Geodesic
A lower bound for risks of non-parametrical estimates of density in the uniform metrics
Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 4, pp. 824-828

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Let $W^{(\beta)}(L,[a,b])$ be the class of functions satisfying (3) for $x_i\in[a,b]$, $\beta=r+\alpha$. Estimators $\hat{f}_n$ for which the sequence (4) is uniformly (in $f\in W^{(\beta)}(L,[a,b])$) bounded in probability were constructed in [11], [12]. It is proved in this paper that sequence (4) does not tend to zero in probability for any other estimator. More precisely, inequality (5) is proved for an arbitrary strictly increasing function $l\colon R^1\to R^1$. 
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     author = {R. Z. Has'minskiǐ},
     title = {A lower bound for risks of non-parametrical estimates of density in the uniform metrics},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {824--828},
     publisher = {mathdoc},
     volume = {23},
     number = {4},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1978_23_4_a10/}
}
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R. Z. Has'minskiǐ. A lower bound for risks of non-parametrical estimates of density in the uniform metrics. Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 4, pp. 824-828. http://geodesic.mathdoc.fr/item/TVP_1978_23_4_a10/