Multidimensional strong large deviation results
A strong large deviation theorem in the spirit of \textit{R. R. Bahadur} and \textit{R. Ranga Rao} [Ann. Math. Stat. 31, 1015--1027 (1960; Zbl 0101.12603)] is established for sequences \((\mathbf{Z}_n)\) of random vectors generalizing a similar theorem for the unidimensional case [\textit{C. Joutard}, Math. Methods Stat. 22, No. 2, 155--164 (2013; Zbl 1275.60031)]. Building on methods of proof by \textit{N. R. Chaganty} and \textit{J. Sethuraman} [Ann. Probab. 21, No. 3, 1671--1690 (1993; Zbl 0786.60026)], it is shown that under assumptions on the normalized cumulant generating functions (c.g.f.), the probability \(\mathbb{P}(\mathbf{Z}_n \geq \mathbf{a})\) can be approximated apart from a factor \((1+o(1))\) by an expression depending on the differentiable limit of the normalized c.g.f. and the \(n\)th element of the normalizing sequence. The theorem is applied to vectors of independent sampling variances, Gaussian multiple linear regression and the multivariate Nadaraya-Watson estimator. For the first two applications, the approximation is numerically compared to the true probabilities.
Gaussian multiple linear regression model
scientific article; zbMATH DE number 6814688
scientific article