Abstract
This study aims to model the joint probability distribution of periodic hydrologic data using meta-elliptical copulas. Monthly precipitation data from a gauging station (410120) in Texas, US, was used to illustrate parameter estimation and goodness-of-fit for univariate drought distributions using chi-square test, Kolmogorov–Smirnov test, Cramer-von Mises statistic, Anderson-Darling statistic, modified weighted Watson statistic, and Liao and Shimokawa statistic. Pearson’s classical correlation coefficient r n , Spearman’s ρ n, Kendall’s τ, Chi-Plots, and K-Plots were employed to assess the dependence of drought variables. Several meta-elliptical copulas and Gumbel-Hougaard, Ali-Mikhail-Haq, Frank and Clayton copulas were tested to determine the best-fit copula. Based on the root mean square error and the Akaike information criterion, meta-Gaussian and t copulas gave a better fit. A bootstrap version based on Rosenblatt’s transformation was employed to test the goodness-of-fit for meta-Gaussian and t copulas. It was found that none of meta-Gaussian and t copulas considered could be rejected at the given significance level. The meta-Gaussian copula was employed to model the dependence, and these results were found satisfactory.
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Acknowledgments
This work was financially supported by the National Science Council, Republic of China (Grant No. NSC-50579065 and NSC-50879070). The authors appreciate Dr. Mehmet Ozger for supplying data.
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Appendices
Appendix 1: p-dimensional symmetric elliptical type distributions
A p-dimensional random vector Z is said to have an elliptically contoured distribution (or simply called elliptical distribution or ECD) with parameters μ(p × 1) and ∑(p × p) if it has the stochastic representation (Fang and Fang 2002):
where r ≥ 0 is a random variable, u is uniformly distributed on the unit sphere in R p and is independent of r, A is a p × p constant matrix such that AA T = ∑ and the sign \( \mathop = \limits^{d} \) means that both sides of the equality have the same distribution. In particular, if r has a density, then the density of Z is of the form
where g(·) is a scale function uniquely determined by the distribution of r and is referred to as the density generator. Common p-dimensional symmetric elliptical type distributions are given in Table 1.We use the notation Z ~ EC p (μ, ∑, g). Without loss of generality, we shall consider only the case Z ~ EC p (0, R, g), where
All the marginal distributions of Z have an identical pdf:
and cdf
Note q g (z) = q g (−z) and Q g (z) = 1 − Q g (−z) for z > 0 (Kotz and Nadarajah 2001; Nadarajah and Kotz 2005; Nadarajah 2006). Let X = (X 1, X 1,…,X p )T be a random vector with each component X i having a given continuous pdf f i (x i ) and cdf F i (x i ). Let the random vector Z = (Z 1, Z 2,…, Z p )T ~ EC p (0, R, g). Suppose
where Q −1 g is the inverse of Q g . Then the density function of X is given by
where ϕ is the p-variant density weight function:
The n-dimensional random vector X is said to have a meta-elliptical distribution, if its density function is given by Eq. 12. Denote X ~ ME p (0, R, g; F 1, F 2,…,F p ). The function \( \phi \{ {\text{Q}}_{g}^{ - 1} [F_{1} (x_{1} ),{\text{Q}}_{g}^{ - 1} ]F_{2} (x_{2} ), \ldots ,Q_{g}^{ - 1} F_{p} (x_{p} )\} \) is referred to as the density weighting function (Fang et al. 2002).
Appendix 2: Bivariate meta-Gaussian distribution
A meta-Gaussian distribution with marginal distribution F 1(z 1) and F 2(z2)
where u = F 1(z1), v = F 2(z2); Φ−1(·) is inverse of the normal distribution. Its pdf is
The marginal pdf and cdf of Z are
2.1 Bivariate meta-t v distribution
A meta-t v distribution with marginal distribution F 1(z 1) and F 2(z2)
where u = F 1(z1), v = F 2(z2); \( {\text{t}}_{\upsilon }^{ - 1} \left( \cdot \right) \) is the inverse function for t distribution with υ degrees of freedom. Its pdf is
The marginal pdf and cdf of Z are:
2.2 Bivariate meta-Cauchy distribution
A meta-Cauchy distribution with marginal distribution F 1(z 1) and F 2(z2)
where u = F 1(z1), v = F 2(z2); \( \omega^{ - 1} \left( \cdot \right) \) is inverse function for Cauchy distribution.
The marginal pdf and cdf of Z are
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Song, S., Singh, V.P. Meta-elliptical copulas for drought frequency analysis of periodic hydrologic data. Stoch Environ Res Risk Assess 24, 425–444 (2010). https://doi.org/10.1007/s00477-009-0331-1
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DOI: https://doi.org/10.1007/s00477-009-0331-1