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Goodness-of-fit procedures for copula models based on the probability integral transformation. (English) Zbl 1124.62028
I.i.d. copies of a \(d\)-variate random vector \(X=(X_1,\dots,X_d)\) are observed with CDF \(H\) which has the copula representation \[ H(x_1,\dots,x_d)=C(F_1(x),\dots,F_d(x_d)). \] Goodness-of-fit tests are considered for the hypothesis \(H_0:\;C=C_\vartheta\), \(\vartheta\in O\in R^m\). The tests are based on comparison of \(K(\vartheta,t)=\mathbf{P}\{H(X)<t\}\) with the empirical counterpart \(K_n(t)\). The asymptotic behaviour of the empirical process \[ K_{n,\vartheta}(t)=\sqrt{n}(K_n(t)-K(\vartheta,t)) \] is investigated. The proposed test statistics are \[ S_n=\int_0^1(K_{n,\vartheta_n}(t))^2k(\vartheta_n,t)dt,\quad T_n=\sup_{0\leq t\leq 1}| K_n(t)|, \] where \(k\) is the density of \(K(\vartheta,t)\), and \(\vartheta_n\) is an estimate for \(\vartheta\). Two bootstrap procedures are proposed to implement the test. Archimedean, bivariate extreme-value, Fréchet and bivariate Farlie-Gumbel-Morgenstern copulae are considered as examples.

MSC:
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
62G30 Order statistics; empirical distribution functions
62H05 Characterization and structure theory for multivariate probability distributions; copulas
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