1. Introduction
The existence of asymmetric dependence, both within and between extreme returns in bivariate scenarios across diverse market conditions, is not only a crucial factor in asset and risk management but also a primary focus of market supervision. In times of financial crises, there is a noticeable amplification of cross-sectional co-movements in the (lower) tails of return distributions within financial markets [
1,
2,
3]. This amplification accentuates the likelihood of simultaneous extreme events. In devising investment strategies, this phenomenon should be considered through timely and appropriate asset reallocations, such as capitalizing on arbitrage trading opportunities, and making judicious adjustments to hedging decisions [
2].
Alternatively, in adverse market conditions, risk managers and market supervisors may find it necessary to establish larger capital buffer requirements if the inclination towards joint occurrences of extreme losses increases during market distress. Standard linear dependence measures prove inadequate in such cases, necessitating the exploration of alternative statistical models, such as the Gaussian copula, which serves as a convenient tool for modeling dependence near the mean of multivariate distributions [
1]. However, it is essential to recognize that the Gaussian copula lacks the capability to measure dependence at the tails, underscoring the need to explore alternative methodologies in extreme situations [
4,
5].
A tail copula is a function derived from complete tail dependence, and the use of empirical tail copulas provides flexibility, mitigating potential risks associated with parametric misspecification. This approach stands in contrast to established methods that solely estimate and compare scalar summary measures of extreme dependence, like the tail dependence coefficient. Quantifying the degree of (tail) non-exchangeable dependence [
6] poses a significant challenge in insurance and risk management literature. Tail-dependence coefficients often underestimate this degree and fail to capture non-exchangeable tail dependence as they assess the limiting tail probability solely along the diagonals [
5].
This paper focuses on quantifying the degree of tail non-exchangeability for a bivariate random vector with identical marginal distributions. The concept of tail non-exchangeability in this context relies on limiting properties of bivariate copulas. The paper introduces a meaningful measure to quantify the strength of tail non-exchangeability, providing details for constructing non-exchangeable bivariate copula families based on commonly-used approaches. Various non-exchangeable copulas have been explored in the literature, such as the Marshall-Olkin copula, the generalized Clayton copula [
7], and copulas constructed through comonotonic latent variables [
8]. However, this paper specifically considers Khoudraji’s device [
9] for generating non-exchangeable copulas, leaving the exploration of other methods, such as using the non-exchangeable Pickands function for extreme value copulas, for future studies.
For a two-dimensional random vector
with its continuous marginal distributions
for
, the dependence is characterized by the copula
, (i.e., the distribution function of
. In extreme value analysis, a key focus is on assessing the level of dependence at the extremes. This involves measuring the inclination of variables
and
to simultaneously exhibit extreme (either large or small) values. Hua and Joe (2011) [
10] propose a method to characterize the lower tail dependence of a pair of random variables, denoted as
. They introduce the concept of tail order represented by
, where
ranges from 0 to 1, indicating various levels of dependence. Additionally, they establish a condition for the tail dependent parameter
to ensure the the following condition:
where
is slowly varying function with
as
, for functions
. Therefore,
. For
, the tail-dependence parameter
becomes the tail dependence coefficient [
11,
12]. While widely used, the tail dependence coefficient tends to underestimate the extent of tail dependence. This is due to its focus on measuring the rate of decline of joint tail probability exclusively along the main diagonal of
[
5]. Furthermore, the tail dependence coefficient falls short in capturing non-exchangeable tail dependence situations, where
holds true, and
represents the copula of the random variables
[
1,
6]. Furman (2015) [
7] confronts these challenges by proposing adjustments to the tail dependence coefficient, opting to substitute the diagonal with the path that optimizes joint tail probability. Yet, the estimation of these tail indices proves to be notably challenging, primarily due to the intricate nature of determining the path that maximizes dependence for a copula
within a broad context. A parallel endeavor to quantify non-exchangeable tail dependence was undertaken by Genest and Jaworski (2021) [
13].
One reasonable approach involves examining the disparity in certain conditional quantities when transitioning between
and
. Without loss of generality, assuming identical nonnegative random variables
and
, we leverage the asymptotic behavior of
as
t approaches infinity to investigate the robustness of tail non-exchangeability. In the work by Hua et al. (2014) [
14], the expressions
and
are employed to analyze the intensity of tail dependence as
t tends to infinity. Furthermore, Bernard et al. (2015) [
15] utilize conditional quantiles to assess the strength of tail dependence.
Notations
In this section, we elucidate the notation and symbols employed throughout this paper. We define distribution functions as
, survival functions as
, and density functions as
, assuming their existence wherever utilized. Furthermore, we introduce the concept of the survival copula, denoted as
, derived from an ordinary copula. To facilitate discussion, we use
as the copula pre-transformation, following the non-exchangeable approach proposed by Khoudraji (1996) [
9]. Post-transformation, the survival copula is denoted as
. As the survival copula itself is a copula function, our focus in this paper is primarily on discussing the corresponding survival copulas directly. To compute the first-order derivative, we introduce
. For any given
,
serves as a univariate cumulative distribution function (cdf). When addressing tail behavior and calculating conditional expectations through Laplace approximation, we require the second-order derivative of our survival copula. Hence, we define
.
4. Test of Tail Non-Exchangeability
In preceding sections, our analytical exploration has delved into the interplay between univariate marginals and tail non-exchangeability, particularly concerning the Type I and II expectation ratios. Notably, we discovered the utility of this ratio in instances featuring Pareto marginal distributions, thus directing our focus in this section. When endeavoring to construct a statistical test for tail non-exchangeability, the Type I ratio emerges as the more apt choice. This is attributed to the comparative ease in estimating the conditional tail expectation of the form within the framework of the Type I ratio.
Assuming we have observed pairs
drawn from the joint distribution of
and
, a statistical test for bivariate tail non-exchangeability can be formulated when the marginal distributions adhere to a Pareto density. This involves constructing a test based on the estimator for
, wherein the empirical version of this estimator is
where
,
, and
indicating the cardinality of the set
for
. Under the null hypothesis of tail non-exchangeability, the estimator
is expected to approximate 1 for large values of
t. This characteristic forms the foundation for an approximate statistical test. Given that the numerator and denominator of
involve essential sample means, establishing asymptotic normality becomes straightforward.
Proposition 5. Let and for with unbounded support on for fixed . If and are exchangeable thenwhere , andsuch that, , , and . To apply Proposition 5 in constructing a statistical test, one can set a fixed value for
t and compare
with a normal rejection region. Here,
represents an estimation of
, derived either from its empirical form or an alternative method, such as the bootstrap. In the case of a bootstrap-based estimator, both parametric and nonparametric bootstrap techniques can be readily adjusted [
6].
For the parametric bootstrap, resampling from the joint distribution under the assumption of exchangeability can be accomplished by initially simulating
using a fitted copula. Subsequently, transformations
and
for
are applied [
6]. Here,
denotes the estimated common marginal distribution based on the combined observed data from each margin.
For the nonparametric bootstrap, generating a resample under the assumption of exchangeability involves simulating
for
and
. In this expression,
is a simulated observation from the empirical distribution of
,
is a simulated observation from the empirical distribution of
, and
Q is a Bernoulli random variable with a success probability of
, independent of
and
[
6]. It is important to note that
is also independent of
.
Given the emphasis on tail exchangeability, it is reasonable to choose a large value for t. Nevertheless, the selection of t should be carefully weighed against the availability of data beyond t, as this factor influences the standard error of . The reliance on a specific value of t in the methodology is unattractive due to the potential for test results to vary based on this choice. As an alternative approach, we propose aggregating the test over a range of t values and using the maximum difference as the test statistic.
In particular, let
and
, where
, represent two estimated marginal quantiles determined by observed data. Compute
for
, where
forms a grid of values. The test statistic for assessing the null hypothesis of exchangeability is then expressed as
The rejection region for this test statistic can be calculated using the bootstrap method.
The effectiveness of the weak convergence, as outlined in Proposition 5, may necessitate considerably large sample sizes. This stems from the fact that the effective sample size diminishes, given that only data in the upper tail is utilized in computing . Moreover, the nature of being a ratio implies that skewness may persist in the sampling distribution until sample sizes reach substantial magnitudes. These observations are validated by a small-scale empirical study conducted on a computer.
In this study, we generated 1000 samples of sizes , 500, 750, and 1000 from an exchangeable joint distribution with Pareto marginals (shape parameter = 3, scale parameter = 1), exponential marginals (rate = 2), and Weibull marginals (shape parameter = 2). We computed for each simulated sample, where t was chosen to be the 0.75 quantile of the marginal distribution. The results of this investigation underscore the need for ample sample sizes to observe the intended convergence.
Figure 5 and
Figure 6 present histograms and normal quantile plots corresponding to each sample size. These visual representations not only illustrate the occurrence of convergence to a normal distribution but also underscore the relatively gradual pace of this convergence. These findings reinforce the notion of resorting to the bootstrap method in practical scenarios, emphasizing its utility in situations where convergence is not swiftly achieved.
5. Real Data Analysis
To illustrate the practical application of this test, we leverage the dataset from Cook et al. (1986) [
20], which encompasses the observed log-concentration values of seven chemical elements in 655 water samples collected near Grand Junction, Colorado. Our focus centers on exploring the joint distributions of Uranium and Lithium, as well as Uranium and Titanium. The original data sets are visually depicted in
Figure 7 and
Figure 8. Recognizing that the original data likely do not share identical marginal distributions, and aiming for demonstrative clarity, we undertake three distinct transformations for each dataset. Initially, we employ a joint rank transformation on the data, scaling the results to the unit interval. This preprocessing step sets the stage for a nuanced analysis of the interplay between Uranium and the selected chemical elements. That is, we computed (
) for
where
where
is the rank of
relative to
for
and
. To facilitate a meaningful comparison, we proceeded by transforming the previously scaled ranks to achieve identical marginal distributions through the probability integral transformation. In this endeavor, we explored three distinct marginal models: a Pareto distribution characterized by a shape parameter of 5, an exponential distribution with a rate set at 2, and a Weibull distribution featuring a shape parameter of 2. The outcomes of these transformations are visually depicted in
Figure 7 and
Figure 8, showcasing the resulting joint distributions for the transformed data.
The assessment of tail exchangeability was conducted for each case through the bootstrap methodology detailed earlier. The outcomes of these calculations are visually presented in
Figure 9,
Figure 10,
Figure 11,
Figure 12,
Figure 13 and
Figure 14. In the left panel of each figure, the values of
are depicted for a range of
t values corresponding to the interval between the 75th and 95th percentiles of the observed marginal distribution of the combined data. The grey lines represent the
values computed across 500 nonparametric bootstrap resamples.
Moving to the right panel in each figure, a histogram displays the distribution of test statistic
values computed for each of the 500 resamples. If present, a vertical line indicates the observed value of
from the sample. The bootstrap-generated
p-values, signaling the test of the null hypothesis of Type I exchangeability, are detailed in
Table 1. From the estimated
p-values, a clear pattern emerges: there is substantial evidence opposing exchangeability for the joint distributions of Uranium and Titanium, whereas there is limited evidence against exchangeability for the joint distributions of Uranium and Lithium.
Given the results outlined in the sections above, it may seem surprising at first to observe notable outcomes in the exponential and Weibull scenarios. However, this outcome appears to be linked to the nature of the test statistic . The emphasis of on the relative ranks of the two observed marginal distributions, rather than the gaps between the actual values, explains this unexpected pattern. This inclination is evident in the consistent behavior observed in the curves of across different marginal distributions. The impact of relative ranks, rather than absolute values, seems to contribute to the observed significance in these specific cases.
6. Discussion
Throughout this manuscript, our primary focus is on quantifying the extent of tail non-exchangeability. We commence by introducing metrics specifically crafted to measure the strength of tail non-exchangeability, leveraging conditional tail expectations. Subsequent to this, we present theoretical outcomes for non-exchangeable bivariate copulas generated using Khoudraji’s device, in conjunction with three distinct types of univariate marginals. Our findings underscore the heightened significance of tail non-exchangeability when Pareto marginals are chosen. Consequently, we advocate for the transformation of each marginal distribution to adhere to a Pareto distribution. In the pursuit of detecting tail non-exchangeability, we put forward a graphical tool grounded in theoretical insights, accompanied by a statistical test. We have used the R package CopulaModel to perform the copula related simulations and boot to do the empirical studies.
The proposed method in this study introduces a test statistic derived from the ratio of conditional tail expectations to effectively identify instances of tail non-exchangeability. It acknowledges the challenge of establishing limiting properties empirically, especially when concentrating on extreme values that involve fewer data points, leading to a potential loss of information from the entire dataset.
To mitigate this limitation, the study adopts a strategy involving a series of upper sets of data when testing non-exchangeability. The asymptotic normality of the proposed test statistic has been rigorously demonstrated under mild conditions as the sample size tends towards infinity. For practical applications with unknown variability, the paper suggests the use of bootstrap methods.
Focusing specifically on non-exchangeability in the context of positive dependence, the methodology can be extended to bivariate copulas exhibiting tail negative dependence. However, it is worth noting that the approach is tailored for bivariate cases. To apply it to cases with tail negative dependence, one can easily transform one of the marginals to ensure positive dependence, facilitating the direct application of the proposed method. Addressing tail non-exchangeability in multivariate scenarios requires employing the approach for all pairwise bivariate marginals. Nevertheless, it is crucial to highlight that pairwise exchangeability does not necessarily imply mutual exchangeability. As a result, further research is warranted to explore and understand the implications of mutual exchangeability in multivariate cases.
This form of tail non-exchangeability holds significance in the realm of time series analysis, particularly when assessing how two random variables evolve over time. Such occurrences are evident when examining the exchangeability of market shares between two distinct companies operating in the same industry. If these shares exhibit exchangeability, it becomes possible to predict future share prices for one company based on the knowledge of the other. Looking ahead, we can explore the presence of tail exchangeability among different soccer positions across various teams [
21]. For instance, our method can be applied to test the exchangeability of goal dynamics between two strikers from different clubs in the European Football League. If these dynamics prove to be exchangeable, it implies that a club can seamlessly replace one striker with another once the first striker’s contract expires. A similar analysis can be conducted for different batting positions in cricket matches involving various teams.
In the context of infectious disease modeling, the widely used susceptibility-infection-recovery (SIR) framework comes into play [
22,
23]. Our method allows for testing exchangeability by examining different SIR datasets between two regions. If exchangeability is established, it suggests that a specific vaccination strategy could lead to recovery from infectious diseases in those regions. This versatile approach extends its applicability to diverse fields, showcasing its potential for extracting meaningful insights from various types of data.
These tests of non-exchangeability serve a crucial role during financial upheavals like the Great Depression (1930–1937), the Oil crisis (1968–1970 to 1972–1978), Black Monday (1987), and the series of crises from Asia to the millennium leading up to the Dot-Com crisis (1995–2003). It is widely acknowledged that during crises, losses tend to escalate dramatically, leading to a heightened correlation among extreme losses and, consequently, an uptick in the co-movement of extreme gains. This phenomenon serves as a form of compensation for investors who face substantial downside risks across various sectors [
1]. In simpler terms, when significant losses occur more frequently, one can anticipate a corresponding increase in significant gains happening simultaneously. However, in contrast, the 2007–2009 financial crisis stands out due to a temporary spike in tail asymmetries, contrasting with the overall declining trend in tail asymmetries since the mid-1990s. Some may argue that only losses exhibiting tail dependence were impacted, while the tail dependence between gains remained unaffected [
1]. This highlights the particularly devastating nature of the subprime crisis, as investors did not encounter as much potential for extreme upside gains. In the literature of econometrics and time series, the main focus of risk management is on Value-at-Risk (VaR), and other measures designed to estimate the probability of large losses, leading to a demand for flexible models of the dependence between sources of risk [
24]. Therefore, the non-exchangeable structures might exist.
In clinical research, when censoring occurs due to competing risks or patient withdrawal, there’s always a concern regarding the accuracy of treatment effect estimates derived under the assumption of independent censoring [
25]. Since identifying dependent censoring requires additional information, the most we can do is conduct a sensitivity analysis to evaluate how parameter estimates change under varying assumptions regarding the relationship between failure and censoring. Such an analysis proves particularly valuable when insights into this relationship are available through literature reviews or expert opinions [
25]. In regression analysis, the repercussions of mistakenly assuming independent censoring on parameter estimates are unclear. Neither the direction nor the extent of potential bias can be easily anticipated. It is assumed that the joint distribution of failure and censoring times is a function of their distributions, with this function represented as a copula function. Under this assumption, one can examine dependencies at the tails to assess the non-exchangeability.