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Convergence rate of the distributions of normalized maximum likelihood estimators for irregular parametric families. (Russian, English) Zbl 1032.62012
Sib. Mat. Zh. 44, No. 3, 521-541 (2003); translation in Sib. Math. J. 44, No. 3, 411-427 (2003).
Let $$X_1,\dots, X_n$$ be a sample of independent random vectors with common density $$f(x,\theta)$$ in $$\mathbb R^d$$ depending on an unknown parameter $$\theta\in\Theta\subset\mathbb R^m$$. Suppose that this density is continuous in $$x=(x_1,\dots,x_d)$$ except for the points of some set $$K_\theta$$. Suppose also that $$K_\theta$$ is a smooth manifold of dimension $$d-1$$ for all $$\theta\in\Theta$$, and the sets $$\Omega_{\theta}^1$$ and $$\Omega_{\theta}^2$$ are defined so that, first, their combination with $$K_\theta$$ generates a partition of $$\mathbb R^d$$ and, second, the density $$f(x,\theta)$$ has at points $$y\in K_\theta$$ discontinuities of the first kind along directions specified by the sets $$\Omega_{\theta}^1$$ and $$\Omega_{\theta}^2$$. Let $$\theta_0$$ be the true value of the parameter and let $Y_n(u)= \sum _{i\leq n}\ln f(X_i,\theta_0+u/n)f^{-1}(X_i,\theta_0)$ be the logarithmic likelihood ratio process. In the authors’ article, Sib. Math. Zh. 42, No. 2, 275-288 (2001; Zbl 1001.62010), the asymptotic behavior of the process $$Y_n(u)$$ was studied and some asymptotic expansion for $$Y_n(u)$$ was given. The paper under review continues these studies. Now the authors give the convergence rate of the distributions of normalized maximum likelihood estimators.
##### MSC:
 62E20 Asymptotic distribution theory in statistics 62F12 Asymptotic properties of parametric estimators 60F05 Central limit and other weak theorems
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