Assumption 2.1. R t follows the location-scale model:

where m ( A t − 1 , β 0 ) and σ ( A t − 1 , β 0 ) denote the conditional mean and the conditional standard deviation of the return R t given the information set A t − 1 , respectively. The specific functional forms of these two functions are known, but they depend on an unknown parameter vector β 0 . The innovation ε t is independent of the information set A t − 1 and forms an independent and identically distributed (i.i.d.) sequence of continuous random variables with mean zero and variance one. The information set at time t is defined as:

that is, the σ -algebra generated by the historical innovation sequence { ε t − i } i = 1 ∞ and the possibly existing additional random vector sequence { η t − i } i = 1 ∞ . Here, { η t } t = − ∞ ∞ is a sequence of random vectors independent of the innovation sequence { ε t } t = − ∞ ∞ . Furthermore, the functions m ( A t − 1 , β 0 ) and σ ( A t − 1 , β 0 ) are measurable with respect to A t − 1 .

We suppose that ε t follows a distribution with Pareto-type heavy tails. Let F ε ( ⋅ ) = Pr ( ε t ≤ ⋅ ) and F − ε ( ⋅ ) = Pr ( − ε t ≤ ⋅ ) be the distribution functions of ε t and − ε t . Then

and

where γ R > 0 and γ L > 0 denote the extreme value indices corresponding to the right and left tails, respectively. Note that under Assumption 2.1, the innovation ε t is assumed to possess unit variance. This condition means that γ R < 1 / 2 and γ L < 1 / 2 .

By model (2.1), we have

where Q ε ( ⋅ ) is the quantile function of ε t . Assume that for each type ∈ { U , D , R } , we have estimators type-CVaR ˜ and type-CES ˜ for the true type-CVaR and type-CES quantities, respectively. Given a tail region [ τ l , τ u ] satisfying 0 < τ l ≤ τ u , we define the maximum absolute log-ratios as

Suppose that there is a consistent estimator β ^ of β 0 . Let A ˜ t − 1 be the truncated information set, which is generated by feasible information up to time t − 1 . The truncation is necessary when the information set relies on infinite past observations. For example, when A t − 1 = { R t − 1 , R t − 2 , ⋯ , R 0 , R − 1 , ⋯ } , the truncated information set is A ˜ t − 1 = { R t − 1 , R t − 2 , ⋯ , R 1 } . We obtain the standardized residuals ε ^ t = R t − m ( A ˜ t − 1 , β ^ ) σ ( A ˜ t − 1 , β ^ ) , t = 1 , ⋯ , n . For d n < n and d n → ∞ as n → ∞ , we discard the residuals for t < d n and work with ε ^ t for t = d n , ⋯ , n . This discarding eliminates the effect of information truncation. Similar discarding can be applied to other information. Denote by F ε ← the left continuous inverse of F ε . Then, for x > 0 , the ( 1 − 1 / x ) -quantile of F ε is

By Theorem 1.2.1 and Corollary 1.2.10 of de Haan & Ferreira [10], (2.2) is equivalent to

Similarly, let F − ε ← be the left continuous inverse of F − ε and U − ε ( x ) ≡ F − ε ← ( 1 − 1 / x ) , for x > 0 . Then (2.3) is equivalent to

We first consider estimating the CVaR and CES of the right tail, namely Q ε ( 1 − τ ) and E [ ε | ε > Q ε ( 1 − τ ) ] . For positive integers k 1 , it satisfies that k 1 → ∞ and k 1 / n → 0 as n → ∞ . By (2.9), for a small τ implies

To obtain an estimate of Q ε ( 1 − τ ) , we need to estimate γ R and U ε ( n k 1 ) . A suitable estimator for U ε ( n / k 1 ) is given by ε ^ n − k 1 , n , commonly referred to as the intermediate order statistic de Haan & Ferreira [10]. For estimation of γ R > 0 , we use

with α > 0 (Gomes and Martins 2001; Segers 2001), where Γ ( ⋅ ) is the gamma function and

(2.13)

By substituting ε ^ n − k 1 , n and γ ^ R into equation (2.11), the following estimate of Q ε ( 1 − τ ) can be obtained:

By Proposition 4.1 of Pan, Leng, and Hu (2013) [11], we have

Since τ is close to 0, the estimation of E [ ε | ε > Q ε ( 1 − τ ) ] is

Estimation of the left tail proceeds in the same way as what is done for the right tail. Note that U − ε ( 1 / τ ) = − Q ε ( τ ) . Therefore, estimation of Q ε ( τ ) is equivalent to estimation of − U − ε ( 1 / τ ) . Then, for positive integers k 2 , it satisfies k 2 → ∞ and k 2 / n → 0 as n → ∞ , following similar arguments leading to (2.14) and (2.16), we obtain the estimator

for Q ε ( τ ) , and the estimator

for E [ | ε | | ε < Q ε ( τ ) ] , where γ ^ L is estimator of γ L based on { − ε ^ i } , defined as

and

Plugging the above estimators into (2.4)-(2.7), we have estimators of the risk measures

and

And apparently, we have estimators of the relative risk measures

Assumption 3.1.The estimator β ^ of β 0 satisfies ‖ β ^ − β 0 ‖ = O p ( n − υ 0 / 2 ) for some positive υ 0 .

Assumption 3.2.Let B 0 be a neighborhood of β 0 .

(i) sup β ∈ B 0 m ( A n , β 0 ) = O p ( 1 ) .

(ii) inf β ∈ B 0 σ ( A t − 1 , β ) > c for some c > 0 for all t = d n , ⋯ , n + 1 .

(iii) Both m ( A t − 1 , β ) and σ ( A t − 1 , β ) are differentiable with respect to β in B 0 for all t = d n , ⋯ , n + 1 .

(iv) E [ sup β ∈ B 0 ‖ ∂ m ( A t − 1 , β ) ∂ β ‖ υ 1 ] < ∞ and E [ sup β ∈ B 0 ‖ ∂ σ ( A t − 1 , β ) ∂ β ‖ υ 1 ] < ∞ for some υ 1 > 2 / υ 0 for all t = d n , ⋯ , n + 1 , where υ 0 is defined in Assumption 3.1.

Assumption 3.3.

(i) For each β ∈ B 0 , m ( A ˜ t − 1 , β ) and σ ( A ˜ t − 1 , β ) are measurable with respect to A t − 1 .

(ii) sup β ∈ B 0 | m ( A ˜ n , β ) − m ( A n , β ) | = O p ( n − υ 0 / 2 ) and sup β ∈ B 0 | σ ( A ˜ n , β ) − σ ( A n , β ) | = O p ( n − υ 0 / 2 ) , where υ 0 is defined in Assumption3.1.

(iii) ∑ t = d n n E [ sup β ∈ B 0 | m ( A ˜ t − 1 , β ) − m ( A t − 1 , β ) | ] = o ( 1 ) and ∑ t = d n n E [ sup β ∈ B 0 | σ ( A ˜ t − 1 , β ) − σ ( A t − 1 , β ) | ] = o ( 1 ) .

Assumption 3.4.

(i) There exist ρ R < 0 and a function A R ( ⋅ ) which is eventually positive or negative with lim x → ∞ A R ( x ) = 0 such that

lim x → ∞ U ε ( x s ) U ε ( x ) − s γ R A R ( x ) = s γ R s ρ R − 1 ρ R ,for all s > 0 ,

where k 1 1 / 2 A R ( n / k 1 ) → 0 as n → ∞ .

(ii) There exist ρ L < 0 and a function A L ( ⋅ ) which is eventually positive or negative with lim x → ∞ A L ( x ) = 0 such that

lim x → ∞ U − ε ( x s ) U − ε ( x ) − s γ L A L ( x ) = s γ L s ρ L − 1 ρ L , for all s > 0 ,

where k 2 1 / 2 A L ( n / k 2 ) → 0 as n → ∞ .

Assumption 3.5.

(i) τ l ≤ τ u satisfies τ l → 0 as n → ∞ and τ u = C τ l for some constant C ≥ 1 .

(ii) There exist two constants a 0 > 0 and a 1 > 0 such that lim n → ∞ k 1 / k 2 = a 0 and lim n → ∞ n τ l / k 1 = a 1 .

(iii) lim n → ∞ k 1 − 1 / 2 log ( n τ l / k 1 ) = 0 and k 1 = o ( n δ 0 ) for some 0 < δ 0 < υ 0 , where υ 0 is defined in Assumption3.1.

iv) d n = o ( k 1 1 / 2 ) .

Our theoretical findings are grounded in the following three propositions from Proposition 3.1 and 3.2 of Li et al. [2].

Proposition 3.1.Under Assumptions2.1 and3.1 - 3.5, we have, uniformly in τ ∈ [ τ l , τ u ] ,

and

Proposition 3.2.Under Assumptions2.1 and 3.1 - 3.5, we have,uniformly in τ ∈ [ τ l , τ u ] ,

and

Proposition 3.3.Under Assumptions 2.1 and3.1 - 3.5, as n → ∞ ,

is asymptotically four-dimensional standard normal where V ˜ ( α ) = 1 α 2 [ Γ ( 2 α + 1 ) Γ 2 ( α + 1 ) − 1 ] .

Proof. Let ε 1 , ⋯ , ε n be i.i.d. random variables satisfying Assumptions 3.4(i) and (ii). Define

By Lemma 1.5 in the supplementary material of Li, Peng and Song [2],

and the two components are asymptotically independent. Gomes and Martins [6] proved that for any α > 0 ,

Therefore, to obtain the joint convergence of the four-dimensional vector, it suffices to establish the asymptotic independence among the four components.

Using the probability integral transformation, set U i = F ε ( ε i ) ; then U i ~ U ( 0 , 1 ) are independent. Define the Pareto variables ζ i = 1 / ( 1 − U i ) , which are i.i.d. with distribution F ζ ( y ) = 1 − 1 / y ( y ≥ 1 ) , and we have ε i = U ε ( ζ i ) , where U ε ( t ) = Q ε ( 1 − 1 / t ) . The order statistics satisfy ε i : n = U ε ( ζ i : n ) . For the left tail, let V i = F − ε ( − ε i ) = 1 − U i ; then − ε i = U − ε ( 1 / ( 1 − V i ) ) and − ε i : n = U − ε ( ζ i : n − ) with ζ i : n − = 1 / ( 1 − V i : n ) .

By Rényi’s representation [12], there exist i.i.d. Exp ( 1 ) random variables E 1 , ⋯ , E n such that

Consequently,

and the ratios

depend only on E 1 , ⋯ , E k 1 and are independent of ζ n − k 1 : n (because ζ n − k 1 : n also depends on E k 1 + 1 ). Similarly, for the left tail, ζ k 2 + 1 : n − depends on E n − k 2 , ⋯ , E n . Since k 1 + k 2 < n for sufficiently large n , the sets { E 1 , ⋯ , E k 1 + 1 } and { E n − k 2 , ⋯ , E n } are disjoint; hence the right-tail block ( ζ n − k 1 : n , { ζ n − i : n / ζ n − k 1 : n } i = 1 k 1 ) and the left-tail block ( ζ k 2 + 1 : n − , { ζ i : n − / ζ k 2 + 1 : n − } i = 1 k 2 ) are independent.

Notice that γ ^ ε , R depends only on { log ( ζ n − i : n / ζ n − k 1 : n ) } i = 1 k 1 , so γ ^ ε , R is independent of ζ n − k 1 : n . Therefore,

are asymptotically independent. The same holds for the two standardized statistics on the left tail, and the left and right tails are independent of each other. Combining marginal asymptotic normality with the Cramér-Wold theorem yields

Now consider the standardized residuals ε ^ t defined in the proposition. By Lemma S1.9(ii) in the supplementary material of Li, Peng and Song [2],

and analogous relations hold for the left tail. Hence, replacing the true residuals with the standardized residuals changes each component of the four-dimensional vector by o p ( 1 ) , so the limiting distribution remains unchanged.

Moreover, the left-tail quantile in the proposition uses ε ^ k 2 + d n , n , whereas the lemma uses ε k 2 + 1 , n . By Assumption 3.5(iv) and the fact that k 2 is of the same order as k 1 , we have d n = o ( min ( k 1 1 / 2 , k 2 1 / 2 ) ) , thus

so replacing k 2 + 1 with k 2 + d n does not affect the asymptotic distribution. This completes the proof of Proposition 3. □

The following theorem establishes the uniform convergence of the maximum absolute log-ratios under the intermediate scenario a 1 > 0 .

Theorem 3.1.Suppose that Assumptions2.1 and3.1 - 3.5 hold. As n → ∞ ,

sup u ∈ [ 1 , C ] | k 1 1 / 2 { log [ type-CVaR ˜ ( u τ l ) type-CVaR ( u τ l ) ] } | → sup u ∈ [ 1 , C ] | G 1 , type ( u ) | and

in distribution, where G 1 , type ( u ) and G 2 , type ( u ) are centered Gaussian processes with variance-covariance functions γ 1 , type ( u 1 , u 2 ) = Cov ( G 1 , type ( u 1 ) , G 1 , type ( u 2 ) ) and γ 2 , type ( u 1 , u 2 ) = Cov ( G 2 , type ( u 1 ) , G 2 , type ( u 2 ) ) satisfying that

and