Adv Syst Sci Appl 2022; 03:106–123
Published online at https://ijassa.ipu.ru.

Asymptotic Properties of the Block-Type Statistics

Natalia M. Markovich1*, Marijus Vaičiulis2

1V.A. Trapeznikov Institute of Control Sciences Russian Academy of Sciences, Moscow, Russia,
2Vilnius University Institute of Data Science and Digital Technologies, Vilnius University, Vilnius, Lithuania

Abstract:
Extreme value theory is an issue extensively applied in many different fields. One of the central
points of this theory is the estimation of a positive extreme value index. In this paper we introduce
a new family of block type statistics related to this estimation. A weak consistency of the
introduced statistics is proved. A bivariate central limit theorem for newly introduced statistics
is derived. We provide the new family of semi-parametric estimators for the positive extreme
value index. Asymptotic normality of the introduced estimators is proved. It is shown that new
estimators have better asymptotic performance comparing with several block-type estimators over
the whole range of parameters presented in the second order regular variation condition. An
application to the estimation of the positive valued extreme value index for several real data sets
is provided.

Keywords: asymptotic normality, extreme value index, block-type estimator

1. INTRODUCTION

Let X1, X2, . . . , Xn be a sample of independent identically distributed (i.i.d.) random
variables (r.v.s) with an unknown distribution function (d.f.) F . In the present paper we will
formulate our assumptions in terms of a quantile type function U associated to F , which is
defined by

U(t) =

{
0, 0 < t ≤ 1
inf {x : F (x) ≥ 1− (1/t)} , t > 1.

The main assumption is that U belongs to the class of regularly varying functions at
infinity, with γ > 0 (shortly, U ∈ RVγ), i.e., for all x > 0,

lim
t→∞

U(tx)

U(t)
= xγ. (1.1)

The parameter γ in (1.1) is a positive extreme value index (EVI) that is the primary
parameter of large extreme events. Models satisfying (1.1) are quite common in many
application areas such as biostatistics, computer science, finance, insurance and social
sciences, among others.

To exploit the block approach in the extreme value theory one has to divide the
observations X1, . . . , Xn into 1 ≤ s ≤ n non-overlapping blocks of size m = [n/s]:

Bi = {X(i−1)m+1, . . . , Xim}, 1 ≤ i ≤ s,

∗Corresponding author: nat.markovich@gmail.com



ASYMPTOTIC PROPERTIES OF THE BLOCK-TYPE STATISTICS 107

where [·] denotes the integer part. Let X(i)
1,m ≤ X

(i)
2,m ≤ · · · ≤ X

(i)
m,m denote the order statistics

of m observations in the ith block.
The aim of the paper is to investigate asymptotic properties, including a weak consistency

and an asymptotic normality, of the block-type statistics

Qn(s, ℓ, r) =
1

s

s∑
i=1

1

ℓ

ℓ∑
j=1

hr

(
X

(i)
m−j+1,m

X
(i)
m−ℓ,m

)
.

Here, ℓ ∈ N and the family of functions hr, r ∈ R is defined for x > 0 as follows:

hr(x) =

{
(xr − 1) /r, r ̸= 0,
ln(x), r = 0.

Note that the statistics Qn(s, ℓ, r) are applicable particularly when only the largest
observations {X(i)

m−j,m, 0 ≤ j ≤ ℓ, 1 ≤ i ≤ s} are available for the inference. Such type of
data can be found in [4], where battle deaths in major power wars between 1495 and 1975
were analyzed. For more such data see [10] and references therein.

The statistics Qn(s, ℓ, r) generalize two families of statistics. Firstly, by taking r = 0 we
turn back to the estimator for the positive EVI, proposed in [18]. Secondly, the statistics
Qn(s, ℓ, r) generalize the statistics Qn(s, 1, r), introduced in [17], see also [21].

Note that the statistics Qn(s, ℓ, r) are scale-free, that is, Qn(s, ℓ, r) do not change when
observations X1, . . . , Xn are replaced by cX1, . . . , cXn with c > 0.

The paper is organized as follows. In Section 2 our main results are presented. In the next
section, we introduce a new family of estimators for positive EVIs and compare the latter
estimators with several known estimators. In Section 4 the application to several real data
sets is discussed. We finalize with conclusions. The last section contains proofs of the results.

2. MAIN RESULTS

Let d→ and
p→ denote the convergence in distribution and in probability, respectively. The

equality in distribution will be denoted by d
=.

Theorem 2.1:
Suppose that X1, . . . , Xn are i.i.d. r.v.s with the d.f. F , whose quantile type function U satisfies
(1.1) with some γ > 0. Let ℓ ∈ N. Let r ∈ R be such that γr < 1 holds. Let the block number
s = sn and the block width m = mn tend to infinity as n → ∞. Then

Qn(s, ℓ, r)
p→ Λγ(γr), n → ∞,

where Λγ(t) := γ/(1− t), t < 1.
The second order parameter ρ rules the rate of convergence in the first order condition

(1.1). Moreover, ρ is a non-positive parameter that appears in the limiting relation

lim
t→∞

1

A(t)

(
U(tx)

U(t)
− xγ

)
= xγhρ(x), (2.2)

which we assume to be held for all x > 0. Here, A(t) is a measurable function with a constant
sign near infinity and which is not identically zero, and A(t) → 0 as t → ∞. It is known that
(2.2) implies that |A(t)| ∈ RVρ, see [11].

Let Z = Z(t), t < 1/2 be a Gaussian random process with mean 0 and covariance
function σ2(t1, t2) := E (Z(t1)Z(t2)), where σ2(t1, t2) = (1− t1 − t2)

−1.

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



108 N. M. MARKOVICH, M. VAIČIULIS

Theorem 2.2:
Suppose that X1, . . . , Xn are i.i.d. r.v.s with the d.f. F , whose quantile type function U satisfies
the second order condition (2.2) with some γ > 0 and ρ ≤ 0. Let the block number s = sn
and the block width m = mn satisfy the assumptions of Theorem 2.1 and, in addition,

lim
n→∞

√
sA(m) = µ ∈ (−∞,+∞) (2.3)

hold. Let ℓ ∈ N and ri ∈ R, i = 1, 2 be such that γri < 1/2 holds. Then it holds
√
s (Qn(s, ℓ, r1)− Λγ (γr1) , Qn(s, ℓ, r2)− Λγ (γr2))

d→
(
µν(ℓ, ρ, r1) +

Λγ (γr1)√
ℓ

Z(γr1), µν(ℓ, ρ, r2) +
Λγ (γr2)√

ℓ
Z(γr2)

)
(2.4)

as n → ∞, where

ν(ℓ, ρ, ri) =
Γ(1 + ℓ− ρ)

Γ(1 + ℓ)
· 1

(1− γri)(1− γri − ρ)
, i = 1, 2,

and Γ(t) =
∫∞
0

xt−1 exp{−x}dx, t > 0 is the gamma function.

Remark 1. If in conditions of Theorem 2.2 assumption (2.3) is replaced by

lim
n→∞

√
sA(m) = ∞, (2.5)

then it holds
1

A(m)
(Qn(s, ℓ, r1)− Λγ (γr1))

p→ ν(ℓ, ρ, r1), n → ∞. (2.6)

The extension of Theorem 2.2 to the multivariate case is quite trivial. Thus, Theorem
2.2 gives us a lot of possibilities to form asymptotically normal estimators of γ > 0 using
statistics Qn(s, ℓ, r) with different r. Considering the simplest case we propose the following
family of the block-type estimators for γ > 0:

γ̂(1)
n (s, ℓ, r) =

{
Qn(s, ℓ, 0), r = 0,
Qn(s, ℓ, r) (1 + rQn(s, ℓ, r))

−1 , r ̸= 0.

It should be noted that the family of estimators γ̂
(1)
n (s, ℓ, r) generalizes the family of

estimators given in [18] (for r = 0) and the family of estimators proposed in [17] (for ℓ = 1).
We establish the asymptotic normality of γ̂

(1)
n (s, ℓ, r) by applying the univariate form of

Theorem 2.2.
Corollary 2.3:
Let the conditions of Theorem 2.2 be fulfilled for univariate case. Then it holds

√
s
(
γ̂(1)
n (s, ℓ, r)− γ

) d→N
(
µν1(ℓ, ρ, γr), γ

2σ2
1(ℓ, γr)

)
, n → ∞,

where N denotes a normal distribution and

ν1(ℓ, ρ, R) =
Γ(1 + ℓ− ρ)

Γ(1 + ℓ)
· 1−R

1−R− ρ
, σ2

1(ℓ, R) =
(1−R)2

ℓ(1− 2R)
, R < 1/2.

Taking r = 0 in Corollary 2.3 we turn back to Theorem 1 in [18]. It should be noted
that the asymptotic bias is written in a slightly different form in [18]. Another partial case
of Corollary 2.3 can be found in [17] (see (2.12) therein). There, assuming more simple
second order asymptotic condition than (2.3), the asymptotic normality of the estimators
1/γ̂

(1)
n (s, 1, r) is obtained.

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



ASYMPTOTIC PROPERTIES OF THE BLOCK-TYPE STATISTICS 109

3. A NEW FAMILY OF THE BLOCK-TYPE ESTIMATORS FOR POSITIVE EVI

Let r ̸= 0 satisfy γr < 1, while s = sn and m = mn satisfy the assumptions of Theorem 2.1.
Then Qn(s, ℓ, 0)/Qn(s, ℓ, r)

p→1− γr holds as n → ∞. So,

γ̂(2)
n (s, ℓ, r) =

Qn(s, ℓ, r)−Qn(s, ℓ, 0)

rQn(s, ℓ, r)
, r ̸= 0

presents a family of weakly consistent estimators for positive EVI. Let us define
γ̂
(2)
n (s, ℓ, 0) := limr→0 γ̂

(2)
n (s, ℓ, r). By using the L’Hospital’s rule, it follows that

γ̂(2)
n (s, ℓ, 0) =

Q̃n(s, ℓ)

2Qn(s, ℓ, 0)
,

where

Q̃n(s, ℓ) =
1

s

s∑
i=1

1

ℓ

ℓ∑
j=1

ln2

(
X

(i)
m−j+1,m

X
(i)
m−ℓ,m

)
.

An application of Theorem 2.2 yields the following result.

Theorem 3.1:
Let the conditions of Theorem 2.2 be fulfilled with r1 = 0 and r2 = r. Then it holds

√
s
(
γ̂(2)
n (s, ℓ, r)− γ

) d→N
(
µν2(ℓ, ρ, γr), γ

2σ2
2(ℓ, γr)

)
, n → ∞, (3.7)

where

ν2(ℓ, ρ, R) =
Γ(1 + ℓ− ρ)

Γ(1 + ℓ)
· 1−R

(1−R− ρ)(1− ρ)
, σ2

2(ℓ, R) =
2(1−R)

ℓ(1− 2R)
, R < 1/2.

Having the asymptotic normality of the introduced estimators γ̂(2)
n (s, ℓ, r) we can discuss

an optimal choice of s, ℓ and r.
We assume further that ρ < 0 and µ ̸= 0. Keeping ℓ and r fixed, the limiting mean squared

error (MSE) for γ̂(2)
n (s, ℓ, r) is, approximately,

inf
s

{
γ2σ2

2(ℓ, γr)

s
+ A2

(n
s

)
ν2
2(ℓ, ρ, γr)

}
,

where the sequence s = sn satisfies the assumptions of Theorem 2.1.
It is known that there exists a positive decreasing function b ∈ RV2ρ−1 such that A2(t) ∼∫∞

t
b(x)dx, t → ∞, see Here and below, we write an an ∼ cn if an/cn → 1 as n → ∞. Let

b← denote the inverse function of b. Following the lines in [12] we find

MSE
(
γ̂(2)
n (s∗2, ℓ, r)

)
∼ 1− 2ρ

−2ρ

(
ν2
2(ℓ, ρ, γr)

(
γ2σ2

2(ℓ, γr)
)−2ρ)1/(1−2ρ) b←(1/n)

n
, (3.8)

as n → ∞, where the optimal choice of the number of blocks s∗2 satisfies the relation

s∗2 ∼
(
γ2σ2

2(ℓ, γr)

ν2
2(ℓ, ρ, γr)

)1/(1−2ρ)
n

b←(1/n)
, n → ∞.

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



110 N. M. MARKOVICH, M. VAIČIULIS

Next, we minimize the right hand side of (3.8) with respect to ℓ and r. For this it is
sufficient to minimize the product

ν2
2(ℓ, ρ, γr)

(
γ2σ2

2(ℓ, γr)
)−2ρ

=
2−2ρ

(1− ρ)2
(φ(ℓ, ρ))2 ϕ(ρ, γr)

with respect to ℓ and r. Here, we have

φ(ℓ, ρ) =
ℓρΓ(ℓ+ 1− ρ)

Γ(ℓ+ 1)
, ϕ(ρ,R) =

(1−R)2−2ρ

(1−R− ρ)2(1− 2R)−2ρ
, R < 1/2.

By using the standard technique of derivatives we obtain that the function ϕ(ρ,R) attains
its minimum with R∗2 = ρ. Whence, the optimal choice of parameter r for γ̂(2)

n (s∗2, ℓ, r) has
the form r∗2 = ρ/γ.

Before we proceed, let us introduce the following lemma regarding the function φ(ℓ, ρ).

Lemma 3.2:
For any ℓ ∈ N and ρ < 0, φ(ℓ, ρ) > φ(ℓ+ 1, ρ) holds.

By Lemma 3.2, for any ℓ ∈ N, the estimator γ̂(2)
n (s∗2, ℓ, r

∗
2) has a bigger asymptotic MSE

comparing with γ̂
(2)
n (s∗2, ℓ+ 1, r∗2). Thus, there is no an optimal estimator among estimators

γ̂
(2)
n (s∗2, ℓ, r

∗
2), ℓ ∈ N. The same conclusion holds for the estimators γ̂

(1)
n (s∗1, ℓ, r

∗
1), ℓ ∈ N,

where

s∗1 ∼
(
γ2σ2

1(ℓ, γr)

ν2
1(ℓ, ρ, γr)

)1/(1−2ρ)
n

b←(1/n)
, n → ∞

and r∗1 = (2γ)−1
(
2− ρ− (2− 4ρ+ ρ2)

1/2
)

, see [17]. As for MSE
(
γ̂
(1)
n (s∗1, ℓ, r)

)
, it has

the same asymptotic as in (3.8) with replacing ν2(ℓ, ρ, γr) and σ2
2(ℓ, γr) by ν1(ℓ, ρ, γr) and

σ2
1(ℓ, γr), respectively.

Let us find the limit of the ratio of MSE
(
γ̂
(i)
n (s∗i , ℓ+ 1, r∗i )

)
and MSE

(
γ̂
(i)
n (s∗i , ℓ, r

∗
i )
)

:

lim
n→∞

MSE
(
γ̂
(i)
n (s∗i , ℓ+ 1, r∗i )

)
MSE

(
γ̂
(i)
n (s∗i , ℓ, r

∗
i )
) = χ(ℓ, ρ), i = 1, 2,

where

χ(ℓ, ρ) =

((
ℓ

ℓ+ 1

)ρ
ℓ+ 1− ρ

ℓ+ 1

)2/(1−2ρ)

.

For the well-known heavy-tailed models, like the Frechet and the Student’s t, condition
(2.2) holds with ρ = −1. We represent the numerical approximation of χ(ℓ,−1) for several
values of ℓ: χ(1,−1) ≈ 0.825, χ(2,−1) ≈ 0.924, χ(3,−1) ≈ 0.958, χ(4,−1) ≈ 0.973 and
χ(5,−1) ≈ 0.981. Since the values of χ(ℓ,−1) are close to 1 when ℓ ≥ 3, we recommend to
take 3 ≤ ℓ ≤ 5 dealing with real data sets.

At the end of this section, we compare the estimators γ̂
(1)
n (s∗1, ℓ, r

∗
1) and γ̂

(2)
n (s∗2, ℓ, r

∗
2).

Denoting

lim
n→∞

MSE
(
γ̂
(1)
n (s∗1, ℓ, r

∗
1)
)

MSE
(
γ̂
(2)
n (s∗2, ℓ, r

∗
2)
) = χ̃(ρ),

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



ASYMPTOTIC PROPERTIES OF THE BLOCK-TYPE STATISTICS 111

it is not difficult to get that

χ̃(ρ) =

(
(1− 2ρ)1−1/ρ(1−R∗1)

2−1/ρ

2(1− ρ)(1−R∗1 − ρ)−1/ρ(1− 2R∗1)

)−2ρ/(1−2ρ)
,

where R∗1 = γr∗1. Let us note that

(χ̃(ρ))(1−2ρ)/(−2ρ) > 1 (3.9)
for all −∞ < ρ < 0 (see Section 6 for the proof). Since (3.9) yields χ̃(ρ) > 1, ρ < 0,
we conclude that for any ℓ ∈ N the new estimator γ̂

(2)
n (s∗2, ℓ, r

∗
2) dominates the estimator

γ̂
(1)
n (s∗1, ℓ, r

∗
1) in the whole region of parameters {(γ, ρ) : γ > 0, ρ < 0}. Whence the

conclusion related to the estimators introduced in [17] follows, namely, γ̂
(2)
n (s∗2, 1, r

∗
2)

outperforms γ̂(1)
n (s∗1, 1, r

∗
1) in the whole region of parameters {(γ, ρ) : γ > 0, ρ < 0}.

Comparing the estimators γ̂(1)
n (s∗1, ℓ, 0) (proposed in [18]) and γ̂

(2)
n (s∗2, ℓ, 0) we have

lim
n→∞

MSE
(
γ̂
(1)
n (s∗1, ℓ, 0)

)
MSE

(
γ̂
(2)
n (s∗2, ℓ, 0)

) = ((1− ρ)2ρ)2/(1−2ρ) .

Whence we get that the estimator γ̂
(2)
n (s∗2, ℓ, 0) has a smaller asymptotic MSE in the

area {(γ, ρ) : γ > 0,−1 < ρ < 0}, while the estimator γ̂
(1)
n (s∗1, ℓ, 0) is better in the area

{(γ, ρ) : γ > 0, ρ < −1}.
We end this section with comparison of γ̂(2)

n (s∗2, ℓ, r
∗
2) and the Hill’s estimator ( [14]) that

has the form

γ(3)
n (k) =

1

k

k∑
i=1

ln

(
Xn−i+1,n

Xn−k,n

)
,

where X1,n ≤ · · · ≤ Xn,n denote the ascending order statistics of the observations
X1, . . . , Xn.

By [12] we know that the optimal choice of the sample fraction k satisfies the relation

k∗ ∼
(
γ2(1− ρ)2

)1/(1−2ρ) n

b←(1/n)
, n → ∞,

while

MSE
(
γ(3)
n (k∗)

)
∼ 1− 2ρ

−2ρ

(
γ−4ρ

(1− ρ)2

)1/(1−2ρ)
b←(1/n)

n
, n → ∞

holds. Thus, we get immediately

lim
n→∞

MSE
(
γ
(3)
n (k∗)

)
MSE

(
γ̂
(2)
n (s∗2, ℓ, r

∗
2)
) = υ(ℓ, ρ),

where

υ(ℓ, ρ) =

((
ℓ

2

)−2ρ(
1− 2ρ

1− ρ

)2−2ρ(
Γ(1 + ℓ)

Γ(1 + ℓ− ρ)

)2
)1/(1−2ρ)

.

Functions υ(3, ρ) and υ(5, ρ) are shown in Fig.3.1. Whence one can deduce that for
ℓ ≥ 3 there exists ρ̃ = ρ̃(ℓ) < 0 such that the estimator γ̂(2)

n (s∗2, ℓ, r
∗
2) outperforms the Hill’s

estimator γ(3)
n (k∗) within the area (γ, ρ) ∈ (0,∞)× (ρ̃, 0).

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



112 N. M. MARKOVICH, M. VAIČIULIS

-1.0 -0.8 -0.6 -0.4 -0.2
ρ

0.90

0.92

0.94

0.96

0.98

1.02

1.04

Fig. 3.1. Graph of of the functions υ(3, ρ) (grey) and υ(5, ρ) (black)

4. APPLICATIONS

We apply the estimators γ̂
(i)
n (s, ℓ, 0), i = 1, 2 with ℓ = 1, 3, 5 to real data sets. We analyze

absolute log returns of two daily data sets: (i) natural gas prices (dollars USA per cubic feet)
between 7 January 1997 and 1 September 2020; (ii) Europe Brent Spot Prices (dollars USA
per barrel) between 20 May 1987 and 28 August 2020. The data sets contain n = 5389 and
n = 8261 non-zero log returns, respectively. Let us denote the corresponding absolute log
returns by r1,t and r2,t. Fig. 1 displays graphs of r1,t (on the left) and r2,t (on the right).

0 1000 2000 3000 4000 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2000 4000 6000 8000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 4.2. Graphs of r1,t, 1 ≤ t ≤ 5389 (on the left) and r2,t, 1 ≤ t ≤ 8261 (on the right)

Let r(1)i ≥ r
(2)
i ≥ . . . , i = 1, 2 be the order statistics in the decreasing order. We use QQ

plots

T (i)(n, k) =

{(
− ln

(
j

k

)
, ln

(
r
(j)
i

r
(k)
i

))
, 1 ≤ j ≤ k

}
, i = 1, 2

provided in [6] for the preliminary analyze. By Prop. 4.1 in [6] and assuming that the
distribution of the non-zero ri,t belongs to the class RV−1/γ and a sequence k = kn is such
that k → ∞, n/k → ∞ as n → ∞, the random sets T (i)(n, s) converge in probability to the
half-line T (i) =

{
(x, γ(i)x) : 0 ≤ x < ∞

}
. Fig. 2 shows the graph of T (1) (n, k) (on the left)

and T (2) (n, k) (on the right) based on the upper 5% of the absolute log returns. The dotted
least squares (LS) half-lines are presented in the corresponding graph. By Fig. 2, the absolute
log returns for both data sets show a classic power law shape, with a slightly worse reflection
at the largest values.

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



ASYMPTOTIC PROPERTIES OF THE BLOCK-TYPE STATISTICS 113

1 2 3 4 5

0.5

1.0

1.5

2.0

1 2 3 4 5 6

0.5

1.0

1.5

2.0

2.5

Fig. 4.3. Graphs of T (1)
(
n(1), n(1)/20

)
(on the left) and T (2)

(
n(2), n(2)/20

)
(on the right) with dotted LS

half-lines

By considering the first data set we provide plots (suggested in [20])

P1 (n, ℓ) =
{(

θ, γ̂(1)
n

([
nθ
]
, ℓ, 0

))
, 0 < θ ≤ 0.7

}
,

P2 (n, ℓ) =
{(

θ, γ̂(2)
n

([
nθ
]
, ℓ, 0

))
, 0 < θ ≤ 0.7

}
for several ℓ values in Fig. 3-5. In practice, a stable region is visually detected in each plot and
then an estimate of the parameter γ > 0 is taken as a mean over the stability region (shortly,
the SR). The SR and the estimate of γ for each considered value of ℓ is given in Tab. 4.1.
Moreover, the estimate of γ is presented in Fig. 3-5 as a dotted line.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

θ

0.1

0.2

0.3

0.4

0.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

θ

0.1

0.2

0.3

0.4

0.5

Fig. 4.4. Plots P1 (n, 1) (on the left), P2 (n, 1) (on the right) and the estimate of γ (dotted line)

The corresponding plots for the second data set are not represented due to their similarity
to the plots in Fig. 3-5, but the corresponding results obtained by the SR and the estimate of
γ are summarized in Tab. 4.2.

Table 4.1. Estimation results for the first data set with the size n = 5389

Plot P1 (n, ℓ) P2 (n, ℓ)
ℓ 1 3 5 1 3 5

SR of θ [0.48,0.6] [0.34,0.56] [0.36,0.54] [0.48,0.6] [0.34,0.56] [0.36,0.54]
γ estimate 0.201 0.255 0.278 0.177 0.234 0.258

We end this section with several notings. Firstly, the estimates of γ > 0 with ℓ = 1 are
in close correspondence for both data sets, see Tab. 4.1 and 4.2. From the other hand, by

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



114 N. M. MARKOVICH, M. VAIČIULIS

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

θ

0.1

0.2

0.3

0.4

0.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

θ

0.1

0.2

0.3

0.4

0.5

Fig. 4.5. Plots P1 (n, 3) (on the left), P2 (n, 3) (on the right) and estimate of γ (dotted line)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

θ

0.1

0.2

0.3

0.4

0.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

θ

0.1

0.2

0.3

0.4

0.5

Fig. 4.6. Plots P1 (n, 5) (on the left), P2 (n, 5) (on the right) and estimate of γ (dotted line)

Table 4.2. Estimation results for the second data set with the size n = 8261

Plot P1 (n, ℓ) P2 (n, ℓ)
ℓ 1 3 5 1 3 5

SR of θ [0.32,0.64] [0.32,0.56] [0.34,0.52] [0.32,0.64] [0.32,0.55] [0.34,0.52]
γ estimate 0.204 0.228 0.244 0.180 0.212 0.220

comparing the graphs in Fig. 1 one can observe that the left graph contains more intermediate
size peaks than the right graph. This should be reflected on estimates of γ > 0. Such
insensitivity of estimators γ̂

(i)
n (s, 1, 0), i = 1, 2 in the practical applications is one more

reason to use γ̂
(i)
n (s, ℓ, 0) with ℓ > 1. Secondly, summarizing the results on stability region

of parameter θ, we have θ ∈ [1/3, 1/2] for both data sets. This gives that the stability region
for block number s is

[
[n1/3], [n1/2]

]
. Consequently,

[
[n1/2], [n2/3]

]
is stability region for the

block width m. Keeping in mind that [53891/2] = 73 and [82611/2] = 90, we believe that the
block width is large enough in its stability region to take ℓ = 5 for both data sets. Finally, we
estimate the second order parameter ρ by the estimator proposed in [8] and get ρ̂ = −0.69
and ρ̂ = −0.73 for absolute log returns of the first and second data set, respectively. Since the
estimator γ̂(2)

n (s, 5, 0) is better than the estimator γ̂(1)
n (s, 5, 0) when −1 < ρ < 0, we conclude

that 0.258 and 0.220 are the estimates of γ for absolute log returns of the data sets under
consideration.

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



ASYMPTOTIC PROPERTIES OF THE BLOCK-TYPE STATISTICS 115

5. CONCLUSIONS

The new family of block-type statistics Qn(s, ℓ, r) is introduced. Under the classical
assumptions on quantile type function U , block number s = sn, block width m = mn and
additionally on the parameter r, the weak consistency and the asymptotic normality (the
bivariate central limit theorem) of the statistics Qn(s, ℓ, r) are proved.

The statistics Qn(s, ℓ, r) can be used to construct new block-type estimators for positive
EVI. We list practical reasons for using the block type estimators. Firstly, as it is mentioned
in Introduction, the block type estimators are applicable when only a few largest values are
observed within blocks. Secondly, due to the recursiveness of the block-type estimators (see,
pg. 17 in [15]) they are well suited for the on-line estimation of the positive EVI.

We proposed the new family of semiparametric estimators γ̂
(2)
n (s, ℓ, r) for positive EVI.

This family is based on statistics Qn(s, ℓ, 0) and Qn(s, ℓ, r), r ∈ R. Hence, the asymptotic
normality of γ̂(2)

n (s, ℓ, r) is quite simple application of our bivariate central limit theorem for
the statistics Qn(s, ℓ, r), r ∈ R. The optimal choices s∗2 and r∗2 for s = sn and r are proposed.
Also, it is proved that for any ℓ ∈ N, the estimator γ̂

(2)
n (s∗2, ℓ, r

∗
2) has a bigger asymptotic

MSE comparing with γ̂
(2)
n (s∗2, ℓ+ 1, r∗2). Despite that there is no an optimal estimator among

estimators γ̂
(2)
n (s∗2, ℓ, r

∗
2), ℓ ∈ N, we gave some heuristic argument that it is enough to take

3 ≤ ℓ ≤ 5 considering a real data set.
Moreover, a new family of semiparametric estimators γ̂(1)

n (s, ℓ, r) is proposed. It includes
several known families of estimators for positive EVI. New estimators γ̂(2)

n are shown to be
better than γ̂

(1)
n in the sense of the asymptotic MSE.

The performance of the estimators γ̂
(1)
n (s, ℓ, 0) and γ̂

(2)
n (s, ℓ, r) is demonstrated by

considering real data sets. We left numerous open questions related to the behavior of new
estimators for middle size samples. We admit that quite extensive Monte-Carlo simulations
are needed and we hope to fill this gap in the nearest future.

6. PROOFS

Proof of Lemma 3.2.

The inequality φ(ℓ+ 1, ρ) < φ(ℓ, ρ) is equivalent to

−ρ

ℓ+ 1
<

(
1 +

1

ℓ

)−ρ
− 1. (6.10)

If ρ = −1, then (6.10) holds obviously. Let 0 < −ρ < 1. By using a symmetric form of the
Bernoulli’s inequality (see, pg.5 in [3]) we obtain that the right hand side of (6.10) is not less
than (−ρ/ℓ)(1 + 1/ℓ)−1−ρ. It remains to check that for any ℓ ∈ N the last quantity exceeds
−ρ/(ℓ+ 1).

Let us consider the case −ρ > 1. By applying the symmetric form of the Bernoulli’s
inequality one more time we get(

1 +
1

ℓ

)−ρ
− 1 >

−ρ

ℓ
>

−ρ

ℓ+ 1
, ℓ ∈ N.

This completes the proof of Lemma 3.2.

Let Y1, Y2, . . . , Ym be i.i.d. Pareto r.v.s with P(Y1 ≥ x) = 1/x, x ≥ 1. Let Yℓ,m, 1 ≤ ℓ ≤
m denote, as usual, the ℓ-th ascending order statistic. If ℓ ∈ N, then by combining Example

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



116 N. M. MARKOVICH, M. VAIČIULIS

8.3.3 and Theorem 8.4.1 in [1], we get that m−1Ym−ℓ,m converges in distribution to some
non-degenerate distribution as m → ∞. The next lemma establishes the asymptotic behavior
of the moments of m−1Ym−ℓ,m.
Lemma 6.1:
Let ζ ∈ R satisfy ζ < ℓ+ 1, where ℓ ∈ N. Then it holds

E

(
Ym−ℓ,m

m

)ζ

→ Γ(ℓ− ζ + 1)

Γ(ℓ+ 1)
, m → ∞. (6.11)

Proof
The trivial case ζ = 0 does not require a proof. Let ζ > 0. By using (2.2.2) in [1] we derive
that the density function of (m−1Ym−ℓ,m)

ζ is

p(x) =
m−(ℓ+1)m!

ζℓ!(m− ℓ− 1)!
x−(ℓ+1)/ζ−1

(
1− 1

mx1/ζ

)m−ℓ−1

, m−ζ ≤ x < ∞.

From the above density function we obtain the expectation of (m−1Ym−ℓ,m)
ζ :

E

(
Ym−ℓ,m

m

)ζ

=
m−(ℓ+1)m!

ζℓ!(m− ℓ− 1)!

∫ ∞
m−ζ

x−(ℓ+1)/ζ

(
1− 1

mx1/ζ

)m−ℓ−1

dx

=
m−ζm!

ℓ!(m− ℓ− 1)!
B(ℓ− ζ + 1,m− ℓ),

where B(a, b) =
∫ 1

0
xa−1(1− x)b−1dx, a > 0, b > 0 is the beta function. The gamma and

beta functions are related as B(a, b) = Γ(a)Γ(b)/Γ(a+ b). Thus, we have

E

(
Ym−ℓ,m

m

)ζ

=

{
m−ζΓ(m+ 1)

Γ(m− ζ + 1)

}
Γ(ℓ− ζ + 1)

Γ(ℓ+ 1)
.

By using the Stirling’s formula for the gamma function one can verify that the quantity in
the curly brackets tends to 1 as m → ∞. For the rest case ζ < 0 the proof of (6.11) is similar
and thus, it is omitted. Lemma 6.1 is proved.

Proof of the Theorem 2.1.
We use the Potter’s bounds (see, e.g., Prop. B.1.9 in [13]). Namely, for any ε > 0 and δ > 0
there exists t0 = t0(ε, δ), such that for t ≥ t0 and x ≥ 1,

(1− ε)xγ−δ ≤ U(tx)

U(t)
≤ (1 + ε)xγ+δ

holds. Without loss of generality let us assume that 0 < ε < 1. Since the function hr is strictly
increasing at (0,∞) for any r ∈ R, we get

hr

(
(1− ε)xγ−δ) ≤ hr

(
U(tx)

U(t)

)
≤ hr

(
(1 + ε)xγ+δ

)
,

or equivalently,

xγrhr

(
(1− ε)x−δ

)
≤ hr

(
U(tx)

U(t)

)
− hr (x

γ) ≤ xγrhr

(
(1 + ε)xδ

)
. (6.12)

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



ASYMPTOTIC PROPERTIES OF THE BLOCK-TYPE STATISTICS 117

Keeping in mind
U(Y

(i)
j )

d
=X

(i)
j , 1 ≤ j ≤ ℓ, 1 ≤ i ≤ s, (6.13)

where Y
(1)
1 , . . . , Y

(1)
ℓ , . . . , Y

(s)
1 , . . . , Y

(s)
ℓ are i.i.d. Pareto r.v.s with P

(
Y

(1)
1 > x

)
= 1/x,

x ≥ 1, we have

Qn(s, ℓ, r)
d
=

1

s

s∑
i=1

1

ℓ

ℓ∑
j=1

hr

U
(
Y

(i)
m−j+1,m

)
U
(
Y

(i)
m−ℓ,m

)
 .

The right endpoint of the Pareto distribution is equal to infinity. Thus, for any 1 ≤ i ≤ s,
Y

(i)
m−ℓ,m

p→+∞ as m → ∞. Substituting

t = Y
(i)
m−ℓ,m, x =

Y
(i)
m−j+1,m

Y
(i)
m−ℓ,m

, (6.14)

into (6.12) we get

hr

U
(
Y

(i)
m−j+1,m

)
U
(
Y

(i)
m−ℓ,m

)
− hr

((
Y

(i)
m−j+1,m

Y
(i)
m−ℓ,m

)γ)

≥

(
Y

(i)
m−j+1,m

Y
(i)
m−ℓ,m

)γr

hr

(1− ε)

(
Y

(i)
m−j+1,m

Y
(i)
m−ℓ,m

)−δ , (6.15)

hr

U
(
Y

(i)
m−j+1,m

)
U
(
Y

(i)
m−ℓ,m

)
− hr

((
Y

(i)
m−j+1,m

Y
(i)
m−j,m

)γ)

≤

(
Y

(i)
m−j+1,m

Y
(i)
m−ℓ,m

)γr

hr

(1 + ε)

(
Y

(i)
m−j+1,m

Y
(i)
m−ℓ,m

)δ
 . (6.16)

By summing inequalities (6.15), (6.16) over 1 ≤ j ≤ ℓ and then over 1 ≤ i ≤ s we obtain

Q̃n(s, ℓ, r)−∆(−)
n (s, ℓ, r) ≤ Qn(s, ℓ, r) ≤ Q̃n(s, ℓ, r) + ∆(+)

n (s, ℓ, r),

where

Q̃n(s, ℓ, r) =
1

s

s∑
i=1

1

ℓ

ℓ∑
j=1

hr

((
Y

(i)
m−j+1,m

Y
(i)
m−ℓ,m

)γ)
,

∆(±)
n (s, ℓ, r) =

1

s

s∑
i=1

1

ℓ

ℓ∑
j=1

(
Y

(i)
m−j+1,m

Y
(i)
m−ℓ,m

)γr

hr

(1± ε)

(
Y

(i)
m−j+1,m

Y
(i)
m−ℓ,m

)±δ .

Now it suffices to prove that

Q̃n(s, ℓ, r)
p→ Λγ(γr), (6.17)

∆(±)
n (s, ℓ, r)

p→ 0 (6.18)

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



118 N. M. MARKOVICH, M. VAIČIULIS

as n → ∞. The Rényi’s representation ( [19]) enables us to write

1

s

s∑
i=1

1

ℓ

ℓ∑
j=1

g

(
Y

(i)
m−j+1,m

Y
(i)
m−ℓ,m

)
d
=
1

s

s∑
i=1

1

ℓ

ℓ∑
j=1

g
(
Y

(i)
j

)
, (6.19)

where g is any measurable function, see (4.14) in [9]. By applying (6.19) we get

Q̃n(s, ℓ, r)
d
=

1

s

s∑
i=1

ηi, ηi :=
1

ℓ

ℓ∑
j=1

hr

((
Y

(i)
j

)γ)
.

Note that ηi, 1 ≤ i ≤ s is a sequence of i.i.d. r.v.s with Eη1 = Λγ(γr). Thus, by applying
the Khintchine weak law of large numbers the claim (6.17) follows.

Let δ > 0 be small enough that such γ(r + δ) < 1 holds. Similarly as in (6.17) we find

∆(±)
n (s, ℓ, r)

p→
{

h0(1± ε)± δ, r = 0,
hr(1±ε)(1−γr)±γδ/r
(1−γ(r±δ))(1−γr) , r ̸= 0. (6.20)

Let us recall that for any r ∈ R, hr is a continuous function on (0,∞). This, together with
hr(1) = 0, gives hr(1± ε) → 0 as ε ↓ 0. So, for any r ∈ R, the right hand side of (6.20) tends
to 0 as ε ↓ 0 and δ ↓ 0. Thus, the claim (6.18) follows. Theorem 2.1 is proved.

Proof of the Theorem 2.2.
By using (6.13) one more time we get that for any (θ1, θ2) ∈ R2,

θ1 (Qn(s, ℓ, r1)− Λγ (γr1)) + θ2 (Qn(s, ℓ, r2)− Λγ (γr2))
d
= Tn

holds, where

Tn :=
1

s

s∑
i=1

1

ℓ

ℓ∑
j=1

θ1

(
hr1

(
U(Y

(i)
m−j+1,m)

U(Y
(i)
m−ℓ,m)

)
− Λγ (γr1)

)

+θ2

(
hr2

(
U(Y

(i)
m−j+1,m)

U(Y
(i)
m−ℓ,m)

)
− Λγ (γr2)

)
.

By the Cramér-Wold Theorem (see, e.g., pg. 16 in [22]), the claim (2.4) will be proved if
we show that

√
snTn

d→ θ1

(
µν(ℓ, ρ, r1) +

Λγ (γr1)√
ℓ

Z(γr1)

)
+ θ2

(
µν(ℓ, ρ, r2) +

Λγ (γr2)√
ℓ

Z(γr2)

)
(6.21)

as n → ∞.
The relation (2.2) is equivalent to

lim
t→∞

hr ((tx)
−γU(tx))− hr (t

−γU(t))

(t−γU(t))r A(t)
= hρ(x), (6.22)

see [16] for details. By applying the Drees inequality ( [7], see also Prop. 2.1 in [5]), the
relation (6.22) implies that there exists the function A0(t), such that A0(t) ∼ (t−γU(t))

r
A(t),

t → ∞ and for all ε > 0 and δ > 0 there is t0 = t0(δ, ϵ) such that for t ≥ t0 and x ≥ 1,∣∣∣∣hr ((tx)
−γU(tx))− hr (t

−γU(t))

A0(t)
− hρ(x)

∣∣∣∣ ≤ εxρ+δ

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



ASYMPTOTIC PROPERTIES OF THE BLOCK-TYPE STATISTICS 119

holds, and consequently, it follows

hr

(
U(tx)

U(t)

)
≥ hr (x

γ) + xγrhρ(x)A1(t)− εxγr+ρ+δ |A1(t)| , (6.23)

hr

(
U(tx)

U(t)

)
≤ hr (x

γ) + xγrhρ(x)A1(t) + εxγr+ρ+δ |A1(t)| , (6.24)

where A1(t) = (tγU(t))−r A0(t). We note that A1(t) ∼ A(t) as t → ∞. This, together with
|A(t)| ∈ RVρ, implies |A1(t)| ∈ RVρ.

Next, we apply (6.23), (6.24) with substitutions (6.14). Then we get, as in the proof of
Theorem 2.1,

T (1)
n + θ1T

(2)
n (r1) + θ2T

(2)
n (r2)− ε|θ1|T (3)

n (r1)− ε|θ2|T (3)
n (r2) ≤ Tn, (6.25)

T (1)
n + θ1T

(2)
n (r1) + θ2T

(2)
n (r2) + ε|θ1|T (3)

n (r1) + ε|θ2|T (3)
n (r2) ≥ Tn, (6.26)

where

T (1)
n =

1

s

s∑
i=1

1

ℓ

ℓ∑
j=1

θ1

(
γhγr1

(
Y

(i)
m−j+1,m

Y
(i)
m−ℓ,m

)
− Λγ (γr1)

)

+θ2

(
γhγr2

(
Y

(i)
m−j+1,m

Y
(i)
m−ℓ,m

)
− Λγ (γr2)

)
,

T (2)
n (r) =

1

s

s∑
i=1

A1

(
Y

(i)
m−ℓ,m

)
Υi(r),

Υi(r) =
1

ℓ

ℓ∑
j=1

(
Y

(i)
m−j+1,m

Y
(i)
m−ℓ,m

)γr

hρ

(
Y

(i)
m−j+1,m

Y
(i)
m−ℓ,m

)
,

T (3)
n (r) =

1

s

s∑
i=1

∣∣∣A1

(
Y

(i)
m−ℓ,m

)∣∣∣ 1
ℓ

ℓ∑
j=1

(
Y

(i)
m−j+1,m

Y
(i)
m−ℓ,m

)γr+ρ+δ

.

Let us prove the relation

√
snT

(1)
n

d→ θ1
Λγ(γr1)√

ℓ
Z(γr1) + θ2

Λγ(γr2)√
ℓ

Z(γr2), n → ∞. (6.27)

By applying (6.19) we get

T (1)
n

d
=

1

s

s∑
i=1

1

ℓ

ℓ∑
j=1

θ1

(
γhγr1

(
Y

(i)
j

)
− Λγ (γr1)

)
+ θ2

(
γhγr2

(
Y

(i)
j

)
− Λγ (γr2)

)
.

(6.28)
The summands over i in the right hand side of (6.28) present a sequence of the i.i.d. zero

mean r.v.s. Moreover, under assumptions γr1 < 1/2 and γr2 < 1/2 we have

E

(
1

ℓ

ℓ∑
j=1

θ1

(
γhγr1

(
Y

(1)
j

)
− Λγ (γr1)

)
+ θ2

(
γhγr2

(
Y

(1)
j

)
− Λγ (γr2)

))2

=
1

ℓ

(
θ21Λ

2
γ(γr1)σ

2(γr1, γr1) + 2θ1θ2Λγ(γr1)Λγ(γr2)σ
2(γr1, γr2)

+θ22Λ
2
γ(γr2)σ

2(γr2, γr2)
)
.

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



120 N. M. MARKOVICH, M. VAIČIULIS

Thus, the relation (6.27) follows by applying the Lindeberg-Lévy central limit theorem
(see, e.g., pg. 16 in [22]).

We claim that under assumption γr < 1/2,
√
snT

(2)
n (r)

p→ µ · ν(ℓ, ρ, r), n → ∞, (6.29)

and, assuming additionally, 0 < δ < 1− γr − ρ,

√
snT

(3)
n (r)

p→ |µ|
1− γr − ρ− δ

, n → ∞. (6.30)

The relations (6.29), (6.30), together with (6.27) prove (6.21).
Let us prove that (6.29) holds. By the definition of the function A, there exists t1, such

that the function A1(t) has a constant sign when t ≥ t1. Let us recall that |A1| ∈ RVρ holds.
By applying the Drees inequalities for regularly varying functions (see, e.g., Prop. B.1.10
in [13]) we get that for any ε̃ > 0 and δ̃ > 0 there exists t2 = t2(ϵ̃, δ̃), such that for t ≥ t1 ∨ t2
and tx ≥ t1 ∨ t2, it holds

−ε̃max
{
xρ+δ̃, xρ−δ̃

}
≤ A1(tx)

A1(t)
− xρ ≤ ε̃max

{
xρ+δ̃, xρ−δ̃

}
.

We apply the latter inequality with t replaced by m and x replaced by Y
(i)
m−ℓ,m/m. Put

T (4)
n (ρ, r) =

1

s

s∑
i=1

(
Y

(i)
m−ℓ,m

m

)ρ

Υi(r).

As in the proof of Theorem 2.1, we get

−ε̃A1(m)max
{
T (4)
n (ρ− δ̃, r), T (4)

n (ρ+ δ̃, r)
}
+ A1(m)T (4)

n (ρ, r) ≤ T (2)
n (r),(6.31)

ε̃A1(m)max
{
T (4)
n (ρ− δ̃, r), T (4)

n (ρ+ δ̃, r)
}
+ A1(m)T (4)

n (ρ, r) ≥ T (2)
n (r),(6.32)

when A1(m) > 0 for m ≥ t1 ∨ t2 and the reverse inequalities to (6.31), (6.32) hold in the
case A1(m) < 0 for m ≥ t1 ∨ t2.

Since assumption (2.3) implies
√
sA1(m) → µ as n → ∞, the relation (6.29) will be

proved if we show that for any ρ̃ ∈ R satisfying 2ρ̃ < ℓ+ 1,

T (4)
n (ρ̃, r)

p→ ν(ℓ, ρ̃, r), n → ∞. (6.33)

As for relation (6.33), it follows from

T (4)
n (ρ̃, r)− E

(Y
(1)
m−ℓ,m

m

)ρ̃

Υ1(r)

 p→ 0, (6.34)

E

(Y
(1)
m−ℓ,m

m

)ρ̃

Υ1(r)

 → ν(ℓ, ρ̃, r), n → ∞. (6.35)

We will prove

E

T (4)
n (ρ̃, r)− E

(Y
(1)
m−ℓ,m

m

)ρ̃

Υ1(r)

2

→ 0, n → ∞, (6.36)

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



ASYMPTOTIC PROPERTIES OF THE BLOCK-TYPE STATISTICS 121

which, in turn, will complete the proof of (6.34), see Section 8.3 in [2]. By using the Renyi’s
representation one can get that for any 1 ≤ i ≤ s, Y (i)

m−ℓ,m and Υi(r) are independent r.v.s.
Thus, the left hand side of (6.36) is equal to

1

s

E

(
Y

(1)
m−ℓ,m

m

)2ρ̃

E (Υ1(r))
2 − E2

(
Y

(1)
m−ℓ,m

m

)ρ̃

E2 (Υ1(r))


≤ 2

s
E

(
Y

(1)
m−ℓ,m

m

)2ρ̃

E (Υ1(r))
2 .

By our assumptions s = sn → ∞ holds as n → ∞. This, together with

E
(
Y

(1)
m−ℓ,m/m

)2ρ̃
→ Γ(ℓ− 2ρ̃+ 1)/Γ(ℓ+ 1) as n → ∞ (see Lemma 6.1) and

E (Υ1(r))
2 < ∞ yields (6.36). To prove that Υ1 is square integrable r.v. it is enough

to use (6.19) and then to apply a direct integration.
It rests to prove (6.35). Since Y

(1)
m−ℓ,m and Υ1(r) are independent r.v.s we get that the left

hand side of (6.35) is equal to E
(
Y

(1)
m−ℓ,m/m

)ρ̃
E (Υ1(r)). Now (6.35) follows by applying

Lemma 6.1 and (6.19) one more time.
The proof of (6.30) is similar to the proof of (6.29) and thus, it is omitted. This completes

the proof of Theorem 2.2.

Proof of Remark 1.
With the notation of the previous proof, we have T

(1)
n /A(m) =

√
sT

(1)
n / (

√
sA(m)), where

we take θ1 = 1 and θ2 = 0 in the definition of T
(1)
n . By combining (2.5) and (6.27)

we get T
(1)
n /A(m)

p→ 0 as n → ∞. From the proof of Theorem 2.2 it follows that
T

(2)
n (r1)

p→ ν(ℓ, ρ, r1), n → ∞ and the sequence T
(3)
n (r1), n = 1, 2, . . . is bounded in

probability. Thus, keeping in mind inequalities (6.25), (6.26), the relation (2.6) follows. This
completes the proof.

Proof of Theorem 3.1.
Let r ̸= 0. We have

γ̂(2)
n (s, ℓ, r)− γ =

(1− γr) (Qn(s, ℓ, r)− Λγ(γr))− (Qn(s, ℓ, 0)− Λγ(0))

rQn(s, ℓ, r)
. (6.37)

By Theorem 2.2, the numerator of the right hand side of (6.37), multiplied by
√
sn,

converges in distribution to

N
(

µγr

(1− ρ)(1− γr − ρ)
· Γ(1 + ℓ− ρ)

Γ(1 + ℓ)
,

2γ4r2

ℓ(1− γr)(1− 2γr

)
.

As the denominator, it tends in probability to rΛγ(γr) as n → ∞. Now (3.7) follows by
applying the Slutsky’s lemma (see, e.g., pg. 11 in [22]).

Let r = 0. The similar argument used in proving (2.4) shows that

√
s
(
Qn(s, ℓ, 0)− γ, Q̃(s, ℓ)− 2γ2

)
d→ µΓ(1 + ℓ− ρ)

(1− ρ)Γ(1 + ℓ)

(
1,

2γ(2− ρ)

1− ρ

)
+ (Π1,Π2)

holds as n → ∞. Here, µ is the same as in (2.3), while (Π1,Π2) is a Gaussian random vector
with zero means and EΠ2

1 = γ2/ℓ, EΠ2
2 = 20γ4/ℓ, E (Π1Π2) = 4γ3/ℓ. Now, by using the

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



122 N. M. MARKOVICH, M. VAIČIULIS

identity

γ̂(2)
n − γ =

(
Q̃(s, ℓ)− 2γ2

)
− 2γ (Qn(s, ℓ, 0)− γ)

2Qn(s, ℓ, 0)
,

it is not difficult to verify that (3.7) holds with r = 0. This ends the proof.

Proof of Corollary 2.3.
This follows by a similar argument as used in proving Theorem 3.1.

Proof of the inequality (3.9).
Note that (χ̃(ρ))(1−2ρ)/(−2ρ) can be rewritten as follows

(χ̃(ρ))(1−2ρ)/(−2ρ) =

(
(χ(ρ) + ρ)2

2

)1−1/ρ(
1

2
+

χ(ρ)

2(1− ρ)

)
,

where χ(ρ) = (ρ2 − 4ρ+ 2)
1/2. It is easy to check that χ(ρ) >

√
2− ρ and, consequently,

χ(ρ) > 1− ρ hold for any ρ < 0. Thus, (3.9) follows.

ACKNOWLEDGEMENTS

The reported study was funded by the Russian Science Foundation RSF, project number
22-21-00177 (recipient N.M. Markovich, writing-review and editing).

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Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)


	Introduction
	Main results
	A new family of the block-type estimators for positive EVI
	Applications
	Conclusions
	Proofs