Consideration of tail dependence is a very important part of risk analysis in many applied sciences that is measured in order to estimate the risk of simultaneous extreme events. Usually the tail dependence coeficient is the measurement in question. Pearson correlation coeficient unfortunately is not a suitable measure for estimating dependencies between two quantities in the context of simultaneous occurrence of extreme events when these events are of interest for the researcher because it takes extreme events into account with the same weight as it takes “normal” events although dependence of extreme values may slightly difer. Present work emphasizes the importance of taking into account tail dependencies in bivariate statistical analysis using copulas. Due to increasing frequency of environmental cataclysms the issue of analyzing risks (e.g. economic losses) and their consequences comes to the fore. Moreover, researchers should take into consideration consequences of their joint occurrence. Three non-parametric estimators of tail dependence coeficients were compared in order to estimate correlation between daily cumulative rainfall totals recorded in central European part of Russia. The majority of existing estimators depends on threshold and thus there is a trade-of between variance and bias during the calculation of the best value for . For balancing an algorithm is presented that is based on using moving average filter and then searching the “stable” part of tail dependence coeficient. Estimate of tail dependence coeficient is assumed to be equal to mean value on the “stable” part.
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In: K. E. Samouylov, L. A. Sevastianov, D. S. Kulyabov (eds.): Selected Papers of the VIII Conference “Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems”, Moscow, Russia, 20-Apr-2018, published at http://ceur-ws.org
One of the most important parts of multivariate extreme value analysis is the study of
extremal dependencies [
Using the copula function [
1. Introduction = lim →1− 1 − 2 + (, ) 1 −
. 2.
Main section
(1)
(2)
(3)
(4)
(5)
of bivariate random vector (1, 2). Using their joint distribution function [
)︁ log ^ 1 − , 1 −
)︁ log (1 − ) , , 1 6 < . 1 6 < . where ^ denotes empirical copula.
Note that both estimators depend on choice of threshold and thereafter ℎ order
statistic [
Another estimator for upper tail dependence coeficient is suggested in works [
1 1())︁ ,^2(︁ 2())︁ 2 ⎪⎪⎦ ⎭ ⎫⎤ ⎪ ⎪ ⎬⎥⎥ . estimator.
Main advantage of this equation is that ^ doesn’t depend on . However, copula
(1, 2) must be well approximated with extreme-value copulas for correctness of the
algorithm for finding this “stable” part is presented in paper [
It follows from equations (3), (4) that estimators depend on choice of threshold
which is determined by balancing variance and bias for estimator according to stability
theorem for [
+− 1 ∑︁ =+1 = | − | 6 2, =1 1 ∑︁ +− 1. takes the form of where is standard deviation of ^1, . . . , ^− 2 then the final expression for Example of algorithm realisation using R language is presented below:
If the condition is not satisfied after sequential searching, then = 0. lambda_sec <
− lambda_sec2 <−
2 − ( 1 / k ∗ rank_sum ( msk_rank , spb_rank , l e n g t h ( df$Msk ) , k ) ) ( 1 / k ∗ rank_sum2 ( msk_rank ,
mhzsk_rank , l e n g t h ( df$Msk ) , k ) ) f o r ( i i n 1 : l g t h ) { rank_sum <− f u n c t i o n ( rank1 , rank2 , l g t h , k ) { a = 0 rank_sum2 <− a = 0 ; f o r ( i i n
1 : l g t h ) { a = a + i f e l s e ( ( rank1 [ i ] > l g t h − & ( rank2 [ i ] > l g t h − k ) , 1 , 0 ) } r e t u r n ( a ) }
f u n c t i o n ( rank1 , rank2 , l g t h , k ) { } r e t u r n ( a ) } a = a + i f e l s e ( ( rank1 [ i ] > l g t h −
| ( rank2 [ i ] > l g t h − k ) , 1 , 0 ) b = t r u n c m = t r u n c ( s q r t ( l e n g t h ( df$Msk − 2∗b ) ) ) f o r ( i i n 1 : ( l e n g t h ( ls_ma_na)− 2∗b+m− 1)){ k ) k ) rw < −
ls_ma_na [ i : ( i+m) ] f o r ( l i n
1 :m) sssuum i f ( sd ( rw)>= sssuum /m) {wow <
i < −
sssuum + abs ( rw [ l ]− rw [
0} − mean ( rw )
In this study the precipitation data of the All-Russian Research Institute of Hydrometeorological Information
— the World Data Center of the Russian Federation is used,
which contains data on daily precipitation in 11 cities of the European part of Russia [
Implementation of this algorithm is presented in Fig. 1 [
^ (left) and ^ (right). Black line corresponds to ^(); blue smooth line is ^() after applying moving average filter to it. Pink transparent plateau is the resulting value for ^ where placement of plateu corresponds to indexes of vector ^, . . . , ^+− 1 from the algorithm above.
All three estimators (3), (4), (5) for upper tail dependence coeficient were calculated
for 55 pairs of 11 cities under study [
Finally the attempt to find the correlation between estimators for upper tail
dependence and the distance between cities under study was made [
It is obvious that there must be an inverse relation between the distance ℎ and dependence estimators. Therefore ^ is a bad estimator for the problem under study. Three other estimators (^, ^ and PCC) show roughly the same with some correction, which is why they are considered to be more trustworthy. So it is proposed to take the average of ^ and ^ or just ^ as the resulting estimator for . 3.
Conclusions
This paper highlights the importance of taking into account the tail dependence
coeficient in the context of multivariate frequency analysis using copulas. The three
following nonparametric estimators(^ , ^, ^ ) have been compared. The aim
of this comparison was to choose the best estimator in the context of our application [
Most non-parametric estimators have to deal with the choice of the number of order statistics to be considered in the production of an estimate. This is not an easy task since it requires a trade-of between variance and bias (small values of cause large variance and large values of increase the bias).
Frahm et al. [