If $\mu$ is a positive measure on $\mathbb{R}^n$ with Laplace transform $L_{\mu}$ , we show that there exists a positive measure $\mu$ on $\mathbb{R}^n$ such that det $L_{\mu}^'' = L_{\nu}$. We deduce various
corollaries from this result and, in particular, we obtain the Rao-Blackwell estimator of the determinant of the variance of a natural exponential family on $\mathbb{R}^n$ based on $(n + 1)$ observations. A new proof and extensions of Lindsay's results on the determinants of moment matrices are also given. Finally we give a characterization of the Gaussian law in $\mathbb{R}^n$.
@article{1032298297,
author = {Kokonendji, C\'elestin C. and Seshadri, V.},
title = {On the determinant of the second derivative of a Laplace transform},
journal = {Ann. Statist.},
volume = {24},
number = {6},
year = {1996},
pages = { 1813-1827},
language = {en},
url = {http://dml.mathdoc.fr/item/1032298297}
}
Kokonendji, Célestin C.; Seshadri, V. On the determinant of the second derivative of a Laplace transform. Ann. Statist., Tome 24 (1996) no. 6, pp. 1813-1827. http://gdmltest.u-ga.fr/item/1032298297/