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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.
62E20 Asymptotic distribution theory in statistics
62F12 Asymptotic properties of parametric estimators
60F05 Central limit and other weak theorems
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