equivalently, in (2). The problem of estimating R or other related tail indices has received considerable attention and common applications may be found in a big variety of domains. We consider here an important application in risk theory, namely the estimation of the adjustment coefficient (see Csörgő and Steinebach, 1991). For a comprehensive overview of this subject we refer to Csörgő and Viharos (1998).
Based on least squares considerations, Schultze and Steinebach (1996) proposed three estimators for the exponential tail coefficient R, given as follows. Let Z1,n≤Z2,n≤⋯≤Zn,n denote the order statistics of the sample Z1,Z2,…,Zn and assume that (kn) is a sequence of positive integers satisfying The Schultze and Steinebach estimators are defined by and Recently, Brito and Moreira (2001) have introduced a new estimator of R, , related to and . This estimator arises in a natural way from a geometrical adaptation of the procedure used by Schultze and Steinebach in the construction of , i=1,3. These estimators are motivated by the fact that, for large z, −log(1−F(z)) is approximately linear with slope R, since as z→∞. If the regularly varying function r was constant, say r(z)=ed, d∈R, then We thus expect that the above linear relation approximately holds for the largest observations realized in the sample (Z1,Z2,…,Zn), which we simply denote by z(i)=zn−i+1,n, i=1,…,kn. Approximating F(z(i)) by Fn(z(i)−), where Fn is the empirical d.f., this gives that −log(1−Fn(z(i)−))=log(n/i) is “close” to Rz(i)−d, or z(i) is “close” to , i=1,…,kn. Setting a=R−1 and b=R−1d, a least squares estimator may then be obtained by minimizing , leading to the estimator . In the particular case where d=0, the minimization of f2(a)=f1(a,0) yields the estimator . On the other hand, the direct minimization of f3(R,d)=∑i=1kn(log(n/i)−Rz(i)+d)2, leads to the least squares estimator .
Considering the two points of view simultaneously, by minimizing the global sum of the areas of the rectangles indicated in Fig. 1, we obtain the estimator .
In this way, results from minimizing , and is given by the geometric mean of and , that is Schultze and Steinebach (1996) established the consistency of the estimators , i=1,2,3 and their corresponding asymptotic behaviour was subsequently investigated by Csörgő and Viharos (1997). Independent of these authors, Kratz and Resnick (1996) introduced an equivalent form of , designated by qq-estimator, in reference to the quantile–quantile plots (for this interpretation and application of qq-plots in this estimation problem, see also Beirlant et al., 1996). Kratz and Resnick proved the consistency and the asymptotic normality of the qq-estimator centred at 1/R. Not forcing the centring at 1/R, Csörgő and Viharos (1997) have shown that, for suitable sequences (kn), , i=1,2,3, are universally asymptotically normal over the family (1), in the usual sense, that is, with deterministic centring sequences converging to 1/R. Moreover, for , i=1,3, the norming sequence is kn1/2, and as Csörgő and Viharos (1997) pointed out, these were the first estimators asymptotically normal over the whole family (1), with the ideal factor kn1/2.
The above estimation problem is equivalent to the estimation of the tail index of a Pareto type distribution. In fact, setting Xi=eZi with Zi, i=1,2,… as above, we have where α=1/R and is slowly varying at infinity. The qq-estimator was actually introduced under (7). In this context, several estimators have been proposed. One of the most commonly used estimators for α, is the Hill estimator (1975), defined by where X1,n≤X2,n≤⋯≤Xn,n denote the order statistics of the sample X1,X2,…,Xn (for related estimators, see, e.g. De Haan and Resnick, 1980, Csörgő et al., 1985, Bacro and Brito, 1993). The asymptotic properties of the Hill estimator have been much studied and it is well known that, under certain conditions, Hn(kn) is a strongly consistent estimator (cf. Deheuvels et al., 1988) with asymptotic normal distribution (cf. Haeusler and Teugels, 1985).
In this paper, we investigate the asymptotic properties of the geometric-type estimator . In particular, we shall give conditions which ensure the asymptotic normality of when centred at R. We shall also see that is universally asymptotically normal over the family (1). We recall that this property is not shared by the Hill estimator (see, e.g. Csörgő and Viharos, 1998). Moreover, the norming sequence is again the ideal factor kn1/2. This specific property, jointly with the fact that takes values between those of and , makes the use of the estimator specially attractive for the case where R is expected to be small. The application in risk theory considered here is of this kind. Our results are given in Section 2 and the proofs are collected in Section 3. The application in the estimation of the adjustment coefficient is discussed in Section 4. One complex practical problem is the choice of the number of observations included in the estimation of R. We consider here an heuristic method suggested by Schultze and Steinebach (1996) and adapt it to our estimator . This procedure is applied in a small-scale simulation study and the corresponding results are contained in Section 5.
Section snippets
Results
We begin by considering the consistency of the estimator defined by (6). In the sequel, and stand, respectively, for convergence and equality in distribution. In the same way, denotes convergence in probability. Theorem 1 Assume that F satisfies condition (1) and kn is a sequence of positive integers satisfying (3) and such that . If F−1 is continuous on (s0,1) for some s0∈(0,1), then,
As noted in Section 1, is the geometric mean of the estimators
Proofs
Throughout this section we shall assume that (1) holds. We assume also that U1,U2,… is a sequence of independent uniform U(0,1) random variables. The order statistics of the sample (U1,U2,…,Un) are denoted by U1,n≤U2,n≤⋯≤Un,n. Proof of Theorem 1 Schultze and Steinebach (1996) proved that if kn satisfies (3) and as n→∞, then is a consistent estimator of R. Moreover, if F−1 is continuous on (s0,1) for some s0∈(0,1), then is also a consistent estimator of R. Thus, since
Estimating the adjustment coefficient in risk theory
The problem of estimating the coefficient R in Eq. (1) is motivated by an important problem in risk theory. Consider the Sparre Andersen model for claims arriving at an insurance company, and assume that the sequence C1,C2,… of claims occur at times T1,T1+T2,…, where {Ci} and {Ti} are independent sequences of i.i.d. r.v.’s. Starting with initial capital x and with incoming premiums in the time interval [0,t] equal to γt, the risk reserve is where N(t)=max{n≥0:∑i=1nTi≤t} is
Simulation results
Below we extend the simulation study of Schultze and Steinebach (1996) to the estimator , where samples Z1,Z2,…,Zn have been simulated making use of the exact distribution F of the above example, or more precisely, its quantile function where a=β/α<1. For sake of comparison with related studies (see Schultze and Steinebach, 1996 and references therein) we take , resulting in R=5.8(3)×10−5.
In this section we illustrate the
Acknowledgements
The research of the second author was partially supported by PRODEP III, Action 5.3. The authors also thank the referee for his comments.
References (18)
- et al.
A tail bootstrap procedure for estimating the tail Pareto index
Journal of Statistical Planning and Inference
(1998) - et al.
On the estimation of the adjustment coefficient in risk theory via intermediate order statistics
Insurance: Mathematics and Economics
(1991) - Bacro, J.N., Brito, M., 1993. Strong limiting behaviour of a simple tail Pareto-index estimator. Statistics and...
- et al.
Tail estimation, Pareto quantile plots, and regression diagnostics
Journal of the American Statistical Association
(1996) - Bingham, N.H., Goldie, C.M., Teugels, J.L., 1987. Regular Variation. Cambridge University Press,...
- Brito, M., Moreira, A.C., 2001. Estimação do Coeficiente de Cauda Exponencial. In: Oliveira, P., Athayde, E. (Eds.), Um...
- et al.
Asymptotic normality of least squares estimators of tail indices
Bernoulli
(1997) - Csörgő, S., Viharos, L., 1998. Estimating the tail index. In: Szyszkowicz, B. (Ed.), Asymptotic Methods in Probability...
- et al.
Kernel estimates of the tail index of a distribution
The Annals of Statistics
(1985)
Cited by (11)
On tail index estimation using a sample with missing observations
2012, Statistics and Probability LettersConsistent estimation of the tail index for dependent data
2010, Statistics and Probability LettersEdgeworth expansion for an estimator of the adjustment coefficient
2008, Insurance: Mathematics and EconomicsCitation Excerpt :The general conditions ensuring the asymptotic normality are stated in the theorem below. This result follows directly from Proposition 1 of Brito and Freitas (2003). Now consider the normalized estimator
Weak convergence of a bootstrap geometric-type estimator with applications to risk theory
2006, Insurance: Mathematics and EconomicsThe climate niche of Homo Sapiens
2023, arXivTesting the Dismal Theorem
2022, Journal of the Association of Environmental and Resource Economists