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

Song, Peter Xue-Kun

Scand. J. Stat. 27, No. 2, 305-320 (2000).

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.

Reviewer: R.E.Maiboroda (Kyïv)

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)

Keywords:

dispersion models; Gaussian copula; asymptotic normality; logit models; probit models; Poisson distribution; general linear model