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Estimation of Distortion Risk Measures

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Estimation of Distortion Risk Measures Int. Statistical Inst.: Proc. 58th World Statistical Congress, 2011, Dublin (Session CPS046) p.5035 Estimation of Distortion Risk Measures Hideatsu Tsukahara Department of Economics, Seijo University, Tokyo e-mail address: [email protected] The concept of coherent risk measure was introduced in Artzner et al. (1999). They listed some properties, called axioms of `coherence', that any good risk measure should possess, and studied the (non-)coherence of widely-used risk measure such as Value-at- Risk (VaR) and expected shortfall (also known as tail conditional expectation or tail VaR). Kusuoka (2001) introduced two additional axioms called law invariance and comonotonic additivity, and proved that the class of coherent risk measures satisfying these two axioms coincides with the class of distortion risk measures with convex distortions. To be more speci¯c, let X be a random variable representing a loss of some ¯nancial position, and let F (x) := P(X · x) be the distribution function (df) of X. We denote ¡1 its quantile function by F (u) := inffx 2 R: FX (x) ¸ ug; 0 < u < 1. A distortion risk measure is then of the following form Z Z ¡1 ½D(X) := F (u) dD(u) = x dD ± F (x); (1) [0;1] R where D is a distortion function, which is simply a df D on [0; 1]; i.e., a right-continuous, increasing function on [0; 1] satisfying D(0) = 0 and D(1) = 1. For ½D(X) to be coherent, D must be convex, which we assume throughout this paper. The celebrated VaR can be written of the form (1), but with non-convex D; this implies that the VaR is not coherent. Also note that di®erent authors use di®erent names spectral risk measure or weighted V@R for a distortion risk measure. The most well-known example of coherent risk measure is£ the above-mentioned¤ ex- ES ¡1 ¡ ¡ pected shortfall. Taking distortion of the form D® (u) = ® u (1 ®) +, 0 < ® < 1 yields the expected shortfall as a distortion risk measure: Z 1 1 ¡1 ES®(X) := F (u) du: ® 1¡® The following one-parameter families of distortion yields several classes of coherent risk measures: ² PH ¡ ¡ θ Proportional hazards (PH) distortion: Dθ (u) = 1 (1 u) , ² PO ¡ ¡ Proportional odds (PO) distortion: Dθ (u) = θu=[1 (1 θ)u] ² GA ¡1 Gaussian distortion: Dθ (u) = ©(© (u) + log θ) To implement the risk management/regulatory procedure using risk measures, it is necessary to statistically estimate their values based on data. For a distortion risk measure, its form (1) suggests a natural estimator which is a simple form of an L-statistic. The main theme of this paper is to derive the asymptotic statistical properties of simple estimators of those risk measures based on strictly stationary sequences, and to compare some distortion risk measures and VaR. 1 Int. Statistical Inst.: Proc. 58th World Statistical Congress, 2011, Dublin (Session CPS046) p.5036 Let (Xn)n2N be a strictly stationary process with a stationary distribution F , and denote by Fn the empirical df based on the sample X1;:::;Xn. A natural estimator of ½(X) is given by Z 1 Xn b F¡1 ½n = n (u) dD(u) = cniXn:i; (2) 0 i=1 where cni := D(i=n) ¡ D((i ¡ 1)=n) and Xn:1 · Xn:2 · ¢ ¢ ¢ · Xn:n are the order statistics based on the sample X1;:::;Xn. In what follows, instead of restricting ourselves to the particular form (2) of L-statistic, we consider a general L-statistic of the following form: 1 Xn T = c h(X ); (3) n n ni n:i i=1 where cni's are constants. De¯ne for 0 · u · 1, Z Xn u Jn(u) := cni1((i¡1)=n; i=n](u) + cn11f0g(u); ªn(u) := Jn(v) dv i=1 1=2 Then we have Z Z 1 F¡1 F¡1 Tn = h( n (u))Jn(u) du = h( n (u)) dªn(u): 0 [0;1] Let g := h ± F ¡1, and de¯ne the centering constants Z Z 1 ¹n := g(u)Jn(u) du = g(u) dªn(u): 0 [0;1] Consistency is a basic desirable property of statistical estimators. The following result was proved in van Zwet (1980) for the i.i.d. case, but his proof remains to be valid for the ergodic case. Proposition 1 Suppose that X1;X2;::: forms an ergodic stationary sequence. Let 1 · p q p · 1, 1=p + 1=q = 1, and assume that Jn 2 L (0; 1) for n = 1; 2;:::, and g 2 L (0; 1). If either · 1 j jp 1 (i) 1 < p and supn E( Jn ) < , or (ii) p = 1 and fJn; n = 1; 2;:::g is uniformly integrable, then we have Tn ¡ ¹n ! 0; a.s.. R R 2 t t Further, if there exists aR function J Lp such that limn!1 0 Jn(s) ds = 0 J(s) ds for 2 ! every t (0; 1), then Tn [0;1] J(s) dg(s), a.s. By this result, in particular, our estimator ½bn in (2) of distortion risk measure proves to possess strong consistency under the very general conditions stated above. For the asymptotic normality, we basically draw upon Shorack and Wellner (1986), Chapter 19, for the form of assumptions and the line of argument. First we set out the following assumption on (Xn). 2 Int. Statistical Inst.: Proc. 58th World Statistical Congress, 2011, Dublin (Session CPS046) p.5037 F j (A.1) (Xn)n2N is strongly mixing: Setting i := σ(Xi;:::;Xj), the strong mixing coe±- cient n o j \ ¡ j 2 F k 2 F 1 ¸ ®(n) := sup P (A B) P (A)P (B) : A 1 ;B k+n; k 1 converges to 0 in such a way that p ®(n) = O(n¡θ¡´) for some θ ¸ 1 + 2; ´ > 0 Note that strong mixing assumption is the weakest requirement among various mixing concepts. Next we assume the bounded growth of g and Jn, and smoothness of Jn. (A.2) h is a function of bounded variation: h = h1 ¡ h2, where h1 and h2 are increasing, left-continuous, and satisfy ¡1 jhi(F (u))j · H(u); for all 0 < u < 1; ¡ ¡ where H(u) := Mu d1 (1 ¡ u) d2 . R ± ¡1 For g = h F , let dg be theR integral with respect to the Lebesgue-Stieltjes signed measure associated with g, and djgj be the integral with respect to the total variation measure associated with g. (A.3) There exists a function J which is jgj-a.e. continuous such that Jn converges to J locally uniformly jgj-a.e. ¡b ¡b (A.4) For B(u) := Mu 1 (1 ¡ u) 2 , jJn(u)j · B(u), jJ(u)j · B(u) for all 0 < u < 1 with b1 _ b2 < 1. We note that under (A.2) and (A.4), Z 1 [u(1 ¡ u)]rB(u) djgj(u) < 1 (4) 0 when r > (b1 + d1) _ (b2 + d2) (see Shorack and Wellner (1986), Lemma 19.1.1). Before we state and prove the asymptotic normality of the estimator (2), let us note that it is possible to reduce the argument to the uniform case, as in the i.i.d. case. Namely, there exists a strictly stationary sequence (»n)n2N with the same mixing rate as (Xn) such ¡1 that Xn = F (»n) and »n » U(0; 1) (on a possibly extended probability space; see Lemma 4.2 in Dehling and Philipp (2002)). Let Gn be the empirical df based on »1;:::;»n. Then Z L £ ¤ Tn ¡ ¹n = = g(u) d ªn(Gn(u)) ¡ ªn(u) : (5) [0;1] L Here X = Y means that the random variables X and Y have the same distribution. Let Ck(u; v) := P(»1 · u; »k · v) and put X1 X1 σ(u; v) := u ^ v ¡ uv + [Ck(u; v) ¡ uv] + [Ck(v; u) ¡ uv]: (6) k=2 k=2 When (»n) satis¯es the same mixing rate as in (A.1), it follows from the covariance inequal- ity (see Dehling and Philipp (2002), Lemma 3.9) that the two series on the right-handp side of (6) are absolutely convergent. We de¯ne the empirical process Un(u) := n(Gn(u) ¡ u) as usual. 3 Int. Statistical Inst.: Proc. 58th World Statistical Congress, 2011, Dublin (Session CPS046) p.5038 Theorem 2 Let (Xn)n2N be a strictly stationary sequence satifying (A.1){(A.4) with 2b + 1 1 b + d + i < ; i = 1; 2 (7) i i 2θ 2 Then we have p L 2 n(Tn ¡ ¹n) ¡! N(0; σ ); where Z Z 1 1 σ2 := σ(u; v)J(u)J(v) dg(u)dg(v) < 1 (8) 0 0 Returning to the problem of estimating distortion risk measures, we should set cni = n[D(i=n) ¡ D((i ¡ 1)=n)] and h(x) = x. Then in most cases, the limit of Jn will be d, so applying Theorem 2 we have the following corollary. Corollary 3 Assume (A.1), (A.2) with h(x) = x, (A.3) with J = d, and (A.4). Then, for the estimator ½bn of (2), we have p L 2 n(½bn ¡ ½(X)) ¡! N(0; σ ); where Z Z 1 1 σ2 = σ(u; v)d(u)d(v) dF ¡1(u)dF ¡1(v): 0 0 When we try to construct approximate con¯dence intervals for risk measures, we need to estimate the asymptotic variance (8). Let Z Yn := J(F (x)) dh(x); n 2 Z: [Xn;1) It is then easy to see that σ2 is written as the double-sided in¯nite sum of autocovariance γ(n) of the stationary sequence (Yn).
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