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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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