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We work out a version of the van Trees inequality in a Hajek--Le Cam spirit, i.e., under minimal assumptions that, in particular, involve no direct pointwise regularity assumptions on densities but rather almost-everywhere differentiability in quadratic mean of the model.
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[author] van Trees, H. L.H. L. (1968). Detection, Estimation and Modulation Theory. Wiley & Sons
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[author] Gill, R.R. and Levit, B.B. (1995). Applications of the van Trees inequality: a Bayesian Cramér-Rao bound. Bernoulli 1 59–79
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[author] van der Vaart, A. W.A. W. (1998). Asymptotic Statistics. Cambridge Unversity Press
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[author] Lenstra, A. J.A. J. (2005). Cramér-Rao revisited. Bernoulli 11 263–282
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[author] Pollard, D.D. (2001; 2005). Asymptotia (book in progress), chapter on Hellinger differentiability. Lecture notes from the Paris 2001 statistics semester at IHP, with a final edit in 2005; available at http://www.stat.yale.edu/~pollard/Courses/607.spring05/handouts/DQM.pdf
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[author] Jupp, P. E.P. E. (2010). A van Trees inequality for estimators on manifolds. Journal of Multivariate Analysis 101 1814–1825
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[author] Gassiat, E.E., Rousseau, J.J. and Vernet, E.E. (2018). Efficient semiparametric estimation and model selection for multidimensional mixtures. Electronic Journal of Statistics 12 703–740
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2022
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