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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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