Jackknife Empirical Likelihood Methods for Risk Measures and Related Quantities

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1 Jackkife Empirical Likelihood Methods for Risk Measures ad Related Quatities Liag Peg, Yogcheg Qi 2, Ruodu Wag ad Jigpig Yag 3 Abstract The quatificatio of risks is of importace i isurace. I this paper, we employ the jackkife empirical likelihood methods to costruct cofidece itervals for the risk measures ad some related quatities studied by Joes ad Zitikis (23). A simulatio study shows the advatages of the ew methods over the ormal approximatio method ad aive bootstrap method. Keywords: Cofidece iterval, jackkife empirical likelihood, risk measure. Itroductio I life isurace ad fiace, quatifyig risk is a very importat task for pricig a isurace product or maagig a fiacial portfolio. Geerally speakig, a risk measure is costructed to be a mappig from a set of risks to some real umbers. Some well-kow risk measures iclude coheret risk measures (Yaari (987), Artzer (999)), distortio risk measures, Wag s premium priciple ad proportioal hazards trasform risk measures (Wag, Youg ad Pajer (997); Wag (995), (996), (998); Wirch ad Hardy (999); Necir ad Meraghi (29)). For a risk variable X with distributio fuctio F, Joes ad Zitikis (23) defied a large class of risk measures associated with X as R(F ) = F (t)ψ(t)dt, () where F deotes the geeralized iverse fuctio of F, ad ψ is a oegative fuctio chose for showig the objective opiio about the risk loadig. Differet choices of ψ result i differet risk measures. For example, the proportioal hazards trasform risk measure has ψ(t) = r( t) r School of Mathematics, Georgia Istitute of Techology, Atlata, GA , USA. peg@math.gatech.edu, ruodu.wag@math.gatech.edu 2 Departmet of Mathematics ad Statistics, Uiversity of Miesota Duluth, 7 Uiversity Drive, Duluth, MN yqi@d.um.edu 3 LMAM, Departmet of Fiacial Mathematics, Pekig Uiversity, Beijig 87, Chia. yagjp@math.pku.edu.c

2 ad Wag s premium priciple has ψ(t) = g ( t), where g is a icreasig covex fuctio with derivatives over [, ]; see Joes ad Zitikis (23) for details. Other choices of the fuctio ψ ca be foud i Joes ad Zitikis (27). Joes ad Zitikis (23) also itroduced a related quatity to illustrate the right-tail, left-tail ad two-sided deviatios, which is defied as r(f ) = R(F ) E(X). (2) I this paper, we will focus o the statistical iferece of the risk measure ad its related quatity defied i () ad (2), respectively. How to ifer R(F ) ad r(f ) plays a importat role i the applicatios of risk measures. Recetly, Joes ad Zitikis (23) proposed oparametric estimatio by replacig F ad E(X) by the sample quatile fuctio ad sample mea respectively, ad derived the asymptotic ormality. Therefore, cofidece itervals for R(F ) ad r(f ) ca be costructed via estimatig the asymptotic variace. For comparig two risk measures, we refer to Joes ad Zitikis (25). Joes ad Zitikis (27) ivestigated the oparametric estimatio of the parameter associated with distortio-based risk measures. order to costruct cofidece itervals for R(F ) ad r(f ) without estimatig the asymptotic variace, we ivestigate the possibility of applyig empirical likelihood method i this paper so as to improve the iferece. Empirical likelihood method is a oparametric likelihood approach for statistical iferece, which has bee show to be powerful i iterval estimatio ad hypothesis testig. We refer to Owe (2) for a overview o the method. However, it is kow that empirical likelihood methods are ot effective i dealig with o-liear fuctioals. Recetly, a so-called jackkife empirical likelihood method was proposed by Jig, Yua ad Zhou (29) to deal with oliear fuctioals. The key idea is to formulate a jackkife sample based o estimatig the oliear fuctioal ad the apply the empirical likelihood method for a mea to the jackkife sample. Sice the risk measure R(F ) ad its related quatity r(f ) are o-liear fuctioals, we propose to employ the jackkife empirical likelihood method to obtai iterval estimatio for these two quatities. The paper is orgaized as follows. I Sectio 2, the methodologies ad mai results are preseted. A simulatio study is give i Sectio 3. All proofs are put i Sectio 4. Some coclusios are draw i Sectio 5. I 2

3 2 Methodologies ad mai results Put Ψ(t) = t The the risk measure defied i () ca be writte as R = R(F ) = ψ(s)ds. (Ψ() Ψ(F (t))dt. Let X,, X be idepedet real-valued radom variables with cotiuous distributio fuctio F (x) ad E(X ). Defie the empirical distributio fuctio as F (x) = (X j x). The Joes ad Zitikis (23) proposed to estimate R(F ) ad r(f ) by ˆR = respectively, ad showed that j= (Ψ() Ψ(F (t)))dt, ad ˆr = (Ψ() Ψ(F (t)))dt j= X, j { ˆR R} d N(, σ 2 ) ad {ˆr r(f )} d N(, σ 2 2) (3) uder some regularity coditios, where σ 2 = Q F (Ψ, Ψ), σ 2 2 = µ 2 ( QF (Ψ, Ψ) 2r(F )Q F (Ψ, ) + (r(f )) 2 Q F (, ) ) (4) ad Q F (a, b) = (F (x y) F (x)f (y))a(f (x))b(f (y))dxdy. Based o (3), cofidece itervals for R(F ) ad r(f ) ca be obtaied via estimatig σ 2 ad σ2 2. A alterative way to costruct cofidece itervals is to employ the empirical likelihood method. Sice the risk measure R is o-liear, a commo techique is to liearize the fuctioal by itroducig some lik variables before applyig the profile empirical likelihood method; see the study for variace, ROC curve (Claeskes et al. (23)) ad copulas (Che, Peg ad Zhao (29)). Ufortuately it remais ukow o how to liearize R by itroducig some lik variables. Here we propose to apply the jackkife empirical likelihood method i Jig, Yua ad Zhou (29). This procedure is easy to implemet ad is described as follows. 3

4 Defie F,i = j=,j i (X j x) ad ˆR,i = (Ψ() Ψ(F,i(t)))dt for i =,,. The the jackkife sample is defied as Y i = ˆR ( ) ˆR,i, i =,,. Now we apply the empirical likelihood method to the above jackkife sample. That is, we defie the jackkife empirical likelihood fuctio for θ = R(F ) as L (θ) = sup{ p i : p i, for i =,, ; p i = ; p i Y i = θ}. By Lagrage multiplier techique, we have p i = {+λ(y i θ)} ad 2 log L (θ) = 2 log{+ λ(y i θ)}, where λ = λ(θ) satisfies Y i θ =. (5) + λ(y i θ) The followig theorem shows that the Wilks theorem holds for the proposed jackkife empirical likelihood method. Theorem. Assume that ψ(x) cx α ( x) β, ψ (x) exists ad ψ (x) cx α 2 ( x) β 2 for all < x < ad some costats α > /2, β > /2 ad c >. Further assume E( X i γ ) < for some γ such that γ > /(α /2) ad γ > /(β /2). The we have 2 log L (R ) d χ 2 as, where R deotes the true value of R ad χ 2 freedom. deotes a chi-square distributio with oe degree of Remark. Some well-kow risk measures, such as proportioal hazards trasform risk measure, Wag s right-tail deviatio ad Wag s left-tail deviatio satisfy the assumptios of Theorem ; see Joes ad Zitikis (23). Based o the above theorem, a cofidece iterval for R with level b ca be obtaied as I R b = {R : 2 log L (R) χ 2,b }, where χ 2,b is the b-th quatile of χ2. 4

5 Next we cosider the related quatity r(f ) = R(F )/µ where µ = E(X ). Alteratively, we cosider the quatity R θµ with θ = r(f ). The oe ca estimate this quatity by ˆR θ As before, we defie the jackkife sample as X i = ˆR θ x df (x) = ˆR θ F (x) dx. ( ) ( ) ˆR θ x df (x) ( ) ˆR,i θ x df,i (x) = Y i θx i for i =,,, where Y i s are defied as above. So the jackkife empirical likelihood fuctio for θ = r(f ) is defied as L 2 (θ) = sup{ p i : p i, for i =,, ; p i = ; p i (Y i θx i ) = }. The followig theorem shows that the Wilks theorem holds for the proposed jackkife empirical likelihood method for r(f ). Theorem 2. Assume the coditios of Theorem hold. Further assume EX, ad E(X 2) <. The 2 log L 2 (r ) d χ 2 as, where r deotes the true value of r(f ). Based o the above theorem, a cofidece iterval for r with level b ca be obtaied as I r b = {r : 2 log L 2(r) χ 2,b }. 3 Simulatio study I this sectio we examie the fiite sample behavior of the proposed jackkife empirical likelihood methods i terms of coverage accuracy ad compare with the ormal approximatio method ad the aive bootstrap method. We focus o the proportioal hazards trasform risk measure with ψ(s) = a( s) a. Sice Pareto distributio, log-ormal distributio, Weibull distributio ad Gamma distributio are widely used i fittig the losses data i isurace (Klugma, Pajer ad Willmot (28)), our simulatio study is based o these four distributios. We draw, radom samples of sizes = 3 ad from the followig distributios: 5

6 . Pareto distributio F (x; θ) = x θ for x ; 2. Log-ormal distributio F 2 (x; θ, θ 2 ) = Φ((log x θ )/θ 2 ) for x >, where Φ(x) deotes the stadard ormal distributio; 3. Weibull distributio F 3 (x; θ, θ 2 ) = exp{ (x/θ 2 ) θ } for x > ; 4. Gamma distributio F 4 (x; θ, θ 2 ) = x θ θ 2 Γ(θ ) sθ exp{ θ 2 s}ds for x >. For calculatig the proposed jackkife empirical likelihood itervals for both R(F ) ad r(f ) (JELCI), we employ the proportioal hazards trasform risk measure with ψ(s) = a( s) a, a =.55 ad.85, ad use the R package emplik. For calculatig the cofidece itervals for R(F ) based o the ormal approximatio method (NACI), we use the variace estimatio i Joes ad Zitikis (23). For computig the aive bootstrap cofidece itervals for r(f ) (NBCI), we draw, bootstrap samples with replacemet from each radom sample X,, X. Empirical coverage probabilities are reported i Tables ad 2 for these three cofidece itervals with levels.9,.95 ad.99. From these two tables, we coclude that the proposed jackkife empirical likelihood methods give much more accurate coverage probabilities tha the other two methods. 4 Proofs Throughout we put U i = F (X i ) for i =,,, G (t) = (U i t) ad G,i = ( ) j=,j i (U j t) for i =,,. Sice F is cotiuous, U,, U are idepedet ad uiformly distributed over (, ). Without loss of geerality we also assume o ties i U,, U, ad let U, < < U, deote the order statistics of U,, U. We also use C to deote a geeric costat which may be differet i differet places. First we list some facts which will be employed i the proofs. We assume β α throughout sice proofs for the case of β > α are exactly the same. Therefore we have ψ(x) cx β ( x) β ad ψ (x) cx β 2 ( x) β 2 for all < x <. Sice E X γ < with γ + β < 2, we have which implies P ( X > x) = o(x γ ) as x, (6) (F (x)) β +δ ( F (x)) β +δ dx 2 + C 6 x (β +δ)γ dx < (7)

7 wheever δ ( γ + β, 2 ), ad It follows from the give coditios o ψ that max X j = max F (U j ) = o p ( /γ ). (8) j j Ψ( ) = O( β ) ad Ψ( ) Ψ( ) = O( β ). (9) Lemma. Uder the coditios of Theorem, we have (Y i R ) d N(, σ), 2 () where σ 2 is give i (4). Proof. Write Y i = ( ) = ( ) = ( ) = ( ) + ( ) + ( ) {Ψ(F,i (t)) Ψ(F (t))}dt + ˆR {Ψ(G,i (t)) Ψ(G (t))}df (t) + ˆR {Ψ(G,i (t)) Ψ(G (t))}(u, t < U, )df (t) + ˆR {Ψ(G,i (t)) Ψ(G (t))}(u, t < U,2 )df (t) {Ψ(G,i (t)) Ψ(G (t))}(u,2 t < U, )df (t) {Ψ(G,i (t)) Ψ(G (t))}(u, t < U, )df (t) + ˆR = ( ) {Ψ(G,i (t)) Ψ(G (t))}(u, t < U,2 )df (t) }{{} Z i, + ( ) ψ(g (t)){g,i (t) G (t)}(u,2 t < U, )df (t) }{{} Z i,2 + ψ (ξ,i (t)){g,i (t) G (t)} 2 (U,2 t < U, )df (t) 2 }{{} Z i,3 + ( ) {Ψ(G,i (t)) Ψ(G (t))}(u, t < U, )df (t) + ˆR } {{ } Z i,4 = Z i, + Z i,2 + Z i,3 + Z i,4 + ˆR, 7

8 where for some θ i (t) [, ]. ξ,i (t) = G (t) + θ i (t){g,i (t) G (t)} = G (t) + θ i(t) {G (t) (U i t)} Whe U, t < U,2, we have G (t) = ad G,i(t) = if U i = U, else. () Hece, it follows from (8) ad (9) that Z i, = ( ) {Ψ() Ψ( )}(U, t < U,2 )df (t) + ( ) 2 {Ψ( ) Ψ( )}(U, t < U,2 )df (t) = ( )Ψ( ){F (U,2 ) F (U, )} + ( ) 2 {Ψ( ) Ψ( )}{F (U,2 ) F (U, )} = O(( ) β )o p ( /γ ) + O(( ) 2 β o p ( /γ ) (2) = o p ( /2 β+/γ ) = o p ( ) sice 2 β + γ <. Similarly, we ca show that Sice {G,i(t) G (t)} =, we have Whe t U,2, we have Z i,4 = o p ( ). (3) Z i,2 =. (4) ( ) (U i t) G (t) /( ) 2/ = 2( ), i.e., ξ,i (t) G (t){ 2( ) } uiformly i t U,2. I the same maer, we ca show that the equatio ξ,i (t) ( G (t)){ 8 2( ) }

9 holds uiformly i t < U,. Hece, for large eough, (ξ,i (t), ξ,i (t)) 3 (G (t), G (t)) uiformly for U,2 t < U, ad i. (5) Note that G (t) G (t) sup = O p () ad sup = O p () (6) U,2 t U, t U,2 t U, t (see Page 44 of Shorack ad Weller (986)). It follows from (5) ad (6) that Z i,3 = O p ( ) t β 2 ( t) β 2 {G,i (t) G (t)} 2 (U,2 t < U, )df (t), which coupled with (7) yields Z i,3 = O p ( t β 2 ( t) β 2 ) {G,i (t) G (t)} 2 (U,2 t < U, )df (t) ) = O p ( t β 2 ( t) β 2 ( ) 2 G (t){ G (t)}(u,2 t < U, )df (t) ( ) = O p t β ( t) β (U,2 t < U, )df (t) ( ) = O p t β ( t) β df (t) ( ) = O p δ t β +δ ( t) β +δ df (t) ( ) = O p δ (F (x)) β +δ ( F (x)) β +δ dx = O p ( δ ) (7) for ay δ ( γ + β, 2 ). By Joes ad Zitikis (23), we have { ˆR R} d N(, σ 2 ). (8) Hece, the lemma follows from (2), (4), (7), (3) ad (8). Lemma 2. Uder the coditios of Theorem, we have (Y i R) 2 p σ 2 as. 9

10 Proof. We use the same otatios Z i,j as i the proof of Lemma. The, it follows from () ad (9) that Zi, 2 = ( )2 + = O( = o p (). ( )3 {Ψ() Ψ( )}2 (U, t, t 2 < U,2 )df (t )df (t 2 ) {Ψ( ) Ψ( )}2 (U, t, t 2 < U,2 )df (t )df (t 2 ) ( )2 2β )o p ( 2/γ ( )3 ) + O( 2 2β )o p ( 2/γ ) (9) Similarly, It is easy to check that Zi,2 2 ( )2 = = =2 ψ(g (t ))ψ(g (t 2 )) Zi,4 2 = o p (). (2) {G,i (t ) G (t )}{G,i (t 2 ) G (t 2 )} (U,2 t, t 2 < U, )df (t 2 )df (t ) t ψ(g (t ))ψ(g (t 2 )){G (t t 2 ) G (t )G (t 2 )}(U,2 t, t 2 < U, )df (t 2 )df (t ) ψ(g (t ))ψ(g (t 2 ))G (t 2 ){ G (t )}(U, t, t 2 < U, )df (t 2 )df (t ). } {{ } I By (6), we have ( ) sup ψ(g (t ))ψ(g (t 2 ))G (t 2 ){ G (t )} = O p t β ( t ) β t β 2 ( t 2 ) β t 2 ( t ). U,2 t,t 2 <U, Similar to the proof of (7), we ca show that = t x <. t β ( t ) β t β 2 ( t 2 ) β t 2 ( t )df (t 2 )df (t ) F (x) β ( F (x)) β F (y) β ( F (y)) β F (y)( F (x))dydx By the Gliveko-Catelli theorem, sup <t< G (t) t almost surely. It the follows from the domiated covergece theorem that I p t ψ(t )ψ(t 2 )t 2 ( t )df (t 2 )df (t ).

11 Hece Note that = Z 2 i,2 p {G,i (t ) G (t )} 2 {G,i (t 2 ) G (t 2 )} 2 { G (t ) (U i t ) } 2 { G (t 2 ) (U i t 2 ) ψ(t )ψ(t 2 ){t t 2 t t 2 }df (t 2 )df (t ) = σ 2. (2) = ( ) 4 { 3G2 (t )G 2 (t 2 ) + G 2 (t )G (t 2 ) + G (t )G 2 (t 2 ) + 4G (t )G (t 2 )G (t t 2 ) 2G (t )G (t t 2 ) 2G (t 2 )G (t t 2 ) + G (t t 2 )} = ( ) 4 {3G (t )G (t 2 ) (G (t t 2 ) G (t )G (t 2 )) G }{{} (t ) (G (t t 2 ) G (t )G (t 2 )) }{{} II G (t 2 ) (G (t t 2 ) G (t )G (t 2 )) + ( G }{{} (t ))( G (t 2 ))G (t t 2 ) }{{} II 3 = ( ) 4 {II II 2 II 3 + II 4 }. It follows from (6) that } 2 II 2 II 4 } G (t t 2 ) G (t )G (t 2 ) G (t t 2 )( G (t t 2 ) sup = sup = O p (), U,2 t,t 2 U, t t 2 t t 2 U,2 t,t 2 U, t t 2 ( t t 2 ) which, coupled with (5) ad (6), yields that Z 2 i,3 (( ) 2 =O p t β 2 ( t ) β 2 t β 2 2 ( t 2 ) β 2 4 {G,i (t ) G (t )} 2 {G,i (t 2 ) G (t 2 )} 2 (U,2 t, t 2 < U, )df (t 2 )df (t ) ) =O p ( 2 t β 2 ( t ) β 2 t β 2 2 ( t 2 ) β 2 {II II 2 II 3 + II 4 }(U,2 t, t 2 < U, )df (t 2 )df (t ) ).

12 From the above equatio we ca get that Zi,3 2 U, U, ( =O p 2 t β 2 ( t ) β 2 t β 2 2 ( t 2 ) β 2 t t 2 (t t 2 t t 2 )df (t 2 )df ) (t ) U,2 U,2 }{{} III ( U, U, + O p 2 t β 2 ( t ) β 2 t β 2 2 ( t 2 ) β 2 t (t t 2 t t 2 )df (t 2 )df ) (t ) U,2 U,2 }{{} ( U, U, + O p 2 t β 2 ( t ) β 2 t β 2 2 ( t 2 ) β 2 t 2 (t t 2 t t 2 )df (t 2 )df ) (t ) U,2 U,2 }{{} ( U, + O p 2 U,2 U, U,2 III 2 III 3 t β 2 ( t ) β 2 t β 2 2 ( t 2 ) β 2 ( t )( t 2 )(t t 2 )df (t 2 )df ) (t ) } {{ } III 4 =O p (III ) + O p (III 2 ) + O p (III 3 ) + O p (III 4 ). It is easy to check from (7) that for every δ ( γ + β, 2 ) III 2 + III 3 =2 2 U, U,2 4 2 U, = 2 U,2 =O( 2 U,2 δ ) t U,2 t β 2 ( t ) β 2 t β 2 2 ( t 2 ) β 2 (t + t 2 )t 2 ( t )df (t 2 )df (t ) t β ( t ) β t U,2 t β 2 ( t 2 ) β 2 df (t 2 )df (t ) t β ( t ) β O(U,2 δ + ( t ) δ )df (t ) t β ( t ) β df (t ) + O( 2 ) t β ( t ) β 2 δ df (t ) =O( 2 U,2 δ δ )(U,2 + ( U, ) δ ) + O( 2 )(U,2 δ + ( U, ) 2δ ) =O p ( 2+2δ + +2δ ) =o p (). Similarly, we ca show that III = o p () ad III 4 = o p (). Hece, Zi,3 2 = o p (). (22) Sice ˆR p R, we have ( ˆR R) 2 = o p (). (23) 2

13 It follows from (9), (2), (22) ad (23) that {Z i, + Z i,3 + Z i,4 + ˆR R} 2 = O( Note that Z i,2 {Z i, + Z i,3 + Z i,4 + ˆR R} Therefore, the lemma follows from (9) (25). {Zi, 2 + Zi,3 2 + Zi,4 2 + ( ˆR R) 2 }) = o p (). (24) = o p (). Z 2 i,2 {Z i, + Z i,3 + Z i,4 + ˆR R} 2 Proof of Theorem. First we observe by usig (7) that for ay δ ( γ + β, 2 ) (25) max Z i,2 ψ(g (t))(u,2 t < U, )df (t) i ( ) U, = O p t β ( t) β df (t) U,2 (26) = O p (U δ,2 + ( U, ) δ ) = o p ( /2 ). Similarly we ca show that max Z i,j = o p ( /2 ) for j =, 3, 4. i Hece, max i Y i = o p ( /2 ). By the stadard argumets i the empirical likelihood method (see Chapter of Owe (2)), it follows from Lemmas ad 2 that 2 log L (R) = { (Y i R)} 2 (Y i R) 2 + o p () d χ 2 (). I order to prove Theorem 2, we eed the followig lemmas. Lemma 3. Uder the coditios of Theorem 2, we have ( ) ˆR R(F ) d X i N(, σ 2 ) as, µ where σ 2 = + 2 R(F ) µ ψ(t )ψ(t 2 )(t t 2 t t 2 )df (t )df (t 2 ) + R2 (F ) µ 2 E(X µ) 2 ψ(t )(t t 2 t t 2 )df (t )df (t 2 ). 3

14 Proof. It is kow that there exists a Browia bridge W such that (G (t) t) W (t) sup t t δ ( t) δ = o p () (27) for ay δ (, /2) (see Chapter 4 of Csorgo ad Horvath (993)). The we have { ˆR R(F ) µ X i } = d {Ψ(t) Ψ(G (t))}df (t) + R(F ) {t G (t)}df (t) µ ψ(t)w (t)df (t) R(F ) µ Lemma 4. Uder the coditios of Theorem 2, we have W (t)df (t). ( Y i R(F ) ) µ X d i N(, σ 2 ) as. Proof. It ca be show i a way similar to the proof of Lemma. Lemma 5. Uder the coditios of Theorem 2, we have ( Y i R(F ) ) 2 p µ X i σ 2 as. Proof. It ca be proved i a similar way to the proof of Lemma 2. Proof of Theorem 2. This ca be doe i a way similar to the proof of Theorem. 5 Coclusios This paper proposes jackkife empirical likelihood methods to costruct cofidece itervals for the risk measures ad some related quatities studied by Joes ad Zitikis (23). Ulike the ormal approximatio method, the ew methods do ot eed to estimate the asymptotic variace explicitly ad are easy to implemet by employig the R package emplik. A simulatio study shows that the jackkife empirical likelihood cofidece itervals are more accurate tha the ormal approximatio based cofidece itervals. 4

15 Ackowledgmets Peg s research was supported by NSA grat H , Qi s research was supported by NSA grat H , ad Yag s research was partly supported by the Natioal Basic Research Program (973 Program) of Chia (27CB8495) ad Natioal Natural Sciece Foudatio of Chia (Grats No. 878). Refereces [] P. Artzer (999). Applicatio of coheret risk measures to capital requiremets i isurace. North America Actuarial Joural 3, 29. [2] J. Che, L. Peg ad Y. Zhao (29). Empirical likelihood based cofidece itervals for copulas. J. Multivariate Aal., [3] G. Claeskes, B. Jig, L. Peg ad W. Zhou (23). Empirical likelihood cofidece regios for compariso distributios ad ROC curves. The Caadia Joural of Statistics. 3(2), [4] M. Csorgo ad L. Horvath (993). Weighted Approximatios i Probability ad Statistics. Wiley, Chichester. [5] B. Jig, J. Yua ad W. Zhou (29). Jackkife empirical likelihood. Joural of America Statistical Associatio 4, [6] B.L. Joes ad R. Zitikis (23). Empirical estimatio of risk measures ad related quatities. North America Actuarial Joural 7, [7] B.L. Joes ad R. Zitikis (25). Testig for the order of risk measures: a applicatio of L- statistics i a actuarial sciece. Metro LXIII, [8] B.L. Joes ad R. Zitikis (27). Risk measures, distortio parameters, ad their empirical estimatio. Isurace: Mathematics ad Ecoomics 4, [9] S.A. Klugma, H.H. Pajer ad G.E. Willmot (28). Loss Distributios: From Data to Decisios. Wiley. 5

16 [] A. Necir ad D. Meraghi (29). Empirical estimatio of the proportioal hazard premium for heavy-tailed claim amouts. Isurace: Mathematics ad Ecoomics 45, [] A. Owe (2). Empirical Likelihood. Chapma & Hall/CRC. [2] G.R. Shorack ad J.A. Weller (986). Empirical Processes with Applicatios to Statistics. New York: Wiley. [3] S. Wag (995). Isurace pricig ad icreased limits ratemakig by proportioal hazards trasforms. Isurace: Mathematics ad Ecoomics 7, [4] S. Wag (996). Premium calculatio by trasformig the layer premium desity. ASTIN Bulleti 26, [5] S. Wag (998). A actuarial idex of the right-tail risk. North America Actuarial Joural 2, 88. [6] S. Wag, V.R. Youg ad H.H. Pajer (997). Axiomatic characterizatio of isurace prices. Isurace: Mathematics ad Ecoomics 2, [7] J.L. Wirch ad M.R. Hardy (999). A sythesis of risk measures for capital adequacy. Isurace: Mathematics ad Ecoomics 25, [8] M.E. Yaari (987). The dual theory of choice uder risk. Ecoometrica 55,

17 Table : Coverage probabilities for R(F ) are reported for the itervals based o the proposed jackkife empirical likelihood method (JELCI) ad the ormal approximatio method (NACI). (, a, F ) JELCI NACI JELCI NACI JELCI NACI level.9 level.9 level.95 level.95 level.99 level.99 (3,.55, F (; 4)) (3,.85, F (; 4)) (,.55, F (; 4)) (,.85, F (; 4)) (3,.55, F 2 (;, )) (3,.85, F 2 (;, )) (,.55, F 2 (;, )) (,.85, F 2 (;, )) (3,.55, F 3 (; 4, )) (3,.85, F 3 (; 4, )) (,.55, F 3 (; 4, )) (,.85, F 3 (; 4, )) (3,.55, F 4 (; 4, )) (3,.85, F 4 (; 4, )) (,.55, F 4 (; 4, )) (,.85, F 4 (; 4, ))

18 Table 2: Coverage probabilities for r(f ) are reported for the itervals based o the proposed jackkife empirical likelihood method (JELCI) ad the aive bootstrap method (NBCI). (, a, F ) JELCI NBCI JELCI NBCI JELCI NBCI level.9 level.9 level.95 level.95 level.99 level.99 (3,.55, F (; 4)) (3,.85, F (; 4)) (,.55, F (; 4)) (,.85, F (; 4)) (3,.55, F 2 (;, )) (3,.85, F 2 (;, )) (,.55, F 2 (;, )) (,.85, F 2 (;, )) (3,.55, F 3 (; 4, )) (3,.85, F 3 (; 4, )) (,.55, F 3 (; 4, )) (,.85, F 3 (; 4, )) (3,.55, F 4 (; 4, )) (3,.85, F 4 (; 4, )) (,.55, F 4 (; 4, )) (,.85, F 4 (; 4, ))