Minimax estimation of a bivariate cumulative distribution function
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Minimax estimation of a bivariate cumulative distribution function Rafał Połoczanski ´ 1 · Maciej Wilczynski ´ 1 Received: 19 July 2018 © The Author(s) 2019
Abstract The problem of estimating a bivariate cumulative distribution function F under the weighted squared error loss and the weighted Cramer–von Mises loss is considered. No restrictions are imposed on the unknown function F. Estimators, which are minimax among procedures being affine transformation of the bivariate empirical distribution function, are found. Then it is proved that these procedures are minimax among all decision rules. The result for the weighted squared error loss is generalized to the case where F is assumed to be a continuous cumulative distribution function. Extensions to higher dimensions are briefly discussed. Keywords Minimax estimation · Cumulative distribution function · Loss function
1 Introduction Minimax estimation of a one dimensional cumulative distribution function (c.d.f.) was initiated in 1955 by Aggarwal (1955) and has been extensively studied since then (see the references given in the next paragraph). To the best of our knowledge, extensions of this approach to higher dimensions have not been investigated. In this paper we therefore consider estimating a bivariate c.d.f. and we generalize to this case some known results concerning minimax estimation of a univariate c.d.f. We also briefly discuss a multivariate generalization of these results. Minimax estimation of a univariate c.d.f. F was considered by many authors. Using an invariance structure relative to the group of continuous and strictly increasing transformations, Aggarwal (1955) found the best invariant estimator of a continuous c.d.f.
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Maciej Wilczy´nski [email protected] Rafał Połocza´nski [email protected]
1
Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, 50-370 Wrocław, Poland
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R. Połoczanski, ´ M. Wilczynski ´
= F under the invariant loss L(F, F)
R
r d F, r ≥ 1. Here F stands for an |F − F|
estimate of F, based on a sample from F.Ferguson (1967, pages 191–197) generalized 2 h(F) d F, where h(·) is a contin = (F − F) this result to the case that L(F, F) R
uous and positive function. He also asked whether the best invariant estimates are minimax among the larger class of (not necessarily invariant) procedures. Yu (1992b) established the minimaxity of the best invariant procedure in Ferguson’s setup. In particular he found the minimax estimator of a continuous c.d.f. F under the loss of 2 F −δ (1 − F)−γ d F, where δ, γ ∈ {0, 1} are fixed = (F − F) the form L(F, F) R
numbers. Analog minimaxity findings were obtained by Mohammadi and van Zwet (2002) (entropy loss), Ning and Xie (2007) (Linex loss), and St¸epie´n–Baran Stepien (2010) (strictly convex loss). Phadia and Yu (1992) proved minimaxity of the empir ical distribution function under the Kolmogorov–Smirnov loss supt∈R |F(t) − F(t)|. Jafari Jozani et al. (2014) considered the problem of estimating a continuous d
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