A lower bound for risks of non-parametrical estimates of density in the uniform metrics

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 $l\colon R^1\to R^1$.