Let X 1 , … , X n be independent and identically distributed (i.i.d.) random variables with a common distribution function F. Suppose that the right tail 1 − F is a regularly varying function with index − 1 / γ (written 1 − F ∈ RV − 1 / γ ), that is, for x > 0, (1) lim t → ∞ 1 − F ( t x ) 1 − F ( t ) = x − 1 / γ , where γ > 0 is the positive extreme value index (EVI). Put U ( t ) = 0 , 0 < t ≤ 1 inf x : F ( x ) ≥ 1 − ( 1 / t ) , t > 1 . Condition (1) is equivalent to (2) lim t → ∞ ln U ( t x ) − ln U ( t ) = γ ln ( x ) , x > 0 , see, e.g. pg. 73 in [5]. For a large class of quantile type functions U satisfying (2) there exists a function A ( t ) → 0 of constant sign for large values of t, such that (3) lim t → ∞ ln U ( t x ) − ln U ( t ) − γ ln ( x ) A ( t ) = h ρ ( x ) for every x > 0, where ρ ≤ 0 is a second order parameter and h ρ ( x ) = ln ( x ) , ρ = 0 , ( x r − 1 ) / r , ρ < 0 .

Numerous of the existing estimators of a positive extreme value index (see, e.g. [18, 9, 8]) are constructed by using parameterized statistics M n ( k , r ) = 1 , r = 0 , 1 k ∑ i = 1 k ln X n − i + 1 , n − ln X n − k , n r , r > 0 , where X 1 , n ≤ X 2 , n ≤ ⋯ ≤ X n , n denote the ascending order statistics of X 1 , … , X n . M n ( k , 1 ) is the classical Hill estimator ([18]), while M n ( k , 2 ) was introduced in [8]. The asymptotic properties (including weak consistency and asymptotic normality) of M n ( k , r ) when r is real positive were considered in [13]. To learn more about estimators for a positive extreme value index, we refer to the review [10], where more than one hundred estimators are collected.

Several papers are devoted to the investigation of the asymptotic properties of estimators (with two tuning parameters), defined by (4) γ ˆ n ( k , r 1 , r 2 ) = Γ ( r 1 ) M n ( k , r 1 − 1 ) M n ( k , r 1 r 2 ) Γ ( r 1 r 2 + 1 ) 1 / r 2 , r 1 ≥ 1 , r 2 > 0 , where Γ ( 1 + r ), r ≥ 0 denotes the gamma function defined by the integral Γ ( 1 + r ) = ∫ 0 ∞ t r exp { − t } d t. The estimators (4) were presented in [3]. It is important to note that the family of estimators (4) generalizes several classical estimators. The estimator γ ˆ n ( k , 1 , 1 ) coincides with the Hill estimator, while the estimator γ ˆ n ( k , 1 , 2 ) = M n ( k , r 1 ) / 2 was proposed in [9] as an alternative to the Hill estimator. Also, the estimator γ ˆ n ( k , 2 , 1 ) = M n ( k , 2 ) / 2 M n ( k , 1 ) is a moment ratio estimator, introduced by de Vries; see [6].

For completeness, we recall the result regarding the asymptotic normality of estimators (4).

The subfamily γ ˆ n ( k , 1 , r ), r > 0 of the family of estimators (4) was investigated in [12]. The subfamilies γ ˆ n ( k , r , 2 ), r ≥ 1 and γ ˆ n ( k , r , 1 ), r ≥ 1 of (4) were considered in [4] and [22], respectively.

Recall from Prop. 1 in [4] that if (2) and (5) hold, then for r > 0, (8) M n ( k , r ) → p γ r Γ ( r + 1 ) , n → ∞ . Motivated by the construction of the moment ratio estimator, we consider the ratio M n ( k , r 2 ) / M n ( k , r 1 ), 0 ≤ r 1 < r 2 and by combining (8) with Slutsky’s theorem (see e.g. [24]) we find that M n ( k , r 2 ) / M n ( k , r 1 ) converges in probability to γ r 2 − r 1 Γ ( r 2 + 1 ) / Γ ( r 1 + 1 ) as n → ∞. This leads to the new family of semi–parametric estimators of γ > 0: (9) γ ˆ n ( 1 ) ( k , r 1 , r 2 ) = Γ ( r 1 + 1 ) M n ( k , r 2 ) Γ ( r 2 + 1 ) M n ( k , r 1 ) 1 / ( r 2 − r 1 ) , 0 ≤ r 1 < r 2 . It should be noted that the family of estimators (9) generalizes the Hill estimator ( r 1 = 0, r 2 = 1), the alternative Hill estimator ( r 1 = 0, r 2 = 2) and the moment ratio estimator ( r 1 = 1, r 2 = 2).

Our main result states that the estimators in (9) are asymptotically normal for γ > 0.

Obviously, that (11) ν 1 ( ρ , r 1 , r 2 ) ≠ 0 for any ρ < 0 and 0 < r 1 < r 2 . Recall that A ( t ) → 0 as t → ∞. Thus, under assumption (5), the asymptotic bias ν 1 ( ρ , r 1 , r 2 ) A ( n / k ) tends to zero as n → ∞, but estimators satisfying (11) are referred to as asymptotically biased; see, e.g. [19].

Next, consider three families of asymptotically biased estimators for γ > 0. Namely, by taking r 2 = 2 r 1 in (9) we obtain the estimators γ ˆ n ( 1 ) ( k , r ) = Γ ( r + 1 ) M n ( k , 2 r ) Γ ( 2 r + 1 ) M n ( k , r ) 1 / r , r > 0 . Using a slight modification of the comparison scheme for biased estimators (proposed in [6]), we will compare the estimator γ ˆ n ( 1 ) ( k , r ) with the following estimators: γ ˆ n ( 2 ) ( k , r ) = γ ˆ n ( k , 1 , r ) , r > 0 , γ ˆ n ( 3 ) ( k , r ) = γ ˆ n ( k , r , 1 ) , r ≥ 1 . In [4] it is noted that γ ˆ n ( k , r , 2 ), r ≥ 1 are asymptotically unbiased estimators, and thus we eliminated these estimators from our comparison. We refer to [21] for a discussion of difficulties related to the comparison of an asymptotically unbiased estimator with an asymptotically biased estimator. Taking into account that γ ˆ n ( 2 ) ( k , r ) = γ ˆ n ( 1 ) ( k , 0 , r ), r > 0 and γ ˆ n ( 3 ) ( k , r ) = γ ˆ n ( 1 ) ( k , r − 1 , r ), r ≥ 1, the asymptotic normality of the estimators γ ˆ n ( ℓ ) ( k , r ), ℓ = 1 , 2 , 3 follows directly from Theorem 2.

The paper is organized as follows. In the next Section, we compare estimators γ ˆ n ( ℓ ) ( k , r ), ℓ = 1 , 2 , 3 theoretically. In Section 3, a small-scale simulation study is undertaken. In Section 4, an application to the exchange rate data is presented to illustrate the behavior of the estimator γ ˆ n ( 1 ) ( k , r ). Section 5 contains conclusions, while all proofs are collected in Section 6.

The asymptotic second moment of γ ˆ n ( ℓ ) ( k , r ) is A 2 n / k λ ℓ 2 ( ρ , r ) + γ 2 ς ℓ 2 ( r ) / k, see [6] for the definition of the asymptotic second moment. Firstly, following [20] (see also [1]), we fix r and find the asymptotic behavior (as n → ∞) of the so-called minimal mean squared error (12) MMSE γ ˆ n ( ℓ ) ( k n , ℓ ∗ , r ) = inf k n A 2 n / k n λ ℓ 2 ( ρ , r ) + γ 2 ς ℓ 2 ( r ) k n , where k n , ℓ ∗ is the minimizing sequence.

Further, we will assume the classical restriction ρ < 0. It eliminates the case where | A | is a slowly varying function at infinity. In addition, under assumption ρ < 0 in (3), there exists a positive decreasing function a ∈ RV 2 ρ − 1 such that (13) A 2 ( t ) ∼ ∫ t ∞ a ( τ ) d τ , t → ∞ , see [7].

Note that λ ℓ ( ρ , r ) ≠ 0 in (12) is an essential assumption. It allows us to balance the rate of decay of squared asymptotic bias and asymptotic variance.

By applying Lemma 2.8 in [7], we get that the minimizing sequence k n , ℓ ∗ = k n , ℓ ∗ ( r ) satisfies the relation (14) k n , ℓ ∗ ( r ) ∼ γ 2 ς ℓ 2 ( r ) λ ℓ 2 ( ρ , r ) 1 / ( 1 − 2 ρ ) n a ← ( 1 / n ) , n → ∞ , ℓ = 1 , 2 , 3 , where a ← is the inverse function of a. Following the lines in [6] it can be proven that (15) k n , ℓ ∗ ( r ) A 2 n k n , ℓ ∗ ( r ) ∼ ς ℓ 2 ( r ) − 2 ρ λ ℓ 2 ( ρ , r ) , n → ∞ . Substituting (14)–(15) into (12) we get (16) MMSE γ ˆ n ( ℓ ) ( k n , ℓ ∗ , r ) ∼ 1 − 2 ρ − 2 ρ λ ℓ 2 ( ρ , r ) γ 2 ς ℓ 2 ( r ) − 2 ρ 1 / ( 1 − 2 ρ ) a ← ( 1 / n ) n , as n → ∞. The next step is to minimize the right-hand side of (16) with respect to r, or equivalently, to minimize the product (17) λ ℓ 2 ( ρ , r ) ς ℓ 2 ( r ) − 2 ρ with respect to r. Using Wolfram Mathematica 10.4 the minimization of (17) is performed numerically. If product (17) attains its minimum at a point r = r ℓ ∗ , then r ℓ ∗ is called the optimal choice of the tuning parameter r. The graphs of the optimal choices r ℓ ∗ = r ℓ ∗ ( ρ ), ℓ = 1 , 2 , 3 are provided in Figures 1 and 2.

The graphs of r ℓ ∗ ( ρ ), ℓ = 1 , 2 , 3 are plotted using a set { − i / 100 , 1 ≤ i ≤ 500 } of values of ρ. Consider r ℓ ∗ ( ρ ), ℓ = 1 , 2 , 3 as functions of ρ on [ − i / 100 , − 1 / 100 ], 1 < i ≤ 500. Then the difference r ℓ ∗ ( − 1 / 100 ) − r ℓ ∗ ( − i / 100 ) is the width of the range of r ℓ ∗ ( ρ ) on [ − i / 100 , − 1 / 100 ]. For example, taking i = 100 we find that the widths of the range of r 1 ∗ ( ρ ), r 2 ∗ ( ρ ) and r 3 ∗ ( ρ ) are 0.38, 0.73 and 0.58, respectively. Calculating the widths of the range of r 1 ∗ ( ρ ), r 2 ∗ ( ρ ) and r 3 ∗ ( ρ ) for values i > 100 allows us to conclude that the width of the range of r 1 ∗ ( ρ ) is smaller than the widths of the range of r 2 ∗ ( ρ ) and r 3 ∗ ( ρ ). Thus, r 1 ∗ ( ρ ) is less sensitive to the estimation of the parameter ρ than the other two optimal choices.

Now we can compare the estimator γ ˆ n ( 1 ) ( k n , 1 ∗ ( r 1 ∗ ) , r 1 ∗ ) with the estimators γ ˆ n ( ℓ ) ( k n , ℓ ∗ ( r ℓ ∗ ) , r ℓ ∗ ) , ℓ = 2 , 3 . Following [6], we write RMMSE γ ˆ n ( ℓ ) ( k n , ℓ ∗ ( r ℓ ∗ ) , r ℓ ∗ ) | γ ˆ n ( 1 ) ( k n , 1 ∗ ( r 1 ∗ ) , r 1 ∗ ) , ℓ = 2 , 3 for the limit of the ratio of the minimal mean squared errors of γ ˆ n ( ℓ ) ( k n , ℓ ∗ ( r ℓ ∗ ) , r ℓ ∗ ) and γ ˆ n ( 1 ) ( k n , 1 ∗ ( r 1 ∗ ) , r 1 ∗ ). Using (16) we get that RMMSE γ ˆ n ( ℓ ) ( k n , ℓ ∗ ( r ℓ ∗ ) , r ℓ ∗ ) | γ ˆ n ( 1 ) ( k n , 1 ∗ ( r 1 ∗ ) , r 1 ∗ ) = ϕ ℓ , 1 ( ρ ) , ℓ = 2 , 3 , where ϕ ℓ , 1 ( ρ ) = λ ℓ 2 ( ρ , r ℓ ∗ ) λ 1 2 ( ρ , r 1 ∗ ) ς ℓ 2 ( r ℓ ∗ ) ς 1 2 ( r 1 ∗ ) − 2 ρ 1 / ( 1 − 2 ρ ) . We say that the estimator γ ˆ n ( 1 ) ( k n , 1 ∗ ( r 1 ∗ ) , r 1 ∗ ) outperforms γ ˆ n ( ℓ ) ( k n , ℓ ∗ ( r ℓ ∗ ) , r ℓ ∗ ) on a half-line { ( ρ , γ ) : ρ = ρ 0 , γ > 0 } if ϕ ℓ , 1 ( ρ 0 ) > 1.

Graphs of the functions ϕ 2 , 1 ( ρ ), − 5 ≤ ρ < 0 and ϕ 3 , 1 ( ρ ), − 5 ≤ ρ < 0 are presented in Fig. 3. Whence one can deduce that the estimator γ ˆ n ( 1 ) ( k n , 1 ∗ ( r 1 ∗ ) , r 1 ∗ ) outperforms the estimators γ ˆ n ( ℓ ) ( k n , ℓ ∗ ( r ℓ ∗ ) , r ℓ ∗ ), ℓ = 2 , 3 in the area { ( γ , ρ ) : γ > 0 , − 5 ≤ ρ < 0 }.

We end this Section with a comparison of the estimators γ ˆ n ( k , r 1 , r 2 ) and γ ˆ n ( 1 ) ( k , r 1 , r 2 ) . Theorem 1 in [3] states that for any ρ < 0 and r 2 > 1 there is a value r ¯ 1 = r ¯ 1 ( ρ ) such that ν ( ρ , r ¯ 1 ( ρ ) , r 2 ) = 0, where ν ( ρ , r 1 , r 2 ) is the same as in Theorem 1. As for the case 0 < r 2 ≤ 1, we add the following statement: ν ( ρ , r 1 , r 2 ) ≠ 0 for any ρ < 0 and ( r 1 , r 2 ) ∈ [ 1 , ∞ ) × ( 0 , 1 ], see Lemma 1 in Section 6. With the goal of eliminating unbiased estimators from our consideration, we will compare estimators (9) with γ ˆ n ( k , r 1 , r 2 ), r 1 ≥ 1, 0 < r 2 ≤ 1. In fact, this comparison reduces to the comparison of estimators γ ˆ n ( 1 ) ( K ˜ n ( R ˜ 1 ( ρ ) , R ˜ 2 ( ρ ) ) , R ˜ 1 ( ρ ) , R ˜ 2 ( ρ ) ) and γ ˆ n ( k ˜ n ( r ˜ 1 ( ρ ) , r ˜ 2 ( ρ ) ) , r ˜ 1 ( ρ ) , r ˜ 2 ( ρ ) ), where ( K ˜ n ( R ˜ 1 ( ρ ) , R ˜ 2 ( ρ ) ) , R ˜ 1 ( ρ ) , R ˜ 2 ( ρ ) ) and ( k ˜ n ( r ˜ 1 ( ρ ) , r ˜ 2 ( ρ ) ) , r ˜ 1 ( ρ ) , r ˜ 2 ( ρ ) ) are optimal choices for estimators γ ˆ n ( 1 ) ( k , r 1 , r 2 ) and γ ˆ n ( k , r 1 , r 2 ), respectively.

One can verify that the limit of the ratio of the minimal mean squared errors of γ ˆ n ( k ˜ n ( r ˜ 1 ( ρ ) , r ˜ 2 ( ρ ) ) , r ˜ 1 ( ρ ) , r ˜ 2 ( ρ ) ) and γ ˆ n ( 1 ) ( K ˜ n ( R ˜ 1 ( ρ ) , R ˜ 2 ( ρ ) ) , R ˜ 1 ( ρ ) , R ˜ 2 ( ρ ) ) equals ϕ ( ρ ) = ν 2 ( ρ , r ˜ 1 ( ρ ) , r ˜ 2 ( ρ ) ) ν 1 2 ( ρ , R ˜ 1 ( ρ ) , R ˜ 2 ( ρ ) ) σ 2 ( r ˜ 1 ( ρ ) , r ˜ 2 ( ρ ) ) σ 1 2 ( R ˜ 1 ( ρ ) , R ˜ 2 ( ρ ) ) − 2 ρ 1 / ( 1 − 2 ρ ) . Numerically we obtain that the estimator γ ˆ n ( 1 ) ( K ˜ n ( R ˜ 1 ( ρ ) , R ˜ 2 ( ρ ) ) , R ˜ 1 ( ρ ) , R ˜ 2 ( ρ ) ) dominates the estimator γ ˆ n ( k ˜ n ( r ˜ 1 ( ρ ) , r ˜ 2 ( ρ ) ) , r ˜ 1 ( ρ ) , r ˜ 2 ( ρ ) ) in the area { ( γ , ρ ) : γ > 0 , − 5 ≤ ρ < 0 }, see Fig. 4.

Let γ ˆ n ( ℓ ) ( k n , ℓ ∗ ( r ℓ ∗ ) , r ℓ ∗ ), ℓ = 1 , 2 , 3 be the estimators discussed in Section 2. In this section, we use numerical simulations to verify the theoretical result provided in Fig. 3. For this purpose, the i.i.d. samples X 1 , … , X n of sizes n = 1000 were simulated N = 500 times from Burr distribution with several values of positive extreme value index γ and second order parameter ρ < 0.

We recall that the Burr (type XII) distribution has a distribution function F ( x ) = 1 − 1 + x − ρ / γ 1 / ρ , x ≥ 0, while the appropriate quantile type function has the form U ( t ) = t γ 1 − t ρ − γ / ρ , t > 1.

The Burr distribution belongs to the Hall’s class of Pareto-type distributions ([16, 17]), i.e. its quantile type function U ( t ) satisfies the relation (18) U ( t ) = C t γ 1 + γ β ρ t ρ + o t ρ , t → ∞ with ( C , β ) = ( 1 , 1 ).

It is well-known that under assumption (18) a condition (3) is satisfied with A ( t ) = β γ t ρ . Now, using (13) one can find that (19) a ← ( t ) = − 2 ρ β 2 γ 2 1 / ( 1 − 2 ρ ) t 1 / ( 2 ρ − 1 ) . By substituting (19) into (14) we get (20) k n , ℓ ∗ ( r ) ∼ ς ℓ 2 ( r ) − 2 ρ β 2 λ ℓ 2 ( ρ , r ) 1 / ( 1 − 2 ρ ) n − 2 ρ / ( 1 − 2 ρ ) , n → ∞ , ℓ = 1 , 2 , 3 . Put T n ( k , τ ) = M n ( k , 1 ) τ − M n ( k , 2 ) / 2 τ / 2 M n ( k , 2 ) / 2 τ / 2 − M n ( k , 3 ) / 6 τ / 3 , τ > 0 , ln M n ( k , 1 ) − ln M n ( k , 2 ) / 2 1 / 2 ln M n ( k , 2 ) / 2 1 / 2 − ln M n ( k , 3 ) / 6 1 / 3 , τ = 0 . To estimate the second order parameter ρ we apply the estimator ρ ˆ n ( κ , τ ) = − 3 ( T n ( κ , τ ) − 1 ) T n ( κ , τ ) − 3 , This estimator was introduced in [11]. To decide which value (0 or 1) of the parameter τ to take in the above mentioned estimator, we implemented the algorithm provided in [15]. To estimate the parameter β we use the estimator β ˆ n ( κ ) = κ n ρ ˆ n ( κ , τ ) × × 1 κ ∑ i = 1 κ i κ − ρ ˆ n ( κ , τ ) 1 κ ∑ i = 1 κ V i − 1 κ ∑ i = 1 κ i κ − ρ ˆ n ( κ , τ ) V i 1 κ ∑ i = 1 κ ( i / κ ) − ρ ˆ n ( κ , τ ) 1 κ ∑ i = 1 κ i κ − ρ ˆ n ( κ , τ ) V i − 1 κ ∑ i = 1 κ i κ − 2 ρ ˆ n ( κ , τ ) V i , where V i = i ln ( X n − i + 1 , n ) − ln ( X n − i ) , 1 ≤ i ≤ κ, which was introduced in [14]. Following the recommendations in [2], for both estimators we used κ = [ n 0.995 ], where [ · ] denotes the integer part.

Replacing β and ρ by estimators β ˆ n ( κ ) and ρ ˆ n ( κ , τ ) in (20) we obtain empirical values of k n , ℓ ∗ ( r ). k ˆ n , ℓ ∗ ( r ) = [ ς ℓ 2 ( r ) − 2 ρ ˆ n ( κ , τ ) β ˆ n 2 ( κ ) λ ℓ 2 ( ρ ˆ n ( κ , τ ) , r ) 1 / ( 1 − 2 ρ ˆ n ( κ , τ ) ) × × n − 2 ρ ˆ n ( κ , τ ) / ( 1 − 2 ρ ˆ n ( κ , τ ) ) ] , ℓ = 1 , 2 , 3 .

To calculate the estimators γ ˆ n ( ℓ ) ( k n , ℓ ∗ ( r ℓ ∗ ) , r ℓ ∗ ), ℓ = 1 , 2 , 3 we use the following algorithm:

1. Estimate the parameters β and ρ using the estimators β ˆ n ( κ ) and ρ ˆ n ( κ , τ ), respectively.

2. Find numerically r ℓ ∗ = argmin r > 0 : λ ℓ 2 ( ρ ˆ n ( κ , τ ) , r ) ς ℓ 2 ( r ) − 2 ρ ˆ n ( κ , τ ) for ℓ = 1 , 2 and r 3 ∗ = argmin r > 1 : λ 3 2 ( ρ ˆ n ( κ , τ ) , r ) ς 3 2 ( r ) − 2 ρ ˆ n ( κ , τ ) .

3. Compute k ˆ n , ℓ ∗ ( r ℓ ∗ ) and then find estimate γ ˆ n ( ℓ ) ( k n , ℓ ∗ ( r ℓ ∗ ) , r ℓ ∗ ).

The results of simulations are summarized in Fig. 5. For the Burr distribution, we took parameters ρ and γ from intervals [ − 4 , 0 ) and ( 0 , 2 ], respectively. We divide the rectangle [ − 4 , 0 ) × ( 0 , 2 ] into squares V i , j = [ − 0.1 ( i + 1 ) , − 0.1 i ) × ( 0.1 j , 0.1 ( j + 1 ) ] , i = 0 , 1 , … , 39 , j = 0 , 1 , … , 19. Taking the true values of the parameters ρ and γ as coordinates of the center of each square V i , j , we performed Monte-Carlo simulations. Let E M S E ℓ ( ρ , γ ) denote the empirical MSE of the estimator γ ˆ n ( ℓ ) ( k n , ℓ ∗ ( r ℓ ∗ ) , r ℓ ∗ ) when observations are simulated from the Burr distribution with true parameters ρ and γ. The square V i , j is colored in black, gray and white if E M S E 1 ( − 0.1 i − 0.05 , 0.1 j + 0.05 ) < E M S E ℓ ( − 0.1 i − 0.05 , 0.1 j + 0.05 ) , ℓ = 2 , 3 , E M S E 2 ( − 0.1 i − 0.05 , 0.1 j + 0.05 ) < E M S E ℓ ( − 0.1 i − 0.05 , 0.1 j + 0.05 ) , ℓ = 1 , 3 , E M S E 3 ( − 0.1 i − 0.05 , 0.1 j + 0.05 ) < E M S E ℓ ( − 0.1 i − 0.05 , 0.1 j + 0.05 ) , ℓ = 1 , 2 , respectively.