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Xn, w) is called an M- estimator for P if Also, en = en(Xl,···Xn,w) will be called a sequence of weak (resp. strong) approximate M-estimators for p and P if limn--+oo[PnP(-' en) - infeE8 PnP(', e)] = 0 in outer probability (resp. d. (P). ) M-estimators may not exist, but under mild measurability conditions, approximate M-estimators always exist and can be chosen by measurable selection so that en is universally measurable in (Xl,'" ,Xn ), see Appendix A. Let d be a metric on 8 and let P be a collection of laws (probability measures) on (X,A).

Dudley (Xl + ... + Xn)jn which is strongly consistent for any law P in the class P of all laws P with Plxl < 00, and 80 (P) = Px. If p(x) = lxi, an M-estimator always exists and is a sample median, called a spatial median for k > 1. It is unique if Xl,· .. ,Xn are not all in any line (Haldane, 1948) or if they are and have a unique sample median there, specifically if n is odd. Huber defined further functions, where for each r > 0, Pr(x) = Ixl 2 for Ixl ::::: r and 2rlxl - r2 for Ixl ~ r. Let po(x) := Ixl.

Let po(x) := Ixl. Huber (1981, pp. 43-55) treats Pr for k = 1. For Pn M-estimators are more often unique than medians for k = 1, namely if the interval of sample medians has length::::: 2r. Also, Ppr < 00 if and only if Plxl < 00. (III) Stochastic programming. Here P reduces to a single law P. One wants to find 8 = 80 for which Pp(·, 8) is minimized. g. Shapiro (1989). Facts on stochastic programming can also yield facts on estimation if the hypotheses hold for all P in some class. For some P and p, Pp(·,8) = +00 for all 8.

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A Bayesian Alternative to Parametric Hypothesis Testing by Rueda R.

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