Sankhya: The Indian Journal of Statistics

1995, Volume 57, Series A, Pt. 1, pp. 79--87

NORMAL MIXTURE REPRESENTATIONS OF MATRIX VARIATE ELLIPTICALLY CONTOURED DISTRIBUTIONS

By

 WENCESLAO GONZÁLEZ MANTEIGA, Universidad de Santiago de Compostela

SUMMARY.  Let (X, Y) be a (p+1)-dimensional random vector satisfying the model $Y=\bold {A}^t(\bold {X})\beta_0+\varepsilon$, where A is q-dimensional valued functional and $\epsilon$ is a random error, independent of X, of mean equal to zero, and variance $\sigma_2$. This article proves the strong consistency and asymptotic normality of the family of estimators of $\beta_0$ obtained by minimizing functionals of the form:
                                    $\psi_s(\beta) = \int (\hat{\alpha}_n(\bold x)- (\bold A^{t}(\bold x) \beta)^2 \hat{\bold f}_{n} (\bold x) dx$ where $\hat{\bold }_{n}(\bold x)$ and $\hat{\alpha}_n(\bold x)$ are non-parametric estimators of f(x) (the density of x) and of $\alpha(x)=E(Y | \bold X=\bold x) (the regression of Y on X) respectively. Both $\hat{\bold f}_{n}$ and $\hat{\alpha}_n$ have sample-determined bandwidths.

AMS (1991) subject classification.  Primary 62J05, 62G07.

Key words and phrases. Linear regression, non-parametric estimation, sample-determined non-parametric estimation.

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This article in mathematical reviews.