Summary
Let f(·) be a strictly positive density function defined on (a,b)\( \subseteq \) R 1 with a continuous derivative f′(·) and let \(F(x) = \int\limits_a^x {f(t)} dt\),-∞≦a<x<+∞ be the corresponding distribution function. Define the quantile function Q of F by Q(y)=F −1(y)=inf{x:F(x)≧y}, 0<y<1, the score function (-1)J of the density function f by J(y)=f′(Q(y))/f(Q(y)), and the Fisher information I(f) of f by \(I(f) = \int\limits_0^1 {(J(y))^2 } dy\), assumed to be finite. Given some regularity conditions on F, we propose a sequence of nearest neighbour (N.N.) type estimators J n for J and prove that for all ε∈(0,1/5) there exists an estimator J n,ε of J such that for all δ∈(0,(5ε/18)∧(ε/12+1/40)) we have
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References
Beran, R.: Asymptotically efficient adaptive rank estimates in location models. Ann. Stat. 2, 63–74 (1974)
Csörgő, M., Révész, P.: Strong approximation of the quantile process. Ann. Statist. 6, 882–894 (1978)
Csörgő, M., Révész, P.: Strong Approximations in Probability and Statistics. New York: Academic Press 1981
Csörgő, M., Révész, P.: An invariance principle for N.N. empirical density functions. In: Gnedenko, B.V., Puri, M.L., Vincze, I. (eds.). Coll. Math. Soc. J. Bolyai, 32. Non-parametric Statistical Inference, pp. 151–170. Amsterdam: North-Holland 1982
Hájek, J., Šidák, Z.: Theory of Rank Tests. New York: Academic Press 1967
Huber, P.J.: Robust Statistics. New York: Wiley 1981
Parzen, E.: Nonparametric statistical data modeling. J. Am. Stat. Assoc. 74, 105–131 (1979)
Stone, C.J.: Adaptive maximum likelihood estimators of a location parameter. Ann. Statist. 3, 267–284 (1975)
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This research was supported by a NSERC Canada Grant at Carleton University, Ottawa
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Csörgo, M., Révész, P. A nearest neighbour-estimator for the score function. Probab. Th. Rel. Fields 71, 293–305 (1986). https://doi.org/10.1007/BF00332313
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DOI: https://doi.org/10.1007/BF00332313