On robust tail index estimation for linear long-memory processes

Transcription

1 On robust tail index estimation for linear long-memory processes Jan Beran 1, Bikramjit Das 2 and Dieter Schell 1 1 Department of Mathematics and Statistics, University of Konstanz, Germany 2 RiskLab, Department of Mathematics, ETH Zürich, Switzerland September 5, 2011 Abstract We consider robust estimation of the tail index α for linear longmemory processes X t = j=0 a jϵ t j with i.i.d. innovations ϵ j following a symmetric α-stable law (1 < α < 2) and coefficients a j c j β ( 1 α < β < 1). Estimates based on the left and right tail respectively are obtained together with a combined statistic with improved efficiency, and a test statistic comparing both tails. Asymptotic results are derived. Simulations illustrate the finite sample performance. Keywords: long memory, infinite variance, tail index, M estimation 1 Introduction Long memory and distributions with infinite moments are known to play an important role in many areas of application. There is an extended literature on both topics (see e.g. Samorodnitsky and Taqqu 1994, Beran 1994, Embrechts et al. 1997). However, results on statistical inference for processes 1

2 where both properties occur simultaneously are scarce. Among the most significant contributions in this direction are limit theorems for the empirical distribution function, M-estimators of location and regression (see Koul and Surgailis 2001, Zhou and Wu 2010 and references therein), and asymptotic results for estimators of the long-memory parameter (Kokoszka and Taqqu 1996, Stoev et al. 2002). The present paper grew out of the need to close a further important gap, namely the estimation of the tail index α in the long memory context. The results presented here provide a first step towards a general solution. Many more questions remain and will need to be answered in future. Some of these are discussed briefly in the final section of the paper. For some recent work on tail index estimation for long memory stochastic volatility models and LARCH models, see Jach and Kokoszka (2010), Kulik and Soulier (2011). Consider a time series X t (t Z) generated by a linear process of the form X t = a j ϵ t j (1) j=0 where the innovations ϵ i are iid symmetric α-stable (SαS) random variables with characteristic function Ee itϵ 1 = e t α for some 1 < α < 2, and the weights a j are such that a j D 0 j β (2) as j, with D 0 0 and 1 α < β < 1. (3) Here means that the ratio of both sides converges to one. By definition, X t is a long-memory moving average process with infinite variance, since E X 0 α = and E X 0 r < if r < α. The condition on β makes sure that the process is well defined (see e.g. Kokoszka and Taqqu 1995, Samorodnitsky and Taqqu 1992 and references therein). A typical example is a stationary solution of a FARIMA(p, d, q) process with SαS innovations. Our interest is in estimating the tail index α when we observe X 1, X 2,..., X n. Note that this is also the tail index of the process X t itself. The limiting behavior of the empirical distribution function F n (x) = 1 n n I(X t x) (4) 2

3 and of sums S n (g) = n g(x t ) (5) with g satisfying some regularity conditions has been studied in Koul and Surgailis (2001) and Surgailis (2004). The results for α < 2 differ sharply from the case of a finite variance (α = 2). While for α = 2, the asymptotic distribution of both statistics is characterized by a Hermite process (in the simplest case, fractional Brownian motion; see e.g. Taqqu 1975, 1979, Dobrushin and Major 1979, Surgailis 1982, Dehling and Taqqu 1989, Giraitis and Surgailis 1999, Ho and Hsing 1996, Surgailis 2000, Koul and Surgailis 2002, Csörgő and Kulik 2008, Kulik 2009; also see references in Beran 1994), a SαS law with infinite variance is obtained for 1 < α < 2 (Koul and Surgailis 2001, Surgailis 2004). Note that this is true even if g is a bounded function (such as for example in F n ). The results in Koul and Surgailis (2001) are obtained under the general assumption that the innovations ϵ i have a distribution function G that belongs to the domain of attraction of a SαS law with 1 < α < 2. In the following, we work under similar assumptions but, for simplicity of presentation as well as ease of calculation of certain functionals, we restrict ourselves to the case where G is exactly a standard SαS law. The general case where G is in the domain of attraction of a SαS law is under investigation. Observe that, if the distribution G of ϵ 1 is (standard) SαS, then with lim x x α G(x) = lim x α (1 G(x)) = c α (6) x c α = sin( πα 2 )Γ(α) π = 1 α 2Γ(2 α) cos( πα 2 ) (7) (see Nolan 2011, Theorem 1.12; Samorodnitsky and Taqqu 1994). From (2) and (3) we also have a j =, A := j=0 a j α <. (8) Clearly X t is well-defined (in the sense of convergence in probability) with marginal distribution F satisfying j=0 lim x x α F (x) = lim x α (1 F (x)) = c α A. (9) x 3

4 Koul and Surgailis (2001) proved that if G is in the domain of attraction of a SαS distribution, G is twice differentiable and there exists a constant C < such that for all x, y R with x y < 1, then G (x) C(1 + x ) α, (10) G (x) G (y) C x y (1 + x ) α, (11) n β 1/α (F n (x) F (x)) D(R) c(α)f(x)z (12) where Z is a SαS r.v. with Ee itz = e t α (t R), f(x) = F (x) and Γ(2 α) cos(απ/2) 1 1 α 1/α c(α) = D 0 2D 1 (t s) β + dt ds, (13) 1 α D 1 = lim x x α G(x) = lim x α (1 G(x)). (14) x Here denotes weak convergence and D(R) denotes weak convergence of random processes in the space of càdlàg functions on R = [, ] under the sup-norm metric. Also, throughout the paper we use the notation x + = max (0, x) and x = min (0, x). If G is exactly SαS, then D 1 = c α as defined in (7). Hence (13) reduces to c(α) = D (t s) β + dt α ds 1/α. (15) Furthermore, the stable distribution is infinitely differentiable (Nolan 2011, Theorem 1.9) and (10) and (11) can be verified by an asymptotic expansion of the stable density (Ibragimov and Linnik 1971). Note that the formula for c(α) as given in equation (1.10) of Koul and Surgailis (2001) contains a typo, namely the α-th power for the inner integral is missing. The correct version can be obtained from Astrauskas (1983), Avram and Taqqu (1986) and Zhou and Wu (2010). The paper is organized as follows. M-estimators Tn and T n + of α based on the left and right tail respectively are defined in section 2. Their asymptotic distribution is derived in section 3. An improved estimator combining Tn and T n + is considered in section 4, together with a test for equality of the left and right tail index. The paper concludes with simulations illustrating the asymptotic results and a brief discussion of some open problems. Proofs are given in the Appendix. 4

5 2 Estimation of α Under the assumptions above, the marginal distribution of X t is symmetric α-stable with scale parameter γ = A 1/α. We will use the notation X t S α (0, γ = A 1/α, 0). This also means that X t /A 1/α is (standard) SαS and we often write X t /A 1/α SαS. It is well-known that for a SαS random variable Y, we have P (Y > x) (constant)x α (x ) (16) so that the conditional distribution given Y > c is approximated by a Pareto law for c large enough. A natural estimate of α can therefore be based on a Huberized Pareto score function ψ v,u(x, + α) = [α log(x) 1] u v 1 {x>0} E ( [α log(x t ) 1] u v 1 ) {X t >0} (17) = [α log(x) 1] u v 1 {x>0} C(α, v, u). (18) Here [y] u v = max (v, min (y, u)). The reason for huberization is that the Pareto distribution is only an approximation of the true (marginal) distribution function, not only if the innovations are exactly SαS but also for many other distributions that may occur. By huberizing, the estimate becomes robust against a large class of deviations from this approximation. Since deviations are mainly relevant in the center of the distribution (i.e. far away from the tails), the lower truncation parameter v is more important. For a detailed discussion of these issues see Beran and Schell (2010) who consider the iid case (also see Huber 1981 and Hampel et al for general literature on robust inference). Note also that most estimators discussed in the literature are essentially based on the Pareto score function applied to k upper order statistics with k/n 0. This includes in particular various versions of Hill s estimator (Hill 1975). In the iid and weak dependence setting, such estimators are usually consistent with ˆα α = O p (k 1/2 ) and asymptotically normal (Mason 1982, Hall 1982, Davis and Resnick 1984, Csörgö and Mason 1985, Csörgö et al. 1985, Haeusler and Teugels 1985, Deheuvels et al. 1988, Csörgö and Viharos 1997, de Haan and Peng 1998 and de Haan and Resnick 1998, Hsing 1991, Resnick and Starica 1995, 1998, Dekkers et al. 1989, Rootzen et al. 1998, Drees 2000). A major difficulty is a suitable choice of k for a given data set. An inappropriate value of k can lead to completely misleading estimates (see e.g. Resnick 1997, Embrechts et al. 1997). Asymptotically optimal choice of k is considered for instance in Hall and Welsh (1984), Hall (1990), Csörö et al. (1985), Drees (1998) and Beirlant et 5

6 al. (2006). For an overview on tail index estimation in the iid setting see e.g. de Haan and Peng (1998) (also Resnick 1997, 2007). As an alternative to Hill-type estimation, Beran (1997) proposed to use all data but huberize the Pareto score function at lower quantiles. This method has been investigated in the iid setting in Beran and Schell (2010). The constant C(α, v, u) is defined for any choice of 1 < α < 2 and v < u. In particular, for v = and u = we have where and C(α,, ) = C e(1 α) + log A 1 2 E log X + t = E log X t = C e 2 C e = lim n ( n i=1 (19) ( ) 1 α log A (20) 2α ) 1 i log n = (21) is the Euler constant (Zolotarev 1986, p. 215). An estimator T n + of α can now be defined as a functional of the empirical distribution function F n by setting T n + = [τ 0 ] 2 1 where τ 0 solves the equation λ + F n (τ) = n ψ v,u(x + t, τ) = 0, (22) see Beran and Schell (2010). The restriction of the T n + guarantees a feasible estimate for α (1, 2). The corresponding functional of F is a solution τ 0 of the equation λ + F (τ) = ψ v,u(x, + τ)df (x) = 0. (23) R Clearly if F S α (0, A 1/α, 0), then τ 0 = α is a solution. By symmetry, E ( [α log(x t ) 1] u v 1 {X t >0}) = E ( [α log( Xt ) 1] u v 1 {X t <0}) = C(α, v, u) (24) and an analogous function ψ v,u(x, α) = [α log( x) 1] u v 1 {x<0} E ( [α log(x t ) 1] u v 1 {X t <0}), = [α log( x) 1] u v 1 {x<0} C(α, v, u) (25) 6

7 can be defined for the left tail. Thus, we define the left tail estimator for α as Tn = [τ 0 ] 2 1 where τ 0 solves the equation n λ F n (τ) = ψv,u(x t, τ) = 0. (26) Note that ψ v,u(x, α) = ψ + v,u( x, α). 3 Asymptotic distribution of T n and T + n Denote by H = 1 β + 1 the self-similarity parameter. The function α ψ+ v,u is of bounded variation (as a function of x), continuous and E [ ψ v,u(x + t, α) ] = 0. Results in Koul and Surgailis (2001) imply the limit n n H ψ v,u(x + t, α) c(α)h + Z α, (27) where Z α is a standard SαS random variable, c(α) is as defined in (15) and h + = ψ v,u(x, + α)f (x)dx = f(x)dψ v,u(x, + α) = f(x) x ψ+ v,u(x, α)dx. R R (28) Since ψ + v,u(x, α) = ψ v,u( x, α) and f (x) is an odd function, we also have h = ψ v,u(x, α)f (x)dx = h + and n H n ψv,u(x t, α) c(α)h + Z α (29) i=1 where Z α is the same random variable as for ψ v,u. + Note furthermore that ψ v,u + and ψv,u are continuously differentiable with respect to α almost everywhere. Using the notation ψ v,u(x, ± α) = / α ψ v,u(x, ± α) for the derivative with respect to α we define [ ] µ + (α) = E ψ v,u(x + t, α) = ψ v,u(x, + α)f(x)dx (30) and R [ ] µ (α 0 ) = E ψ v,u(x t, α) = µ + (α). (31) The asymptotic distribution of T n and T + n is given as follows. 7 R

8 Theorem 1 Suppose {X t, t Z} is a linear process as defined in (1), (2) and (3) where ϵ 1 SαS for some α = α 0. Moreover, let v >, define T n + = [τ n + ] 2 1 and T n = [τn ] 2 1 where τ n + and τn are solutions of (22) and (26) respectively and let H = 1 β + α0 1. Then, as n, T n + P α 0 and n 1 H (T + n α 0 ) h+ c(α 0 ) µ + (α 0 ) Z α 0 (32) where c(α 0 ) is given by (15) and Z α0 is a SαS random variable with α = α 0. Similarly, Tn P α 0 and where Z α0 n 1 H (T n α 0 ) h+ c(α 0 ) µ + (α 0 ) Z α 0 (33) is the same random variable as in (32). Finally, n 1 H (T n + T + n ) P 0. (34) 4 An improved estimator and testing equality of tails Under the assumption that X t is generated by a linear process with SαS innovations, Tn = [τn ] 2 1 and T n + = [τ n + ] 2 1 estimate the same parameter α 0. Since on the average each of these statistics uses about half of the data only, an obvious idea for obtaining an improved estimate of α 0 is to take their average, in some sense, i.e. to define ˆα = [ τ n + τ + n 2 This is asymptotically equivalent to setting M estimator τ 1,n solves the equation 2 n λ Fn (τ) = ] 2. (35) 1 [ ] 2 ˆα = T1 2,n = τ 1 2,n where the 1 ψ v,u (X t, τ) = 0 (36) with ψ v,u = ψ v,u+ψ + v,u. The problem with this definition is that the coefficient h in the limit theorem for ψ v,u is equal to h = h + h + = h + + h + = 0. (37) 8

9 Thus, n H ψ v,u (X t, α) converges to zero in probability. Since at the same time µ(α) = E[ ψ v,u (X, α)] = µ (α) + µ + (α) 0, this implies that T 1,n has a 2 faster rate of convergence. This is of course a desirable feature, because T 1 2,n is much more accurate than the individual estimators Tn and T n +. However, up to date, the limiting distribution of sums where h is zero is not known in the setting of long memory with infinite variance (see also remark 2.2 in Koul and Surgailis 2001). Nevertheless, an improved estimator with known asymptotic distribution can be obtained by taking a different convex combination of τ n and τ + n. Thus, for κ [0, 1] \ { 1 2}, define Tκ,n = [τ κ,n ] 2 1 where τ κ,n solves the equation and λ κ,fn (τ) = n ψ κ;v,u (X t, τ) = 0 (38) ψ κ;v,u (x, t) = κψ v,u(x, t) + (1 κ) ψ + v,u(x, t). (39) Note that for κ {0, 1} we obtain T n + and Tn respectively. The constant [ ] µ(α) = E ψ κ;v,u (X t, α) = µ + (α) (40) is the same as before, but h = ψ κ;v,u (x, α)f (x)dx = κh + (1 κ) h + = (1 2κ) h + (41) R is smaller (in absolute value) for κ / {0, 1} by the factor 1 2κ. In the limit we have n 1 H (T κ,n α 0 ) δ κ Z α0 (42) with δ κ = (1 2κ) h+ c(α 0 ) µ + (α 0 ). Thus, for κ / {0, 1} the limiting distribution is more concentrated around zero than for κ {0, 1}. Estimating α for the left and right tail separately can also be used for testing the null hypothesis that the tail index is the same on both sides. Thus, suppose that the observed series can be written as X t = X + 1,t X 2,t 9

10 where X 1,t = a j ϵ 1,t j, X 2,t = j=0 a j ϵ 2,t j (43) are linear long memory processes with a j s satisfying (2) and (3), ϵ 1,i are i.i.d. SαS with α = α 1 and ϵ 2,i are i.i.d. SαS with α = α 2. A natural statistic for testing whether α 1 is equal to α 2 is given by T = T + n T n. The asymptotic distribution of T depends on the joint distribution of X 1,t and X 2,t. The following theorem considers two exemplary cases. Theorem 2 1. Suppose that H 0 : X 1,t = X 2,t (t Z) holds with α = α 1. Then ( ) 2h n 1 H (T n + Tn + c(α 1 ) ) Z µ + α1 (44) (α 1 ) j=0 where Z α1 is a SαS random variable with α = α If H 0 : α 1 = α 2 holds with {X 1,t ; t Z} independent of {X 2,t ; t Z}, then ( 2 n 1 H (T n + Tn 1/α 1 ) h + c(α 1 ) ) Z µ + α1, (45) (α 1 ) where Z α1 is a SαS random variable with α = α 1. 5 Simulation results and open problems A small simulation study was carried out to compare the finite sample performance of T κ,n with the asymptotic results. To preserve comparability, for each parameter setting the coefficients a j were defined such that A = aj α = 1. This means that the marginal distribution of each of the simulated linear processes is exactly (standard) SαS. Before discussing the simulation results in detail, a few general remarks should be made on what can be expected. The natural parameter range for α is the open interval (1, 2). Values of α 1 are excluded, because no such linear process with long-range dependence exists. An extension of hyperbolic tail behavior to α 2 would be possible in principle, but is somewhat arbitrary and implies different limiting properties. Asymptotically, the restriction to the open interval (1, 2) does not cause any problems. However, for finite samples, the event that the root of equation (38) either does not exist or is obtained at the border 10

11 {1, 2} can occur with positive probability. For simulations where this happens ˆα = T κ,n is classified as missing value, i.e. not defined. The simulated standardized values Z n = n 1 H T κ,n α 0 δ κ are therefore either in the range z low = n 1 H 1 α 0 δ κ Z n n 1 H 2 α 0 δ κ = z up or classified as missing. Therefore simulated distributions of Z n have to be compared with the conditional distribution of a SαS variable Z α0, given the condition z low < Z α0 < z up. Asymptotically, the limits z low and z up tend to, however the rate at which this happens is quite slow for large values of H. For instance, consider α 0 = 1.5, v = 1, u = and d = 0.2 (and hence H 0.87). Then the bounds for Z n are z up = z low 4.92 for n = 100 and even for n = the statistic Z n cannot exceed z up This means for example that the two-sided 0.99 quantile q = of the limiting SαS distribution cannot be reached. For the conditional asymptotic distribution, we obtain instead two-sided 0.99 quantiles of 4.38 for n = 100 and 8.09 for n = respectively. Tables 1 and 2 display simulation results for X t defined by (1) with FARIMA(0, d, 0) coefficients ( ) d a j = K α,d ( 1) j Γ (1 d) = K α,d j Γ (j + 1) Γ (1 d j) ( 1)j where K α,d is a constant such that a α j = 1. The estimator considered here is T κ,n with κ = 0.3. The results in tables 1 and 2 are ordered according to the value of the self-similarity parameter H = d + 1/α where d = 1 β. The following quantities are given: 1) simulated mean and median of T κ,n ; 2) simulated and (conditional) asymptotic lower and upper quartiles Q 1, Q 2 of T κ,n ; 3) simulated and (conditional) asymptotic and quantiles of n 1 H (T κ,n α); and 4) the proportion p 0 of simulations where convergence in the open interval (1, 2) was achieved. With respect to p 0 one can observe that convergence is more problematic for high values of H. Also, T κ,n has a considerable finite sample bias when H exceeds 0.8, in particular if at the same time α is small. Generally speaking, strong long memory induces overestimation of α. An intuitive explanation is 11

12 that long memory causes the process to stay at a similar level for a long time, thus giving the observer less opportunity to see how fat the tails are. This is illustrated in Figure 1, with boxplots of X t (t = 1, 2,..., 400) generated by (1) with α = 1.5 and H varying between 0.75 to Comparing the boxplots one can see that, as H increases, extreme points tend to come closer together. The relationship between bias (in absolute value) and the parameters α and H is illustrated further in Figure 2. The radius of each circle (at the corresponding location (α, H)) represents the simulated absolute bias of T κ,n for one of the sample sizes n = 100, 200 and 400 respectively. The dotted line corresponds to the lower limit of the allowed range, H > 1/α. The largest positive bias in the simulations can be observed for α = 1.3 and H = 0.9 (d = H 1/α 0.13). To understand these results intuitively, one needs to bear in mind that the estimate of α is confined to the interval (1, 2). The reported simulation results are conditional on the event that the solution of the estimating equation is between 1 and 2. If the true value of α is close to the lower bound of 1, then the variability of λ Fn is higher due to the heavy tails. Therefore, for small to moderate sample sizes, the probability that the actual solution of the estimation equation is below 1 is not negligible. This leads to a strong positive bias when confining estimates to the range (1, 2). On the other hand, if α is close to the upper bound of 2, then estimation is more accurate due to lighter tails. The bias effect is therefore less pronounced. Figures 3 to 6 show a few examples of simulated histograms of Tn, T n + and T κ,n together with (conditional) density functions based on Theorem 1 and (42). Overall, the simulated histograms are in reasonably good agreement with Theorem 1 and (42). Comparing the histograms and boxplots of Tn, T n + and T κ,n it can be seen that, as expected, the third estimator is more accurate. Although the proposed method appears to work reasonably well for moderate values of H, many open questions remain. First of all, a more accurate assessment of finite sample accuracy would be needed for high values of H and/or very low values of α. One possible approach is bootstrap (Efron 1979). For long-memory processes with a finite variance, several versions of resampling have been shown to work in principle (Lahiri 1993, 2003, Hall et al. 1998, Nordman and Lahiri 2005, Kim and Nordman 2011). The case with long memory and infinite second moments received less attention so far although there is an even greater need for finite sample improvements. A promising approach along this line is introduced in a recent paper by Jach 12

13 et al. (2011), however their method is worked out for stochastic volatility models rather than linear long-memory processes considered here. This will certainly be worth pursuing further in the context of tail index estimation. Another interesting question is how to obtain more efficient estimators of α. In particular, T κ,n with κ = 1 is asymptotically better than for κ 1, 2 2 because it has a faster rate of convergence. However, the asymptotic distribution of T 1,n is not known. The question what kind of limit theorems can 2 be obtained for sums ψ v,u (X t ; α) when the value of h is equal to zero was mentioned as an open problem already in Koul and Surgailis (2001). Tail index estimation, as discussed here, is a further motivation to pursue this question. Furthermore, including an additional scale parameter in the estimation procedure is likely to lead to similar questions. Finally let us note that the results on tail index estimation also relevant when estimating the exponent β (or in other words the long-memory parameter d), since Kokoszka and Taqqu (1996) showed that the rate of convergence and the asymptotic distribution of periodogram-based estimators of β depend on α. The final goal pursued in current research is simultaneous inference on α and β. Acknowledgments We would like to thank a referee and the associate editor for their constructive remarks that lead to an improved presentation of the results. The first author would like thank Paul Embrechts and colleagues at RiskLab for their hospitality and many interesting discussions during his sabbatical at ETH Zurich. We would also like to thank Martin Mächler (ETH Zurich) for sending us a preliminary version of the R-package stabledist that enabled us to carry out the simulations efficiently. 6 Appendix 6.1 Proofs Proof (Theorem 1): The results for T n + and Tn follow the same way. Consider, for instance, T n + = [τ n + ] 2 1. We use Serfling (1980), Lemma A, p.249 to see that τ n + is consistent for α 0. Clearly ψ v,u(x, + α) is a continuous nondecreasing function in α. We need to check that α = α 0 is an isolated root 13

14 of λ + F (α). This follows immediately, since by definition of ψ+ v,u, λ + F (α 0) = E [ ψ + v,u(x t, α 0 ) ] = E [ [α 0 log(x t ) 1] u v 1 {X t >0} E ( [α 0 log(x t ) 1] u v 1 {X t >0})] = 0. and λ + F (α) < 0 for 1 < α < 2. Weak consistency of T n + then follows by standard arguments. Next we find the asymptotic distribution of T n +. We can write 1 n n ψ v,u(x + t, T n + ) = 1 n ψ n v,u(x + t, α 0 ) + (T n + α 0 ) 1 n ψ n v,u(x + t, α 0 ) + (T n + α 0 ) 2 1 n ψ n v,u(x + t, α) = 1 n ψ + n v,u(x t, α 0 ) + (T n + α 0 ) 1 n ψ + n v,u(x t, α 0 ) (T + n α 0 ) 2 C( α, v, u) where α α 0 T n + α 0. Since T n + converges to α 0 (1, 2) in probability, we have P sup C( α, v, u) K 0 P 1 α α 0 T + n α 0 for some 0 < K 0 <. Therefore, T + n 1 n n α 0 = ψ+ v,u(x t, α 0 ) ψ v,u(x + t, α 0 ) + o ( ) p T + n α 0 1 n n = A n B n + o p ( T + n α 0 ). Now, the function ψ + v,u(, α 0 ) is of bounded variation so that Corollary 2.3 in Koul and Surgailis (2001) implies n 1 H A n = n 1 H 0 ψ + v,u(x, α 0 )d (F n (x) F (x)) c(α 0 )h + Z α0. (46) 14

15 Moreover, using Birkhoff s Ergodic Theorem, we have, as n, B n ψ + (x, α 0 )df (x) = µ + (α 0 ) R almost surely. Using Slutsky s theorem we thus obtain Analogous arguments lead to, and n 1 H (T + n α 0 ) h+ c(α 0 ) µ + (α 0 ) Z α 0. n 1 H (T n α 0 ) h+ c(α 0 ) µ + (α 0 ) Z α 0 n 1 H (T n + T + n ) P 0. Proof (Theorem 2): In either case, under the null hypothesis H 0, Theorem 1 implies T n + P α 1 and Tn P α 1 as n. Define the function ψ v,u (x, α 1, α 2 ) = ψ + v,u(x, α 1 ) ψ v,u(x, α 2 ). Under H 0 we also use the notation α for α 1. (1) By assumption, X 1,t = X 2,t and hence X t = X 1,t = X 2,t. The idea of the proof follows a similar route as for Theorem [ 1. Since ] ψ u,v (x, α, α) is a bounded measurable function of α 1 with E ψv,u (x, α) = 0, we have where Z α1 n 1 H n ψ v,u (X t, α, α) hc(α)z α1, is a SαS random variable with α = α 1 and h = ψ v,u (x)f (x)dx = 2h +. R 15

16 Now observe that 1 n n ψ v,u (X t, T n +, Tn ) = 1 n = 1 n n ψ v,u(x + t,1, T n + ) 1 n n ψv,u(x t,2, T n + ) n ψ v,u (X t,1, α, α) + (T n + α) 1 n (T n α) 1 n n ψ v,u(x t,2, α ) n ψ + v,u(x t,1, α ) with α ± α T n ± α. The result then follows by analogous arguments as in Theorem 1. (2) The result follows easily using Theorem 1 for T n + and Tn separately, since the two estimators are independent of one another. 16

17 6.2 Tables 17

18 H = 0.60 α = 1.7, d = n mean, median Q 1 Q 2 q q p , , , , , , , , , , , , , , , , , , , , H = 0.65 α = 1.7, d = , , , , , , , , , , , , , , , , , , , , H = 0.70 α = 1.5, d = , , , , , , , , , , , , , , , , , , , , α = 1.70, d = , , , , , , , , , , , , , , , , , , , , H = 0.75 α = 1.5, d = , , , , , , , , , , , , , , , , , , , , α = 1.7, d = , , , , , , , , , , , , , , , , , , , , Table 1: Simulation results for T κ,n (κ = 0.3) and X t defined by (1) with FARIMA(0,d,0)-coefficients a j such that A = 1. For T κ,n, the mean and the median, and the lower and upper simulated and (conditional) asymptotic quartiles Q 1, Q 2 are listed in the second, third and fourth column respectively. For the standardized statistic n 1 H (T κ,n α 0 ), the simulated and (conditional) asymptotic and quantiles are given in the fifth and sixth column respectively. The proportion p 0 of cases where convergence was reached is provided in the last column. 18

19 H = 0.80 α = 1.3, d = n mean, median Q 1 Q 2 q q p , , , , , , , , , , , , , , , , , , , , α = 1.5, d = , , , , , , , , , , , , , , , , , , , , α = 1.7, d = , , , , , , , , , , , , , , , , , , , , H = 0.85 α = 1.3, d = , , , , , , , , , , , , , , , , , , , , α = 1.5, d = , , , , , , , , , , , , , , , , , , , , α = 1.7, d = , , , , , , , , , , , , , , , , , , , , H = 0.9 α = 1.3, d = , , , , , , , , , , , , , , , , , , , , α = 1.5, d = , , , , , , , , , , , , , , , , , , , , α = 1.7, d = , , , , , , , , , , , , , , , , , , , , Table 2: Simulation results for T κ,n (κ = 0.3) and X t defined by (1) with FARIMA(0,d,0)-coefficients a j such that A = 1. For T κ,n, the mean and the median, and the lower and upper simulated and (conditional) asymptotic quartiles Q 1, Q 2 are listed in the second, third and fourth column respectively. For the standardized statistic n 1 H (T κ,n α 0 ), the simulated and (conditional) asymptotic and quantiles 19 are given in the fifth and sixth column respectively. The proportion p 0 of cases where convergence was reached is provided in the last column.

20 6.3 Figures H Figure 1: Boxplots of X t (t = 1, 2,..., 400) generated by (1), with α = 1.5, H {0.70, 0.75, 0.80, 0.85, 0.95} and FARIMA(0,d,0)-coefficients a j (d = H 1/α) scaled such that A = 1. In order to keep the series comparable, the same seed was used for each value of H. 20

21 H[i] alpha[i] Figure 2: Symbol plot of the simulated absolute bias b = T κ,n α (κ = 0.3) as a function of H and α. Each circle corresponds to one of the sample sizes n {100, 200, 400}, the radius being proportional to b. The dotted line represents the lower boundary of the admissible range 1/α < H < 1. 21

22 T : v= 1, u=inf, alpha=1.7, d=0.012, n=100 T+: v= 1, u=inf, alpha=1.7, d=0.012, n= T T+ T_{kappa}: v= 1, u=inf, alpha=1.7, d=0.012, n=100, kappa= T T+ T_kappa T_kappa T : v= 1, u=inf, alpha=1.7, d=0.012, n=400 T+: v= 1, u=inf, alpha=1.7, d=0.012, n= T T+ T_{kappa}: v= 1, u=inf, alpha=1.7, d=0.012, n=400, kappa= T_kappa 22 T T+ T_kappa Figure 3: Simulated histograms of Tn, T n + and T κ,n (κ = 0.3) for H = 0.6 and α = 1.7, together with an asymptotic approximation (dotted line) obtained from Theorem 1. Also shown are boxplots comparing the three estimators. The sample sizes are n = 100 (upper four figures) and n = 400 (lower four figures) respectively.

23 T : v= 1, u=inf, alpha=1.7, d=0.062, n=100 T+: v= 1, u=inf, alpha=1.7, d=0.062, n= T T+ T_{kappa}: v= 1, u=inf, alpha=1.7, d=0.062, n=100, kappa= T T+ T_kappa T_kappa T : v= 1, u=inf, alpha=1.7, d=0.062, n=400 T+: v= 1, u=inf, alpha=1.7, d=0.062, n= T T+ T_{kappa}: v= 1, u=inf, alpha=1.7, d=0.062, n=400, kappa= T_kappa 23 T T+ T_kappa Figure 4: Simulated histograms of Tn, T n + and T κ,n (κ = 0.3) for H = 0.65 and α = 1.7, together with an asymptotic approximation (dotted line) obtained from Theorem 1. Also shown are boxplots comparing the three estimators. The sample sizes are n = 100 (upper four figures) and n = 400 (lower four figures) respectively.

24 T : v= 1, u=inf, alpha=1.7, d=0.112, n=100 T+: v= 1, u=inf, alpha=1.7, d=0.112, n= T T+ T_{kappa}: v= 1, u=inf, alpha=1.7, d=0.112, n=100, kappa= T T+ T_kappa T_kappa T : v= 1, u=inf, alpha=1.7, d=0.112, n=400 T+: v= 1, u=inf, alpha=1.7, d=0.112, n= T T+ T_{kappa}: v= 1, u=inf, alpha=1.7, d=0.112, n=400, kappa= T_kappa 24 T T+ T_kappa Figure 5: Simulated histograms of Tn, T n + and T κ,n (κ = 0.3) for H = 0.7 and α = 1.7, together with an asymptotic approximation (dotted line) obtained from Theorem 1. Also shown are boxplots comparing the three estimators. The sample sizes are n = 100 (upper four figures) and n = 400 (lower four figures) respectively.

25 T : v= 1, u=inf, alpha=1.7, d=0.162, n=100 T+: v= 1, u=inf, alpha=1.7, d=0.162, n= T T+ T_{kappa}: v= 1, u=inf, alpha=1.7, d=0.162, n=100, kappa= T T+ T_kappa T_kappa T : v= 1, u=inf, alpha=1.7, d=0.162, n=400 T+: v= 1, u=inf, alpha=1.7, d=0.162, n= T T+ T_{kappa}: v= 1, u=inf, alpha=1.7, d=0.162, n=400, kappa= T_kappa 25 T T+ T_kappa Figure 6: Simulated histograms of Tn, T n + and T κ,n (κ = 0.3) for H = 0.75 and α = 1.7, together with an asymptotic approximation (dotted line) obtained from Theorem 1. Also shown are boxplots comparing the three estimators. The sample sizes are n = 100 (upper four figures) and n = 400 (lower four figures) respectively.

26 T : v= 1, u=inf, alpha=1.3, d=0.031, n=100 T+: v= 1, u=inf, alpha=1.3, d=0.031, n= T T+ T_{kappa}: v= 1, u=inf, alpha=1.3, d=0.031, n=100, kappa= T T+ T_kappa T_kappa T : v= 1, u=inf, alpha=1.3, d=0.031, n=400 T+: v= 1, u=inf, alpha=1.3, d=0.031, n= T T+ T_{kappa}: v= 1, u=inf, alpha=1.3, d=0.031, n=400, kappa= T_kappa 26 T T+ T_kappa Figure 7: Simulated histograms of Tn, T n + and T κ,n (κ = 0.3) for H = 0.8 and α = 1.3, together with an asymptotic approximation (dotted line) obtained from Theorem 1. Also shown are boxplots comparing the three estimators. The sample sizes are n = 100 (upper four figures) and n = 400 (lower four figures) respectively.

27 T : v= 1, u=inf, alpha=1.5, d=0.133, n=100 T+: v= 1, u=inf, alpha=1.5, d=0.133, n= T T+ T_{kappa}: v= 1, u=inf, alpha=1.5, d=0.133, n=100, kappa= T T+ T_kappa T_kappa T : v= 1, u=inf, alpha=1.5, d=0.133, n=400 T+: v= 1, u=inf, alpha=1.5, d=0.133, n= T T+ T_{kappa}: v= 1, u=inf, alpha=1.5, d=0.133, n=400, kappa= T_kappa 27 T T+ T_kappa Figure 8: Simulated histograms of Tn, T n + and T κ,n (κ = 0.3) for H = 0.8 and α = 1.5, together with an asymptotic approximation (dotted line) obtained from Theorem 1. Also shown are boxplots comparing the three estimators. The sample sizes are n = 100 (upper four figures) and n = 400 (lower four figures) respectively.

28 T : v= 1, u=inf, alpha=1.5, d=0.183, n=100 T+: v= 1, u=inf, alpha=1.5, d=0.183, n= T T+ T_{kappa}: v= 1, u=inf, alpha=1.5, d=0.183, n=100, kappa= T T+ T_kappa T_kappa T : v= 1, u=inf, alpha=1.5, d=0.183, n=400 T+: v= 1, u=inf, alpha=1.5, d=0.183, n= T T+ T_{kappa}: v= 1, u=inf, alpha=1.5, d=0.183, n=400, kappa= T_kappa 28 T T+ T_kappa Figure 9: Simulated histograms of Tn, T n + and T κ,n (κ = 0.3) for H = 0.85 and α = 1.5, together with an asymptotic approximation (dotted line) obtained from Theorem 1. Also shown are boxplots comparing the three estimators. The sample sizes are n = 100 (upper four figures) and n = 400 (lower four figures) respectively.

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