Nonparametric estimation of the ratios of derivatives of a multivariate distribution density from dependent observations. (English. Russian original) Zbl 0955.62039
Sib. Math. J. 41, No. 2, 229-245 (2000); translation from Sib. Mat. Zh. 41, No. 2, 284-303 (2000).
Authors’ summary: The authors study the properties of the nonparametric estimators of the ratios of derivatives of a distribution density of a random sequence \(\{\varepsilon_n\}\) which is consistent with a nondecreasing net of \(\sigma\)-algebras \(\{{\mathcal F}_n\}\). It is assumed that the variables \(\varepsilon_n\) are identically distributed and are observed with an additive noise \(g_{\lambda,n-1}\) which is consistent with \(\{{\mathcal F}_{n-1}\}\). Here \(\lambda\in{\mathcal A}\) is an unknown nuisance vector parameter, and \({\mathcal A}\) is a set of admissible values of \(\lambda\).
The principal term of the asymptotic mean square deviation of estimators is found with an improved convergence rate which coincides with the principal term of the asymptotic mean square deviation of estimators in the case of independent observations when \(g_{\lambda,n}\equiv 0\). The authors establish convergence with probability 1, uniform in \({\mathcal A}\), asymptotic normality, and convergence in the metric \(L_m\), \(m\geq 2\), of the estimators of density derivatives and their ratios. These results are applied to estimation of the ratios of derivatives of the noise distribution density for linear stochastic regression processes with unknown parameters.
The principal term of the asymptotic mean square deviation of estimators is found with an improved convergence rate which coincides with the principal term of the asymptotic mean square deviation of estimators in the case of independent observations when \(g_{\lambda,n}\equiv 0\). The authors establish convergence with probability 1, uniform in \({\mathcal A}\), asymptotic normality, and convergence in the metric \(L_m\), \(m\geq 2\), of the estimators of density derivatives and their ratios. These results are applied to estimation of the ratios of derivatives of the noise distribution density for linear stochastic regression processes with unknown parameters.
Reviewer: D.A.Korshunov (Novosibirsk)
MSC:
| 62G07 | Density estimation |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62G05 | Nonparametric estimation |
| 62F12 | Asymptotic properties of parametric estimators |
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