Tail Index Estimation of Heavy-tailed Distributions

Transcription

1 CHAPTER 2 Tail Index Estimation of Heavy-tailed Distributions 2.1 Introduction In many diverse fields such as meteriology, finance, hydrology, climatology, environmental sciences, telecommunication, insurance and genetics, heavy-tailed distributions are recommended to model the data, see Embrechts et al. (1997) and hence the problem of estimation of the tail index of a heavy-tailed distribution has been paid much attention in recent years. The tail shape of heavy-tailed distributions resembles to a first approximation the hyperbolic shape of the Pareto distribution characterized by the so-called tail index. Besides, this parameter plays a key role in connection with determination of extreme quantiles, upper tail probabilities, mean excess functions and excess of loss and stop reinsurance premium etc. Small relative errors in the estimation of tail index can produce large relative errors in estimation of such quantities. Moreover, in the context of semi-parametric Some results included in this chapter have appeared in the paper Dais George and Sebastian George (2009). 23

2 modeling, one assumes merely a Pareto type distribution, as an approximate model for the upper observations of a sample, to situations, where the upper tail of the model is not parametrically related to the central parts and thus is estimated separately, using only the uppermost observations. Extreme value models focus attention on the tail of a statistical distribution of events rather than imposing a single functional form to hold for the entire distribution. To estimate the tail index, numerous tail index estimators have been proposed in the literature including earlier contributions by Hill (1975) and Pickands (1975) in which the Hill estimator has become somewhat of a benchmark to which later proposed estimators are compared. Again many of the later proposed estimators of tail index are the alternatives of the Hill estimator. They are the smoohill estimator, Hill estimator in alt scale, smoohill estimator in alt scale, for details, see Resnick (1997) and weighted Hill estimator [Gomes et al. (2008)]. The other popular tail index estimators are the qq-estimator [Kratz and Resnick (1996)], harmonic moment estimator [Henry (2009)] and more recently the tail index estimator for dependant hetrogeneous data introduced by Hill (2010). The Hill estimator, its alternatives and the qq-estimator are methods of estimation by plotting techniques. The traditional Hill plot offers little scope of correctly discerning the value of α, but the alternatives such as smoohill plot, Hill plot in altscale, smoohill in alt scale and qq-estimator seems to reveal the value of α fairly clearly. Though the asymptotic variance of the qq-estimator is larger than that of the Hill estimator, the volatility of the plot always seems to be less than that of the Hill plots. Being aware of the limitations of these estimators, we introduce another estimator in this chapter namely, the smooweighted Hill estimator, for estimating the tail index, which is a better one compared to most of the estimators proposed in the literature [Dais and Sebastian (2009)]. It is an improved version of the weighted Hill estimator proposed by Gomes et al. (2008). This chapter is organized as follows. Section 2 describes the heavytailed distributions. In Section 3, the plotting techniques for estimating the tail index of a heavy-tail distribution ((Hill estimator, its alternatives, weighted Hill estimator and the qq-estimator) are reviewed. A new tail index estimator namely, the smooweighted Hill estimator is introduced in Section 4. In Section 5, we establish the performance of the the smooweighted Hill estimator through a comparative study. Bootstrap confidence intervals and coverage probabilities are computed for the Hill and weighted Hill estimators in Section 24

3 6. In Section 7, we concludes the chapter. 2.2 Heavy-tailed Distributions Heavy-tailed distributions (also known as power-law distributions) have been observed in many natural phenomena including both physical and sociological phenomena. Definition and examples of heavy-tailed distributions are given in Section A useful and tractable model with relatively high probability in the upper tail is the Pareto distribution which is hyperbolic over its entire range and has the probability density function, f(x) = αc α x α 1, α, c > 0, x c. (2.2.1) The cumulative distribution function is given by F (x) = P [X x] = 1 (c/x) α, x c where α is the shape parameter which is a measure of tail heaviness of the distribution and c represents the smallest value, that the random variable can take. As α decreases, an arbitrarily large portion of the probability mass may be present in the tail of the distribution. This means that a random variable that follows a heavy tailed distribution can give raise to extremely large values with non-negligible probability. As c increases only the tail of the distribution is modeled. The Pareto distribution belongs to the one parameter exponential family of distributions for known value of c, as the density function can be written as f(x) = C(α) exp[q(α)t(x)]h(x) where C(α) = αc α, Q(α) = (α + 1), t(x) = lnx and h(x)=1 (lnx denotes logarithm to the base e). For general overviews of the role of Pareto tailed distributions in actuarial science, econometrics, telecommunications and other fields we refer to Arnold (1983), Johnson et al. (1994) and Resnick (2007). New application contexts continue to arise: for example, the cost distribution of combinatorial research algorithms, has recently been shown to exhibit Pareto type tail behavior, see Gomes et al. (1997). 25

4 2.3 Estimating the Tail Index α In this section, we discuss some of the plotting techniques to estimate the parameter α. We also consider various properties of these estimators and possible alternatives The Hill Estimator Hill estimator is one of the most popular tools for detecting the presence of heavy tails of the marginal distribution of stationary sequences of random variables. Suppose X 1, X 2,..., X n are iid random variables from a distribution F. Let X (1) > X (2) >... > X (n) be the order statistics. If X has an exact Pareto distribution with 1 F (x) = x α, x > 1, then lnx follows an exponential distribution with parameter α. Thus lnx 1,..., lnx n is a sample from an exponential density with parameter α. Since the mean of the exponential distribution with parameter α is α 1, the maximum likelihood estimator (MLE) of α 1 is the sample mean and thus n H n = 1 n I=1 lnx (i) is the MLE of α 1. Instead of assuming a Pareto Distribution, we assume that 1 F (x) = x α L(x), x, where L(x) is a slowly varying function, satisfying (1.3.2). For k < n, the Hill estimator [Hill (1975)] of α is defined as H k,n = 1 k ( ) X(i) ln, (2.3.1) X (k+1) where k is the number of upper order statistics used in the estimation. A rough idea behind using only k upper order statistics is that we should only sample from that part of the distribution which looks most Pareto like. 26

5 Properties of Hill Estimator The Hill estimator of the tail index α is a pseudo-maximum likelihood estimator based on the exponential approximation of the normalized log-spacings Y j = jln 2,..., k. The following are some of the important properties of the Hill estimator. ( X(j) X (j+1) ) for j = 1, 1. When X is are iid, Hill estimator is a consistent estimator of α 1, a fact which is equivalent to regular variation of 1-F in a way made precise by Mason (1982). The condition for consistency requires that k = k(n), a function of n, k and (k/n) 0 as n. 2. The Hill estimator can be surprisingly sensitive to changes in location. A shift in location does not theoretically affect the tail index, but may throw the hill estimate way off. 3. The Hill estimator is asymptotically normal [Resnick (2007)]. That is (k)[h (k,n) ˆα 1 d ] N(0, α 2 ). In practice, the Hill estimator is used as follows. We graph {(k, H 1 k,n ), 1 k n} and hope the graph looks stable so that we can pick out a value of α = H 1 k,n from the stable regime of this plot. We have conducted a simulation study to estimate the tail index α based on simulated observations from a Pareto distribution with α = 0.5. Figure 2.1 shows the Hill plot for the simulated Pareto data and the Hill estimate seems to lie between.48 and 0.5. Later it has shown that the Hill estimator could successfully be applied to a variety of dependent sequences of random variables, see Hsing (1991) and Resnick and Starica (1995). Unfortunately, the practical use of this estimator is hampered by the fact that the Hill estimator has optimality properties only when the underlying distribution is close to Pareto. If the distribution is far from Pareto, there may be outrageous bias even for sample sizes such as That is why we sample from that part of the distribution which is most Pareto like. The Hill plot is not always so revealing. One of the difficulty that we are facing using the Hill estimator is that the graph may exhibit considerable volatility and/or the true answer may be hidden in the graph. For the bias problem, though there is no completely 27

6 Figure 2.1: Hill Plot for Pareto, α = 0.5. satisfied resolution, one possibility is the bootstrap method which we consider in the last section. Again the estimate is often highly sensitive to the choice of k, the number of upper order statistics used to calculate H k,n. Resnick (1997) suggested some alternative graphical tools which can minimize these difficulties. They are the smoohill estimator and its alternatives SmooHill Estimator This is a technique for smoothing the Hill estimator [Resnick and Starica (1997)], though it is powerless to correct bias, reduces the volatility of the plot and the uncertainty about how to pick out an estimate of α. Thus, it make use of the Hill estimator more reliable through an averaging technique which reduces the asymptotic variance. As a direct result, the range in which the smoothed estimator varies as a function of k decreases and the successful use of the estimator is made less dependent on the choice of k. The smoohill estimator is defined as SmooH k,n = 1 (u 1)k uk j=k+1 where u > 1 is an integer (usually we pick u between n 0.1 and n 0.2, that is 2 or 3) and uk n where n is the sample size. The asymptotic variance of the Hill estimator is 1/α 2 H j,n 28

7 Figure 2.2: SmooHill Plot for Pareto α = 0.5. where as the asymptotic variance of smoohill estimator is ( ) ( ) ( ( )) 1 2 lnu 1 α 2 u u which is less than 1/α 2. The bigger the u, the more the asymptotic variance is reduced. But for large u fewer points are plotted in smoohill. Here u=3. So we stop averaging, when k reached Figure 2.2 shows the result of smoothing in the Hill estimate which gives smoohill estimate of α as Alt Plotting This is an alternative to the Hill plot using a change of scale. Here we plot {(θ, H 1 [n θ ],n), 0 θ 1}, where we write [n θ ] for the integer part of n θ and θ = ln(k)/ln(n) so that we are getting an alternative to the Hill plot abbreviated as althill. This method was suggested by Resnick and Starica (1995). As in the case of traditional Hill plot, here also we read out the estimated value of α from the stable portion of the graph. Figure 2.3 and Figure 2.4 give views of Hill plot and smoohill plot of Pareto data in alt scale. Here α = 0.49 in both cases, more likely choices. 29

8 Figure 2.3: AltHill Plot for Pareto α = 0.5. Figure 2.4: AltsmooHill Plot for Pareto α =

9 This alternative display is more helpful since the significant part of the graph (the part corresponding to a relatively small number of order statistics) gets to be shown more clearly, covering a bigger portion of the displayed space. That is, the part of the graph corresponding to a high number of order statistics, which covers a disproportionately large part in the traditional Hill plot gets rescaled. Thus, the interpretation of the graph is easier and more accurate. Picking k mid way between 1 and n or picking θ so close to 1 seems unwise Weighted Hill Estimator Gomes et al. (2008) proposed a bias reduced Hill estimator called weighted Hill estimator which gives less weight to the largest than to the intermediate order statistics. Doing this in an adequate way, the dominant component of the asymptotic bias of the Hill estimator can be reduced without increasing the asymptotic variance. Moreover, in statistics of extremes, although all top observations are relevant, by giving less weight to the extreme and more weight to the intermediate order statistics, the performance of the estimator can be increased. Here heavy-tailed models, with a specific regularly varying tail function in Hall s class, that is with a tail ( x 1 F (x) = C where C > 0, ρ < 0 and β 0 are considered. ) 1/γ ( β ( x ) ρ/γ o(x ρ/γ ), as x ρ C Then we are in the domain of attraction for maxima of an extreme value distribution function EV γ (x) = exp( (1 + γx) 1/γ ), 1 + γx 0, γ > 0, and we write F D M (EV γ>0 ). The parameter γ is the extreme value index, one of the primary parameters of extreme or even rare events. In the context of heavy tails, with the notation U(t) = F (1 1/t), t 1 where F (y) = inf{x : F (x) y} and for the class of regularly varying functions with index of regular variation α, usually denoted by RV α F D M (EV γ>0 ) iff F = 1 F RV 1/γ iff U RV γ. 31

10 The second order parameter ρ is the non-positive value that appears in the limiting relation lnu(tx) lnu(t) γlnx lim t A(t) = xρ 1, ρ is assumed to hold for every x > 0, in order to derive the non-degenerate behavior of the estimators. Then, A(t) is of regular variation with index ρ. Here ρ is restricted as ρ < 0 and the parametrization, A(t) = γβt ρ is used [for details, see Gomes et al. (2008)]. For intermediate k (a sequence of integers k = k n between 1 and n), k = k n and k n = o(n), as n, the log-excesses are given by V ik = lnx n i+1:n lnx n k:n, 1 i k < n, where X i:n denotes the i th ascending order statistics, 1 i n, associated to a random sample (X 1, X 2,..., X n ). Under the first order frame work, V ik d = lnu(y n i+1:n ) lnu(y n k:n ) d = lnu(y n k:n Y k i+1:k ) lnu(y n k:n ) lny k i+1:k d = α 1 E k i+1:k. That is, the V ik s, 1 i k are approximately k order statistics from an exponential random sample with mean value α 1. This argument justifies the Hill estimator given by (2.3.1). That is H k,n 1 k ( represents equivalent to ). Under the second order framework, for immediate k (1 i k), V ik V ik = d α 1 lny k i+1:k + Y ρ k i+1:k 1 A(n/k)(1 + o p (1)). ρ 32

11 That is, V ik can be written as This follows that ( V ik = α 1 lny k i+1:k 1 + A(n/k) [ A(n/k) = α 1 exp [ A(n/k) V ik α 1 exp Also for 1 i k, α 1 α 1 Y ρ Y ρ k i+1:k 1 ρlny k i+1:k Y ρ k i+1:k 1 (i/k) ρ 1 ρlny k i+1:k ρln(i/k) α 1 k i+1:k 1 ρlny k i+1:k Y ρ k i+1:k 1 (1 + o p (1)) ρlny k i+1:k ) ] E k i+1:k + o p (A(n/k)). ] E k i+1:k = o p ( Vik α 1 E k i+1:k ). = ψ ik ψ(i/k) ψ ik (ρ) [ψ kk 1], with ψ a limiting function and represents approximately equal to. If instead of assuming that V ik come from an exponential family with mean value equal to α 1 as in the case of Hill estimator, assume that V ik come from an exponential family with mean value not equal to α 1, but depend on i (and k). Thus, α 1 ik = α 1 exp [ A(n/k)ψ ik /α 1], 1 i k, A(t) = α 1 βt ρ. The previous result will leads us to get a less biased estimator of α. Since V ik can be approximately written as V ik α 1 exp [β(n/k) ρ ψ ik ] E k i+1:k, ψ ik = (i/k)ρ 1 ρln(i/k), 1 i k, the likelihood associated to V ik, 1 i k, is [ ( n ) ρ L (α, β, ρ) = exp klnα 1 β ψ ik 1 k α 1 ] ( n ) ρ V ik exp β ψik. k Knowing β and ρ, the maximization of L will lead to the weighted Hill estimator of α 1 33

12 denoted by WH given by ˆ α 1 = W H β,ρ (k) = 1/k exp [ β(n/k) ρ ψ ik ] V ik. That is, to obtain the weighted Hill estimator, the Hill estimator is replaced by a weighted combination of the log-excesses so that W H ˆβ,ˆρ (k) = 1/k ] exp [ ˆβ(n/k)ˆρ ˆψik ln ( Xn i+1:n X n k:n ) where ˆβ and ˆρ are any adequate, consistent estimators of the second order parameters β and ρ respectively such that both { ˆβ β} and (ˆρ ρ)ln(n/k) are o p (1/( ka(n/k))), for the k- values on which the estimation of α is based. These conditions for ˆρ and ˆβ hold true, if we consider levels k such that ka 2 (n/k) λ 0, finite. The key success of the WH-estimator lies in the estimation of β and ρ at a level k 1 such that k = o(k 1 ), where k is the number of top order statistics used for the estimation of α. For more details, see Gomes et al. (2008). Estimation of ρ For estimating ρ, the class of ρ estimators in Alves et al. (2003) is used. Such a class of estimators is parameterized using a tuning parameter τ. In the simulations we consider only the statistics associated to τ = 0 and to τ = 1, which are usually preferable when ρ 1 and ρ > 1 respectively. The ρ estimators are, ˆρ i = 3[T (i) n (k 1 ) 1] [T (i) n (k 1 ) 3], i = 0, 1, where ( [ ]) 2n k 1 = min n 1,. lnlnn 34

13 Here T (τ) n (k) = ( ln ( 1 2 ln M (1) n ( (k) ) 1 2 ) ln ( 1 3 ln ) M n (2) (k)/2 M n (2) (k)/2 M n (3) (k)/6 ( ) M n (1) τ ( ) (k) M n (2) τ/2 (k)/2 ( ) τ/2 ( M n (2) (k)/2 M n (3) (k)/6 ) if τ = 0 ) τ/3 if τ > 0, where M (j) n = 1 k [ ln X ] j n i+1:n, j 1, [M n (1) = H k, n ]. X n k:n These statistics converge towards 3(1 ρ) for every τ 0 whenever the second order (3 ρ) condition holds (k is intermediate and as n, ka(n/k) ). Estimation of β For the estimation of β we consider the estimator of β obtained in Gomes and Martins (2002). It is based on the scaled log-spacings U i = i (lnx n i+1:n lnx n i:n ), 1 i k. The estimator is given by ˆβ U (k; ˆρ) = (k/n)ˆρ ( ( (1/k) (1/k) ) (i/k) ((1/k) ˆρ ) (i/k) ((1/k) ˆρ ) ( U i (1/k) ) (i/k) ˆρ U i ) ( (i/k) ˆρ U i (1/k) ). (i/k) 2ˆρ U i For the simulation study, we considered the simulated data from the Pareto distribution. On the basis of the stability criterion for large k, the estimated value of ρ is -0.9, associated to τ = 0 so that we get ˆβ = 0.5. We graph {(k, W H 1 (k)), 1 k n} and pick out that value of α for which the graph ˆρ, ˆβ looks stable. Figure 2.5 is the weighted Hill plot for Pareto data. The weighted Hill plot for Pareto seems to nail at α = correctly. The weighted Hill estimator is asymptotically normal, with asymptotic variance ˆα 2. That is as n (k) ( W H ˆβ,ˆρ (k) ˆα 1 ) d N ( 0, ˆα 2). 35

14 Figure 2.5: Weighted Hill Plot for Pareto, α = Alternative Estimators qq-plotting This graphical technique is a commonly used method of visually assessing goodness of fit and of estimating location and scale parameters, see Rice (1988) and Castilo (1988). It can also be adapted to the problem of detecting heavy tails and for estimating α [Rensnick (1997)]. It rests on the simple observation that for a sample of size n uniformly distributed over (0,1), the spacings are identically distributed, plotting i/(n + 1) against the i th largest in the sample should yield approximately a straight line of slope 1. The procedure is as follows: Suppose {X 1, X 2,..., X n } are iid with distribution F. Pick k upper order statistics X (1) > X (2) >... > X (k) and neglect the rest. Plot {( ( ) ) } j ln 1, lnx (j), 1 j k. (2.3.2) (k + 1) The data is approximately Pareto or even if 1 F is only regularly varying, this should be approximately a straight line with slope α 1. The slope of the least squares line through the points is an estimator called the qq-estimator [Kratz and Resnick (1996)]. Thus, the 36

15 Figure 2.6: Static qq-plot for Pareto, α = 0.5. qq-estimator is given by ˆα 1 k,n = [ 1 k 1 k ( ( )) ( ) i X(i) ln ln 1 (k + 1) X (k+1) k ( ( )) ] i ln H k,n k + 1 ( ( )) ( 2 i ln 1 ( ) ) 2 i ln 1 (k + 1) k (k + 1) Based on the qq-estimator, two different plots can be drawn. They are [( ) ] 1 1. Dynamic qq-plot obtained from plotting k, α 1 k,n, 1 k n which is similar to Hill plot. 2. The static qq-plot obtained by choosing and fixing k, plotting the points in (2.3.2) and putting the least square line through the points while computing the slope as the estimate of α 1. Figure 2.6 shows the static qq-plots for Pareto data with k = 8000 which is approximately a straight line with slope α 1. Here ˆα = Again Figure 2.7 gives a view of the dynamic qq-plot. Here the graph is stable and the value of ˆα = If k and (k/n) 0, the qq-estimator is consistent and under a second order. 37

16 Figure 2.7: qq-plot for Pareto, α = 0.5. regular variation condition and further restriction on k(n), it is asymptotically normal with asymptotic variance 2α 2 which is larger than the asymptotic variance of the Hill estimator. But the volatility of the qq-plot is less than that of the Hill plot, see Resnick (2007). Based on the above studies, we introduce a better tail index estimator, namely smooweighted Hill estimator and its performance is compared with the above mentioned estimators are considered in the following sections. 2.4 Smooweighted Hill Estimator Our method is based on the smoothing procedure suggested by Resnick and Starica (1997). It is a simple averaging technique which reduces the volatility of the plot. This smoothing procedure consists of averaging the weighted Hill estimator values corresponding to different numbers of order statistics. The smoothed weighted Hill estimator denoted as smooweighted Hill estimator is given by smooweightedhˆρ, ˆβ(k) = 1 (u 1)k uk j=k+1 W Hˆρ, ˆβ(j), where u > 1 is an integer (usually 2 or 3) as in the case of smoohill estimator. Here also, u = 3 so that we stop averaging when k reaches at 3300 (uk < 10000). Since in the case of weighted Hill estimator, we give less weight to largest than to 38

17 Figure 2.8: Smooweighted Hill Plot for Pareto, α = 0.5. intermediate observations in an adequate way, it won t affect the asymptotic variance. So the asymptotic variance of the weighted Hill estimator is the same as that of the Hill estimator and the asymptotic variance of the smooweighted Hill estimator is the same as that of the smoohill estimator. Figure 2.8 shows the smooweighted Hill plot for Pareto data. Here the smooweighted Hill estimate is 0.5 exactly. The smoothing procedure consists of averaging the weighted Hill estimator values W H ˆβ,ˆρ (k) over a broad range of k, the number of order statistics. The order of the number of terms involved in the averaging is k. Therefore, when n, k we will be averaging larger and larger number of weighted Hill estimator values with a consequent reduction in asymptotic variance. The asymptotics of the smooweighted Hill estimator can be proved as in the case of the smoohill estimator. For details, see Resnick and Starica (1997). 2.5 Comparison of Various Tail Index Estimators The performance of the parameter estimation method is compared using the bias and efficiency of the parameter in terms of standard deviation (SD) and root mean squared error (RMSE). Here we compare the various tail index estimators of simulated Pareto data in terms of bias and RMSE. As we can see in Table 2.1, the estimate of α consistently out perform (with regard to bias, SD and RMSE) for smooweighted Hill estimator. We extend our study to find out the bootstrap confidence interval for the Hill estimator 39

18 Table 2.1: Comparison of Tail Index Estimators. Estimate of α k = 100 k = 200 k = 300 Bias Hill SmooHill qq- estimator Weighted Hill SmooWeighted Hill SD Hill SmooHill qq-estimator Weighted Hill SmooWeighted Hill RMSE Hill SmooHill qq-estimator Weighted Hill SmooWeighted Hill and weighted Hill estimator in the following section, based on their asymptotic normality. 2.6 The Bootstrap Confidence Interval for the Hill Estimator of α Bootstrapping is a method of resembling similar to a Monte Carlo experiment. In a Monte Carlo study we generate the random variables from a given distribution such as the normal. The bootstrap takes a different approach, the random variables are drawn from their observed distribution. In essence, the bootstrap uses the plug-in principle-observed distribution of their actual distribution of the random variables is the best estimate of their actual distribution. The idea of bootstrap was developed in Efron (1979). The key point made by Efron is that the observed data set is a random sample of size T drawn from the actual probability distribution generating the data, in a sense, the empirical distribution of the data is the best estimate of the actual distribution of the data. The confidence interval for the means of a heavy-tailed distribution was proposed by Hall and Lepage (1996). Conditional on X 1, X 2,..., X n, let X1, X2,..., Xn denote independently and identically distributed random variables drawn randomly with replacement from X 1, X 2,..., X n. Generate n j, j=1,2,...,n such samples. 40

19 Let α n j = 1 n j α nj = [ 1 k ( ) ] 1 X ln i, Xk+1 n j [ n j ] 1/2 α nj and Sn 1 j = (α nj αn j ) 2. n j j=1 Let, where T n = k(αnj α n ) S n n Denote α n = 1 n αn n j and Sn = j=1 j=1 n j S n j. n n j j=1 x 1 β = sup{x : P ( T n x X 1 X 2 X n ) 1 β}. Then a nominal 1 β level confidence interval for α is I 1 β = α n x 1 β S n k, α n + x 1 β S n k, which has asymptotic coverage probability 1 β under very mild regularity conditions including n j /n 0 as n. In a similar way we can found out the a nominal 1 β level confidence interval for α using weighted Hill estimator by replacing the Hill estimator of α by the weighted Hill estimator of α in the above procedure. We generate 200 pseudo random samples of size n = 1000 from the Pareto distribution with α =.5, for k = 100, 200 and 300. The confidence intervals based on this method with confidence level.95 are computed for both Hill estimator and weighted Hill estimator, given by confidence interval using Hill estimator = ( , ) and confidence interval using weighted Hill estimator = (.49981, ). Confidence interval for various sub samples 41

20 Figure 2.9: Hill Plot for Pareto from the Bootstrap Sub Sample. are calculated and coverage probability is also calculated. It is obtained as.954 and.946 respectively, which is near to the nominal value. For the sub sample bootstrap method we drew 200 sub samples each time. A Hill plot from the bootstrap sub sample is given in Figure Conclusion In this chapter, we suggest a better estimate namely smooweighted Hill estimator for estimating the tail index of heavy-tailed distributions and its performance is compared with several tail index estimators using simulated Pareto data. Also we construct bootstrap confidence intervals for both the Hill estimator and the weighted Hill estimator. Hill plot and static qq-plot made a good agreement when estimating the tail index from the graph. The smoohill plot and Hill plot in alt scale also gives a good agreement. The traditional Hill plot offers little scope of correctly discerning the value of α, but the alternatives smoohill plot, Hill plot in altscale, smoohill in altscale and qq-estimator seems to reveal the value of α fairly clearly. Though the asymptotic variance of the qq-estimator is larger than that of the Hill estimator, the volatility of the plot always seems to less than that of the Hill plots. In the case of smoohill, althill, smoohill in altscale, weighted Hill and smooweighted Hill estimators, the asymptotic variance is less than that of the ordinary Hill estimator and at the same time volatility of the plot is also smaller than the volatility of the Hill plots. Among 42

21 all the above mentioned estimators smooweighted Hill estimator is the best estimator with least bias, least asymptotic variance and least RMSE. Coverage probabilities of the confidence intervals through bootstrap method are nearer to the nominal value for both Hill and weighted Hill estimators. References Arnold, B. C. and Press, S. J. (1983). Bayesian inference for Pareto population, Journal of Econometrics, 21, Castillo, E. (1988). Extreme Value Theory in Engineering, Academic Press, San Diego. Dais George and Sebastian George, (2009). Analyzing the tail heavyness of Web server data, International Journal of Web Applications, 1(4), Efron, (1979). Bootstrap methods, another look at the Jacknife, Annals of Statistics, 7, 1-6. Embrechts, P., Klppelberg, C., Mikosch and Thomas, (1997). Modelling Extremal Events for Insurance and Finance, Berlin: Springer. Fraga Alves, M. I., Gomes, M. I. and de Hann, L. (2003). A new class of semi-parametric estimators of the second order parameter, Portugaliae Mathematica, 60:1, Gomes, C. P., Selman, B., and Crato, N. (1997). Heavy-tailed distributions in combinatorial search, In Smolka, G. (ed.), Principles and Practice of Constraint Programming CP-97, Lecture Notes in Computer Science Springer, New York. Gomes, M. I, Laurens de Hann and Ligia Rodrigues, (2008). Tail index estimation for heavy tailed model: accomodation of bias in weighted log excesses, Journal of Royal Statistical Society:Series B (Statistical Methodology), 70, Gomes, M. I. and Martins, M. J. (2002). Asymptotically unbiased estimators of the extreme value index based on external estimation of the second order parameter, Extremes, 5:1, Hall, P. and Lepage, R. (1996). On bootstrap estimation of the distribution of the standardized mean, Ann Inst Statist Math, 48, Henry 111, J. B. (2009). A harmonic moment tail index estimator, Journal of Statistical Theory and Applications, 8,

22 Hill, B. M. (1975). A simple general approach to interface about the tail of a distribution, Ann Statistics, 3, Hill, J. (2010). On tail index estimation for dependent, heterogeneous data, Econometric Theory, 26. Hsing,T. (1991). On tail estimation using dependent data, Ann. Statist.,19, Johnson, N. I., and Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions 1 (2nd ed), Wiley, New York. Kratz, M. and Resnick, S. (1996). The qq-estimator and heavy tails, Stochastic Models, 12, Mason, D. (1982). Laws of large numbers for sums of extreme values, Ann. Probability,10, Pickands, J. (1975). Statistical inference using extreme order statistics, The Annals of Statistics, 25, Resnick, S. I. (1997). Heavy-tail modeling and teletraffic data, The Annals of Statistics, 25, Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Stochastic Modeling, Springer Science+ Business Media LLC, New York, U.S.A. Resnick, S. I. and Starica, C. (1995). Consistency of Hill estimator for dependent data, Journal of Appl.Probab., 32, Resnick, S. I. and Starica, C. (1997). Smoothing the Hill estimator, J. Appl. Probab., 29, Rice, J. (1988). Mathematical Statistics and Data Analysis, Brookes/Cole, Pacific Grove, CA. 44