## Multivariate dispersion models generated from Gaussian copula.(English)Zbl 0955.62054

Let $$H$$ be an $$m$$-dimensional Gaussian CDF with covariance $$\Gamma$$ and standardized margins $$H_i=\Phi$$. The CDF $C_H(u_1,\dots,u_m)=H(H^{-1}_1(u_1),\dots,H^{-1}_m(u_m)),\quad u_i\in[0,1],$ is called a Gaussian copula. Let $$F_i$$ be one-dimensional dispersion model (DM) CDFs, i.e. they have PDFs of the form $f(y_i;\mu_i,\sigma_i^2)=a(y_i,\sigma_i^2)\exp(-d(y_i;\mu_i)/(2\sigma_i^2)),$ where $$a$$ and $$d$$ are some fixed functions, and $$\sigma=(\sigma_i)_{i=1,\dots,m}$$, $$\mu=(\mu_i)_{i=1,\dots,m}$$ are parameters. Then the distribution $$C_H(F_1(x_1),\dots,F_m(x_m))$$ is called a multivariate dispersion model MDM$$(\mu,\sigma,\Gamma)$$. (Note that the margins of MDM are one-dimensional DMs $$F_i$$).
The author considers logit, probit, Poisson and Gamma MDM. Some asymptotic formulas for these models are obtained, e.g., asymptotic normality of normalised MDM when the covariance matrix tends to zero. Applications to longitudinal data and simulation results are presented.

### MSC:

 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62F12 Asymptotic properties of parametric estimators 62H12 Estimation in multivariate analysis 62J12 Generalized linear models (logistic models)
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