Estimating the variance of bootstrapped risk

Estimating the variance of bootstrapped risk measures Joseph H.T. Kim∗and Mary R. Hardy† University of Waterloo October 20, 2007

Abstract Nonparametric variance estimation for the distortion risk measure can be readily done through the bootstrap or the nonparametric delta method based on the influence function. The same task for the bootstrapped risk measures, however, has been relatively unexplored in the literature. In this paper we analytically derive the influence function of the exactly bootstrapped quantile and later extend this to the L-estimator class. The resulting formula provides an alternative method to estimate the variance of the bootstrapped risk measures, or the whole L-estimator class in an analytic form. A simulation study shows that this new method is comparable to the ordinary resampling-based bootstrap method.

1

Introduction

Estimating a risk measure and its variability from a given sample is an important topic in risk management. Being sensitive to extreme values in the data, the estimated risk measure can substantially vary even for samples from the same distribution, if the sample size is small. Consequently the variance of the estimated risk Joseph Kim acknowledges the support in part by the Ph.D. Grant of the Society of Actuaries and the PGS-D2 grant of the Natural Sciences and Engineering Research Council of Canada † Mary Hardy acknowledges the support of the Natural Sciences and Engineering Research Council of Canada ∗

1

measure needs to be estimated to measure the uncertainty in the estimated risk measure. This paper proposes a new method of estimating the variance of bootstrapped distortion risk measure from a i.i.d.1 random sample. Expressed as a functional mapping from a random variable to a real value ρ : X → R, risk measures are used in, for example, setting required capital, insurance premiums, and portfolio optimization. For the actual form of ρ, the distortion risk measure (DRM) class introduced by Wang (1996) has received much attention. The DRM is defied by ! 0 ! +∞ ρg (F ) = − [1 − g(F¯ (x))]dx + g(F¯ (x))dx −∞ 0 ! 1 = F −1 (u)g # (1 − u)du, (1) 0

where g is an increasing function defined on [0, 1] with g(0) = 0 and g(1) = 1, and F¯ (x) = 1 − F (x). The DRM satisfies translation invariance, positive homogeneity, monotonicity, and additivity for comonotonic losses. If g is concave, the risk measure becomes coherent, in the sense of Artzner et al. (1999), by satisfying subadditivity as well; see Wirch and Hardy (2000) and Dhaene et al. (2006). Examples of the DRM include Value-at-Risk (VaR), the Conditional Tail Expectation (CTE), and the Proportional Hazards Transform (PHT) measure, each of which takes a different g. All of these, except the VaR, are coherent because the corresponding g’s are concave. Turning to the actual estimation of ρg (F ) from a sample, an obvious choice is the empirical risk measure ρg (Fˆ ) where Fˆ is the empirical distribution function. That is ρg (Fˆ ) = c1 X(1) + c2 X(2) + . . . + cn X(n) = c# X:n

(2)

where c = (c1 , c2 , . . . , cn )# , X:n = (X(1) , X(2) , . . . , X(n) )# , and ! i/n ci = g # (1 − u)du, i = 1, 2, . . . , n. (i−1)/n

The variability of a given risk measure can be measured by a confidence interval (see, e.g., Kaiser and Brazauskas (2007)) or a variance estimate. For estimating 1

independent and identically distributed.

2

the variance of t(Fˆ ) we can use non-parametric methods such as the bootstrap or the nonparametric delta method. For the latter method Jones and Zitikis (2003) identified the DRM in (1) is equivalent to the L-estimator class whose standard expression is given by ! 1

F −1 (u)J(u)du,

(3)

0

where J(u) is commonly called the score function defined on [0, 1] in the statistics literature. Thus, by setting J(u) = g # (1 − u), they have an access to solid statistical results, such as the asymptotic variance of the empirical estimate of DRM.

Different choices, however, are possible to estimate ρg (F ). Most notably, Kim and Hardy (2007) investigated the bias of the Value-at-Risk (VaR) and the Conditional Tail Expectation (CTE), using the exact bootstrap (EB) of Hutson and Ernst (2000), and proposed a guideline on how to use the bootstrap without obtaining a compromising increase in variance. Their simulations show that the guideline often favors the bootstrapped risk measure2 over the empirical one for the CTE case, in terms of the mean squared error, but the variance for the bootstrapped risk measures is not considered in their paper. While it is true that bootstrap estimate and the empirical one converges they can be substantially different for a finite sample and there are situations where increasing the sample size is constrained in practice, as discussed in Kim and Hardy (2007). In this paper, we explore two non-parametric methods to estimate the variance of the bootstrapped distortion risk measure; in this sense this paper is a sequel of Kim and Hardy (2007). The first method is the bootstrap itself. We will examine the exact bootstrap method of Hutson and Ernst (2000) that provides an analytic solution, thus eliminating the resampling simulation. The second method is the non-parametric delta method. While this method can be readily applied to empirical DRMs as shown in Jones and Zitikis (2003), little is known about the influence function of the bootstrapped DRM. In this paper we derive the influence function of the bootstrapped distortion risk measure in an analytic form and thus provide an alternative way of estimating its variance. The resulting formula requires only the analytic form of the risk measure and not its influence function. Consequently the computation algorithm is generally simpler than the delta method for the empirical risk measure. The developments in this paper have other applications. For example, part of a question by Efron (1992) is analytically solved for the L-estimator class. 2

The exactly-bootstrapped (EB) risk measure, to be precise.

3

This paper is organized as follows: In Section 2 a brief review of the bootstrap and the delta method is provided as non-parametric variance estimation tools. While the bootstrap is straightforward in estimating the variance of any estimate, there has been no discussion on its relative efficiency compared to the nonparametric delta method counterpart in the actuarial context. In Section 3 we derive the influence function of the bootstrapped quantile and extend this to the L-estimator class. Also a qualitative discussion on statistical aspects of the result follows in this section. Section 4 is devoted for a simulation study. Section 5 wraps up this paper.

2

Non-parametric variance estimation

This section provides a brief review of the two methods with variance estimation in mind.

2.1

The exact bootstrap

The core idea of the bootstrap is to repeatedly resample from the given data with replacement; see Efron and Tibshirani (1993), Shao and Tu (1995), Hall (1992), and Davison and Hinkley (1997) for comprehensive treatment for this subject. For the L-estimator class Hutson and Ernst (2000) derived the exact bootstrap mean and variance estimate. Theorem 2.1 (Hutson and Ernst (2000)) The exact bootstrap (EB) of the estimate of E(X(r) |F ), 1 ≤ r ≤ n is E(X(r) |Fˆ ) = where wj(r) and

n "

wj(r) X(j) ,

j=1

# $% # $ # $& n j j−1 =r B ; r, n − r + 1 − B ; r, n − r + 1 , r n n B(x; a, b) =

!

0

x

ta−1 (1 − t)b−1 dt.

4

2 Theorem 2.2 (Hutson and Ernst (2000)) The exact bootstrap of estimate of σr:n 2 and σrs:n , (1 ≤ r ≤ n and r < s) are

V ar(Xr:n |Fˆ ) = Cov(Xr:n , Xs:n |Fˆ ) =

n " j=1

wj(r) (Xj:n − µ ˆr:n )2 ,

j−1 n " " j=2 j=1 n "

+

j=1

vij(rs) (Xi:n − µ ˆr:n )(Xj:n − µ ˆs:n )

vjj(rs) (Xj:n − µ ˆr:n )(Xj:n − µ ˆs:n )

where the weights are vij(rs) =

!

j/n

(j−1)/n

vjj(rs) =

!

j/n

(j−1)/n

!

i/n

frs:n (ur , us )dur dus ,

(i−1)/n

!

us

frs:n (ur , us )dur dus ,

(j−1)/n

s−r−1 and frs:n (ur , us ) = n Crs ur−1 (1 − us )n−s is the joint distribution of two r (us − ur ) uniform order statistics Ur:n and Us:n with n Crs = n!/(r − 1)!(s − r − 1)!(n − s)!.

Kim and Hardy (2007) presented several bootstrap-related quantities of the L-estimator in matrix form which is convenient in notation and useful for programming. In the same spirit, we provide an alternative expression of the variance formula in matrix form. Let D(wr ) be the diagonal matrix with (i, i)-th element equal to i-th element of wr = (w1(r) , w2(r) , . . . , wn(r) )# , the EB mean weight vector of the r-th order statistic defined in Theorem 2.1. Also denote the upper-triangular matrix {vij(rs) }i≤j by v(rs) . Then the EB variance and covariance terms can be re-written as V ar(Xr:n |Fˆ ) = (X:n − µ ˆ(r) )# D(wr ) (X:n − µ ˆ(r) ) # # = (X:n − wr X:n ) D(wr ) (X:n − wr# X:n ), and Cov(Xr:n , Xs:n |Fˆ ) = (X:n − µ ˆ(r) )# v(rs) (X:n − µ ˆ(s) )# = (X:n − wr# X:n )# v(rs) (X:n − ws# X:n ), 5

for 1 ≤ r < s ≤ n. Note that each (co)variance term of order statistic(s) is a quadratic form; D(wr ) and v(rs) are independent of the sample, so they are recyclable for other samples of the same size. Now let us further denote the EB covariance matrix of order statistics by Σ:n = {Cov(X(r) , X(s) |Fˆ )}ni,j=1 .

(4)

Then the EB variance of empirical DRM estimate c# X:n in (2) is V ar(c# X:n |Fˆ ) = c# Σ:n c. Now we turn to the bootstrapped (EB) risk measure, which is our quantity of interest. As Kim and Hardy (2007) reported, sometimes the EB risk measures, or the bias-corrected EB risk measures, are preferred to the empirical risk measures in terms of the mean squared error. Following their notation, the EB estimate of an L-estimator, c# X:n as in (2), is given by E(ρ(Fˆ )|Fˆ ) = E(c# X:n |Fˆ ) = c# w# X:n ,

(5)

where the matrix w = {wi(j) }ni,j=1 comes from the EB weights for each element of X:n . Similarly, the bias-corrected estimate is ! = c# (2I − w# )X:n . ρ(Fˆ ) − Bias

(6)

The bootstrap variance estimates of these bootstrapped quantities therefore are V ar(c# w# X:n |Fˆ ) = c# w# Σ:n wc, and V ar(c# (2I − w# )X:n |Fˆ ) = c# (2I − w# )Σ:n (2I − w)c. Apart from its more appealing appearance, this matrix expression can be advantageous if, for some reason, different risk measures are to be compared (e.g., WT vs. CTE 95 %). One only needs to change vector c because the other two matrices w and Σ:n are fixed for the given sample. The question is then how hard it is to compute Σ:n . While this formula allows us to compute the EB variance for any L-estimator analytically, a closer look prompts a computational issue which Hutson and Ernst 6

(2000) did not discuss. First notice that Σ:n in (4) is a n × n matrix but computing each element of this matrix involves another n × n matrix. The total number of computations is of order O(n4 ) 3 ; the computational burden increases exponentially as the sample size gets larger. Secondly if each v(rs) is to be stored for future recycling, a huge storage capacity is required4 . Thirdly, even if these have been stored for sample size n, one should recalculate all these matrices as the sample size changes. For these reasons we recommend replacing Σ:n with the ordinary resampling version ˆ :n with R bootstrap samples. For the size of R, Booth and Sarkar (1998) showed Σ that R for variance estimation needs to be bigger than previously thought. In summary using the bootstrap to estimate the variance of both empirical DRM estimate and its EB estimate is a straightforward exercise with resampling simulations only needed for the covariance matrix.

2.2

Nonparametric delta method

An alternative way to estimate the variance of an estimator is using the nonparametric delta method (or just delta method in short) which employs the influence function of the estimator. Estimating the variance through the nonparametric delta method is well known and can be found in standard texts such as Staudte and Sheather (1990) or Hampel et al. (1986). Consider the von Mises expansion of any statistical functional t(G) at F . The first order approximation is ! t(G) ≈ t(F ) + Lt (x|F )dG(x). Here the first derivative of t at F , Lt , is called the influence function (IF). The IF is a function of x given F and t, and defined by t[(1 − #)F + #Hx ] − t(F ) . !→0 #

Lt (x|F ) = lim

where Hx is the distribution function of a degenerated random variable at x, commonly referred to the heaviside function. The IF measures the relative influence on T (F ) of a very small amount of contamination at x and also can be used to estimate the variance of T . For the purpose of estimating the variance from the sample, we For instance, when n = 400 the number of computations is about 1.28×1010 and the computing time takes more than a week in Matlab using a high-performance Unix machine. 4 Hard disk space needs almost 6GB when n = 200; more than 200GB when n = 400 in Matlab. 3

7

set G = Fˆ , which is a choice that makes the approximation reasonably accurate as n increases, and the approximation becomes t(Fˆ ) ≈ t(F ) +

!

n

1" Lt (x|F )dFˆ (x) = t(F ) + Lt (xi |F ) n i=1

Now by applying the central limit theorem, T = t(Fˆ ) has asymptotic normality: d

t(Fˆ ) − t(F ) −→ N (0, vL (F )), (7) ' where vL (F ) = n−1 V ar(Lt (X)) = n−1 L2t (x|F )dF (x). Assuming no information on F in practice, we estimate this variance using the sample version: n 1 " 2 ˆ vL (F ) ≡ 2 L (xj |Fˆ ), n j=1 t

(8)

where xj is the j−th observation of the sample. If we focus on the L-estimator, defined in (3), of which the score function J is bounded and continuous, the IF and the asymptotic variance are known to be (See, e.g., Appendix B in Staudte and Sheather (1990)) ! ∞ ! ∞ Lt (x|F ) = J(F (y))dy − F (y)J(F (y))dy (9) −∞

x

and vL (F ) = n

−1

!

∞

−∞

!

∞

−∞

J(F (y))J(F (z))[F (min(y, z)) − F (y)F (z)]dydz.

(10)

These formulas may reduce to a more convenient form depending on J. For example ' −1 ∞ the CTE at a confidence level of α is defined by (1 − α) xdF and its IF and Qα the asymptotic variance is given by   x − αQα − t(F ) if Qα < x Lt (x|F ) = 1−α  Q − t(F ) if Q ≥ x α

and

vL (F ) =

α

V ar(X|X > Qα ) + α(E[X|X > Qα ] − Qα )2 . n(1 − α) 8

In general the sample estimate of the variance in (10) is given by vL (Fˆ ) = n

−1

# $ & # $ # $% n−1 " n−1 " i i j ij j J min , − (X(i+1) −X(i) )(X(j+1) −X(j) ) J n n n n n n j=1 i=1

(11) This is the form used in Jones and Zitikis (2003) for DRM variance estimation. For most DRMs such as the CTE and PHT-measure, the variance estimate of the empirical estimates can be computed by this formula; see Gourieroux and Liu (2006) for variance estimates of different risk measures. However, this result does not apply to the VaR, the quantile risk measure, because its score function is a discontinuous step function. The IF of the quantile still exists, but poses a practical difficulty. We will return to this point in the next section. Finally we note that while the delta method illustrated in this section works well for the empirical DRM estimator, it cannot be applied directly to the bootstrapped or bias-corrected counterpart because they are not empirical estimates of some continuous functionals.

3

Variance of EB risk measures

In this section we show that the IF of the bootstrapped L-estimator (or DRM) is available in an analytic form. In particular, the influence function of the bootstrapped quantile is first derived and is later extended to the whole L-estimator class, or DRM.

3.1

EB quantile

Let us start with the quantile case. The IF of the standard quantile, defined by t(F ) = F −1 (α), is given by  α−1   , if x < F −1 (α) −1 (α)) f (F Lt (x|F ) = (12) α −1   , if x > F (α) f (F −1 (α))

with the asymptotic variance

vL (F ) = n−1

α(1 − α) . f (F −1 (α))2

9

Thus one needs to estimate f (F −1 (α)), the value of the density at the quantile, to estimate the variance; in fact, if the score function consists of discrete masses the corresponding IF and variance involve the unknown density f . This task however might be impractical because estimating the value of density at a large α from the given sample is not easy. Although there are nonparametric tools available allowing us to estimate the density, such as kernel density estimation, it may not produce satisfactory values for tail regions, especially when the given sample is subject to skewness or (and) kurtosis, which is often true for financial and actuarial data. In practice therefore, using the nonparametric delta method to estimate the variance of VaR, a tail quantile risk measure, can be problematic. This is a long-standing problem in many statistical applications. Here we consider the EB quantile and its variance. A closer look at the EB formula given in Theorem 2.1 sheds a new light. We first note that the EB estimate of the r-th order statistic is given by a linear combination of all the order statistics, again belonging to the L-estimator class, and the weight coefficients wi(r) , i = 1, 2, . . . , n, form a Beta density function with parameter (r +1, n−r) if appended together. This observation suggests that the EB estimate ! 1 n " ˆ E(X(r) |F ) = wj(r) X(j) = Fˆn−1 (u)Jq (u)du 0

j=1

can be thought as the plug-in estimate of a functional ! 1 F −1 (u)Jq (u)du,

(13)

0

where the score function Jq (u) =

ur−1 (1 − u)n−r B(r, n − r + 1)

(14)

is continuous, bounded, and differentiable. We will call functional (13) the EB α-th quantile (α = r/n). We can now apply the result (9) and obtain the following result. Lemma 3.1 The IF of the EB α−th quantile defined in (13) is given by %! x & ! ∞ 1 nα−1 n(1−α) nα−1 n(1−α)+1 F (y) (1−F (y)) dy− F (y) (1−F (y)) dy B(nα, n − r + 1) −∞ −∞ (15) 10

Proof: See Appendix. ! The IF of the EB quantile looks quite different from that of the standard quantile given in (12). It is a function of the sample size n and, more importantly, does not involve the density f , leading to straightforward computation of variance estimate with a sample. Note that it is possible to define an alternative α-th quantile, say, by setting α = r/(n + 1) instead. One might further employ an weighted average; the IF is then the weighted average of each IF in that case. In our case, the resulting variance is then given by (10) with the score function J(u) replaced by Jq (u) from (14). Thus the variance estimate is given by # $ # $% # $ & n−1 n−1 j 1 "" i i j ij Jq Jq min , − (X(i+1) − X(i) )(X(j+1) − X(j) ), (16) n j=1 i=1 n n n n nn where

# $r−1 # $n−r i i # $ 1− i n n Jq = (17) n B(r, n − r + 1) For implementation, the numerator of the right hand side requires an approximation because it often produces 0 for large n and r in mathematical software. We recommend the approximation $n−r # $r−1 # ! i n i i 1− ≈n ur−1 (1 − u)n−r du i−1 n n n because the right hand side is readily available through the incomplete beta function. Then the score function in (17) is rewritten as ! i n r−1 n−r # $ n i−1 u (1 − u) du i n Jq ≈ = nwi(r) , n B(r, n − r + 1) where wj(r) is the (j, r)th element of the EB weight matrix w. Plugging this back into (16) finally yields the variance estimate for the EB quantile in the most computationallyconvenient form: n−1 " n−1 " j=1 i=1

% & ij wj(r) wi(r) min(i, j) − (X(i+1) − X(i) )(X(j+1) − X(j) ), n 11

(18)

3.2

Extension to EB L-estimator

For any fixed vector c = (c1 , . . . , cn )# , which is the weight for each order statistic in the empirical DRM estimate, the EB estimate is given by n n " " ˆ E( ck X(k) |F ) = ck E(X(k) |Fˆ ) k=1

k=1

= =

n "

k=1 n " k=1 1

=

!

0

ck

n "

wj(k) X(j)

j=1

ck

!

1

0

Fˆn−1 (u)Jq (u)du

Fˆn−1 (u)

n "

ck Jq (u)du

k=1

The score function of the EB L-estimator is then can be defined by JL (u) =

n "

ck Jq (u) =

k=1

n " ck uk−1 (1 − u)n−k k=1

B(k, n − k + 1)

,

and we have the following result. Corollary 3.2 The IF of the bootstrapped L-estimator c# w# X:n , with c = (c1 , . . . , cn )# , is given by n " k=1

ck B(k, n − r + 1)

%!

x

−∞

k−1

F (y)

n−k

(1 − F (y))

dy −

!

∞

−∞

k−1

F (y)

n−k+1

(1 − F (y))

dy

&

The variance estimate is given by (10) with J(u) replaced by JL (u), with its sample version given by n−1 n−1

1 "" JL n j=1 i=1

# $ # $% # $ & j i i j ij JL min , − (X(i+1) − X(i) )(X(j+1) − X(j) ) n n n n nn

Again we need an approximation for JL . That is,

12

! i # $k−1 # $n−k n i i ck uk−1 (1 − u)n−k du # $ " 1− n ck n " i−1 i n n n JL = ≈n n B(k, n − k + 1) B(k, n − k + 1) k=1 k=1 =n

n " k=1 i

ck wi(k)

= nw c,

(19) (20)

i = 1, 2, . . . , n − 1,

(21)

where wi is the i-th row vector of the EB weight matrix w = {wi(j) }n1 and c is the weight vector. Thus, the most computationally-convenient form of the variance estimate of a bootstrap DRM is & % n−1 " n−1 " , i -, j ij (X(i+1) − X(i) )(X(j+1) − X(j) ) (22) w c w c min(i, j) − n j=1 i=1 If the bias-corrected estimator is required, one needs to change the score function

to JLBC (u) = 2

n " k=1

ck −

n " ck uk−1 (1 − u)n−k k=1

B(k, n − k + 1)

,

and repeat the argument in the same fashion.

3.3

Further remarks

In this section we have derived the IF of the bootstrapped quantile and extended it to the L-estimator (or equivalently the bootstrapped DRM). There are several comments on the analytic developments. • The use of the delta method requires the knowledge of the IF of the DRM of interest. However the same task for the bootstrapped DRM only needs the form of the DRM, that is the the score function. Because of this the variance estimation algorithm is much easier and straightforward. • The key idea of the nonparametric delta method in (8) states that Lt (x|F ) can be estimated by Lt (x|Fˆ ), based on the convergence of Fˆ to F . Since Fˆ EB also converges to the true F , estimating Lt (x|F ) using Lt (x|Fˆ EB ) can be likewise justified. This gives an alternative way to estimate the variance of a statistic t where applying the nonparametric delta method is difficult for some reason. 13

• It is known that both the IF approach and the bootstrap approach produce similar variance estimates for an arbitrary statistic; see, for example, Section 2.7.4 of Davison and Hinkley (1997). This means that as sample size increases one would expect the two numbers from both methods to get close. This point will be illustrated in the numerical examples of the following section.

4

Numerical example

For the simulation study we consider small sample sizes for this problem, say n ≤ 1000, using different nonparametric methods. The sample size of less than 1000 is common in actuarial loss modelling. For the simulation study we consider the same three parametric models used in Kim and Hardy (2007), namely, the LN put, RSLN2 put, and the Pareto loss. We consider the bootstrapped estimate of the CTE at 95%, VaR at 95%, and PHT measure with r = 0.9, respectively, for each of the three models. The corresponding empirical estimates are given by " CT Eα =

n " 1 X(j) , n(1 − α) j=nα+1

V" aRα = X(nα) , and P" HTr =

n "

cj X(j) ,

j=1

where cj =

!

j n j−1 n

r(1 − x)r−1 dx

and their bootstrapped counterparts can easily be computed by (5). Since our focus is on the nonparametric variance estimation of the bootstrapped DRM, the influence function (IF) based nonparametric delta method developed in the previous section is compared with the variance estimate using the ordinary bootstrap using 1,000 and 2,500 resamplings for each sample of size 400 and 1,000, respectively, under the mean squared error (MSE) criterion. We report the results for sample sizes 400 and 1,000, respectively along with standard errors (s.e.), by repeating 5,000 sets 14

of simulations. Table 1 shows the numbers in percentage of the true variance of each model. From the Table, we have several comments. • First of all, two different methods of variance estimation produce very similar values, except the VaR case. At this point it is unclear why the delta method leads to a bigger variance than the bootstrap for the VaR, but we conjecture this is due to the fact that the bootstrapped VaR has been defined on a single order statistic, which may not be smooth enough. On the other hand we see that the delta method is slightly but consistently better than the bootstrap counterpart. This indicates that the performances of the two different variance estimation greatly depend on the underlying distribution as well as the risk measure itself, which is not surprising. • For the bootstrap, we used the resampling size of 1,000 for each sample of size 400 and of 2,500 for a sample of size 1,000. So the resampling size has been set at 2.5 times the sample size for our simulations. A separate simulations show that if the bootstrap resampling size R is too small, the resulting values from the ordinary bootstrap can suffer an unacceptably big bias5 . • In all cases the MSEs of these variance estimators seem large even for samples of size 1,000 for the Pareto case, making both methods less than satisfactory for thick–tailed distributions in practice. This is particularly the case when the risk measure is heavily weighted is in the tail region. • Since both nonparametric methods presented here rely on asymptotic arguments, we generally cannot consider one over the other without a theoretical justification, and as far as we know there is no such one. Considering that the OB for variance needs more bootstrap resamplings than for mean, we believe that the IF method can sometimes be advantageous because it does not involve any simulation.

5

Concluding remarks

In this paper we derived the influence function (IF) of the EB quantile estimate and showed it exists in an analytic form where no density function needs to be computed. 5

We found that when the sample size 1, 000, the results pose serious bias issue even for R = 1, 000.

15

16 1000

400

1000

400

1000

n 400

Method OB IF OB IF OB IF OB IF OB IF OB IF

EB VaR at α = 95% true val rmse (s.e.) 13.420 39.57% (4.21%) 13.42 50.54% (6.81%) 5.552 33.72% (1.33%) 5.552 41.55% (1.98%) 15.137 39.71% (4.78%) 15.137 50.21% (7.61%) 6.248 32.51% (1.38%) 6.248 40.14% (2.07%) 13.819 58.95% (9.35%) 13.819 71.51% (13.61%) 5.86 39.19% (1.86%) 5.86 46.76% (2.61%)

EB PHT with c = 0.9 true val rmse (s.e.) 0.213 21.96% (0.33%) 0.213 21.40% (0.28%) 0.084 14.26% (0.24%) 0.084 13.88% (0.24%) 0.398 17.76% (0.25%) 0.398 17.19% (0.25%) 0.16 11.18% (0.19%) 0.16 10.90% (0.13%) 0.966 54.02% (0.94%) 0.966 52.55% (0.91%) 0.391 36.37% (0.51%) 0.391 35.29% (0.51%)

Table 1: Variance estimation of three bootstrapped risk measures for different sample sizes (n = 400 and 1000) and models (LN put, RSLN put, and Pareto) : comparison of the ordinary bootstrap (OB) and the delta method (IF)

Pareto

RSLN2 put

Model LN put

EB CTE at α = 95% true val rmse (s.e.) 13.262 31.06% (2.59%) 13.262 31.756% (2.61%) 5.318 21.273% (0.57%) 5.318 21.401% (0.57%) 14.343 31.342% (2.84%) 14.343 31.945% (2.85%) 5.757 20.954% (0.59%) 5.757 21.013% (0.59%) 65.777 111.423% (163.13%) 65.777 111.527% (163.41%) 26.737 67.911% (24.68%) 26.737 67.668% (24.50%)

The result directly extends to the whole L-estimator class. Based on this finding, we conducted a simulation study to estimate the variance of the bootstrapped risk measures and compares it against the ordinary resampling method. The result suggests that these two method would produce comparable results when the corresponding risk measure is smooth; the proposed method shows bigger variance for the VaR case, though. It would be interesting to examine alternative VaR estimators that are weighted average of more than one order statistic, as suggested in Kim and Hardy (2007). These are smoother than the VaR estimator considered in this paper.

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Hutson, A. D. and Ernst, M. D. (2000). The exact bootstrap mean and variance of an L-estimator. Journal of Royal Statistical Society: Series B , 62, 89–94. Jones, B. L. and Zitikis, R. (2003). Empirical estimation of risk measures and related quantities. North American Actuarial Journal , 7(4). Kaiser, T. and Brazauskas, V. (2007). Interval estimation of actuarial risk measures. NAAJ , 10(4). Kim, J. H. T. and Hardy, M. R. (2007). Quantifying and correcting the bias in estimated risk measures. ASTIN Bulletin,, (Forthcoming). Manistre, B. J. and Hancock, G. H. (2005). Variance of the CTE estimator. North American Actuarial Journal , 9(2), 129–156. Shao, J. and Tu, D. (1995). The jackknife and bootstrap. Springer Series in Statistics. Springer, New York. Staudte, R. G. and Sheather, S. J. (1990). Robust estimation and testing. John Wiley & Sons, New York. Wang, S. S. (1996). Premium calculation by transforming the layer premium density. ASTIN Bulletin, 26(1), 71–92. Wirch, J. L. and Hardy, M. R. (2000). Distortion risk measures: Coherence and stochastic dominance. Woking paper .

A

Appendix

Proof of Lemma 3.1 Let us From Theorem 2.1, the EB quantile is defined by ! 1 t(F ) = J(u)F −1 (u)du, 0

where J(u) = ur−1 (1 − u)n−r /B(r, n − r + 1). Then the IF of t(F ) is % & ! 1 ! 1 u − H(F −1 (u) − x) J(u)Lt (x|F )du = J(u) du, f (F −1 (u)) 0 0 18

(23)

from the IF of the quantile in (12). Using the variable transformation y = F −1 (u), this becomes ! ∞ J(F (y))(F (y) − H(y − x))dy −∞ ! x ! ∞ = J(F (y))dy − (1 − F (y))J(F (y))dy. −∞

−∞

Replacing J(u) = ur−1 (1 − u)n−r /B(r, n − r + 1) leads to %! x & ! ∞ 1 r−1 n−r r−1 n−r+1 F (y) (1 − F (y)) dy − F (y) (1 − F (y)) dy B(r, n − r + 1) −∞ −∞ Finally setting α = r/n completes the proof. Since quantiles are not uniquely defined for the empirical distribution function due to its discreteness, other possible α-th quantiles, such as α = r/(n + 1), may also be used equally. In any case, we note that it is easy to verify that the expected value of the above IF is zero using differentiation. We emphasize that the IF is a function of x. Q.E.D.

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