1. Introduction
Extreme Value Theory (EVT) is dedicated to modeling extreme events within a sequence of a large number of independent and identically distributed (i.i.d.) random variables. Its applications are diverse, spanning fields such as finance, insurance, environmental science, and engineering [
8,
14]. Let
be a sequence of i.i.d. random variables with common distribution function (d.f.)
F, and denote by
the sample maxima. The risk
is called to be in the max-domain attraction of
G, if there exist some normalization constants
and a non-degenerate d.f.
G such that (with
convergence in distribution)
The limit distribution
G is the so-called generalized extreme value distribution (GEV), which is of the sample
l-type (namely,
x can be replaced with
for some
) as
We denote this by
. Here the three parameters
are called the shape, location, and scale parameters, respectively. In addition, the tail behavior of the potential risk
X is well classified into Fréchet, Weibull, and Gumbel domains, corresponding to the cases with
, respectively [
3].
Given the wide applications of EVT, many extensive studies of limit theory alike Eq.(
1) have been conducted. [
16] extended first the limit distribution under linear normalization in Eq.(
1) to the power limit laws
, i.e., there exist some power normalization constants
and a non-degenerate d.f.
H such that
with the sign function
equal 1,
and 0 for
x being positive, negative and zero, respectively. It is well-known that
H is of
p-max stable distributions composed of six types of limits, which can be rewritten uniformly as below [
15]. For some
and
(recall
G is the GEV defined in Eq.(
2)),
In what follows, we denote this by
.
Recently, [
5] and [
12] investigated the limit behavior of extremes under linear and power normalization in the scenario of competing risks, with the practical consideration of aggregating multiple sources. Namely, the studied sample maxima
is actually obtained from
k heterogeneous subsamples
from source/population
. This considerate modeling in the big data era is desirable due to the complexity of real applications [
6,
24]. The limit theory of
obtained for the
k multiple sources is the so-called limit theory of max of max since
Clearly, the obtained limit laws of Eq.(
5) extending the classical extreme value theory given in Eqs.(
1) and (
3) are the so-called accelerated
l-max stable and accelerated
p-max stable distributions, see Theorem 2.1 [
5] and Theorem 2.1 [
12]. Note that the key condition in determining accelerated limit theory is the interplay of the sample length and the tail behavior among the multiple competing risks. A natural question is how the extreme law varies in the uncertainty of the sample size involved. This is very common in environmental and financial fields, for instance, the extreme claim size of
claims over a
n-day period and the extreme daily precipitation within a
duration of wet period [
13,
21]. This paper aims to study the limit theory under both linear and power normalization in the framework of competing risks with random sample size.
Many authors refined the extreme limit theory under linear normalization with random sample size for two different cases:
Case I) with independent random sample size. The basic risks
and sample size index
are supposed to be independent and
converges weakly to a non-degenerate distribution function [
9];
Case II) with non-independent random sample size. There exists a positive-valued variable
V such that
converges to
V in probability, allowing the interrelation of the basic risk and sample size index
[
19].
The limit theorems with random sample size were further extended for sample minima [
7], extreme order statistics under power normalization [
2,
18], stationary Gaussian process [
22], stationary chi-process [
23], and recent contributions on multivariate extreme behavior [
11]. This paper will further consider the limit behavior of extremes (both minima and maxima) under linear and power normalization in the competing risk scenario, extending those accelerated
l-max and
p-max stable limit distributions when the sample size sequence
satisfies conditions indicated in Cases I) and II). The theoretical results will be illustrated by numerical studies with typical examples such as
are time-shifted Poisson, (negative) Binomial distributions, which have extensive applications in insurance and hydrology [
19,
20].
The remainder of the paper is organized as follows.
Section 2 presents the main results for maxima of maxima under both linear and power normalization with sample sizes. Extensional results for competing minima and typical examples are discussed in
Section 3. Numerical studies are conducted to illustrate our theoretical findings in
Section 4. The proofs of all theoretical results are deferred to the Appendix.
4. Numerical Studies
We will conduct a Monte Carlo simulation to illustrate Theorems 1 and 2 with
m-shifted random sample size given in Examples 2∼1. In what follows, we take the shift parameter
in all time-shifted random sample size distributions, and the basic risks
from Pareto distributions with parameters
1 and the random sample sizes
are supposed to be mutually independent. In addition, the repeated time is taken as
. We will illustrate our main results specified in Theorems 1 with the three examples given in
Section 3.2 above.
1. Comparison of Pareto competing extremes with determinant sample size and Poisson distributed random sample size.
In
Figure 1, we will demonstrate that the competing extremes with Poisson distributed sample size are similar to the case with nonrandom sample size case. Let
follow
m-shifted Poisson with mean parameters
. We then generate competing Pareto extremes with basic risks following Pareto
with
It follows from Theorems 1, 2 and Example 1 together with Example 4.6 by [
12] that (recall
the Fréchet distribution)
- (1)
For
or
with
, we have
- (2)
For
with
, we have
Noting that the power normalized extremes will behave similarly to the linear normalized ones up to a power transformation. We show only the behavior of linear normalization for the numerical studies below.
In
Figure 1, we take
and
with
to show the above two cases. Overall, the competing Pareto extremes are well fitted by the accelerated GEV distribution for the non-randomized sample size, where the latter is slightly better than the randomized sample size cases. Further, the accelerated GEV approximation (
Figure 1 (a, c)) is relatively closer to the empirical competing extremes than the dominated case.
2. Comparison of Pareto competing extremes with Geometric distributed and negative Binomial distributed sample size. We consider the max of maxima
with both basic risks
with random sample size
following
m-shifted negative Binomial distribution with probability
and
. It follows by Example 4.6 by [
12], Theorem 1(a, b) and Example 3 that, with
- (1)
For or with , we have
- (2)
For with , we have
Thus, its density function is given by
In
Figure 2, we set
with
in (a, c) and (b, d), respectively. Meanwhile, the random sample size follows a 5-shifted negative Binomial distribution with
in (a, b) (namely geometric distribution), and
in (c, d), and successful probability
. Meanwhile, we take
in the Pareto basic risks. Consequently, the sub-maxima are completely competing when
, resulting in the accelerated mixed extreme limit distributions as shown in
Figure 2 (a, c). In contrast, the dominated limit behavior is given in
Figure 2 (b, d) as
.
In general, our theoretical density curve given by Eq.(
16) approximates the histogram very well (
Figure 2). Further, we see that the approximation with geometric distributed random size is slightly better than the negative binomial case. In addition, the approximation for the dominated case (
Figure 2 (d)) is slightly better than the accelerated case when negative Binomial random size applies.
Author Contributions
Conceptualization, L.B., K.H. and C.L.; Methodology, K.H. and C.L.; software, L.B.; validation, L.B., Z.T. and C.L.; formal analysis, L.B., K.H. and C.L.; investigation, K.H.; Writing, original draft preparation, C.L. and K.H.; Writing, review and editing, C.W., Z.T. and C.L.; Visualization, C.L.; Supervision, C.L.; Project administration, C.L.; Funding acquisition, C.L. All authors have read and agreed to the published version of the manuscript.