Asymptotic nonequivalence of nonparametric experiments when the smoothness index is \(1/2\)
There have recently been several papers demonstrating the global asymptotic equivalence of certain nonparametric problems. See especially \textit{L. D. Brown} and \textit{M. G. Low} [ibid. 24, No. 6, 2384-2398 (1996; Zbl 0867.62022)], who established global asymptotic equivalence of the usual white-noise-with-drift problem to the nonparametric regression problem, and \textit{M. Nussbaum} [ibi., 2399-2430 (1996; Zbl 0867.62035)], who established global asymptotic equivalence to the nonparametric density problem. In both these instances the results were established under a smoothness assumption on the unknown nonparametric drift, regression or density function. In both cases such functions were assumed to have smoothness coefficient \(\alpha>1/2\), for example, to satisfy \[ \bigl|f(x)-f(y)\bigr|\leq M|x-y|^\alpha \tag{1} \] for all \((x,y)\) in their domain of definition. This note contains an example which shows that such a condition is necessary, in the sense that global asymptotic equivalence may fail between any pairs of the above three nonparametric experiments when (1) fails in a manner that the nonparametric family of unknown functions contains functions satisfying (1) with \(\alpha=1/2\) but not with any \(\alpha>1/2\). \textit{S. Efromovich} and \textit{A. Samarov} [Stat. Probab. Lett. 28, No. 2, 143-145 (1996; Zbl 0849.62023)] have already shown that asymptotic equivalence of nonparametric regression and white noise may fail when \(\alpha< 1/4\) in (1). The present counterexample to equivalence is somewhat different from theirs and carries the boundary value \(\alpha=1/2\).
scientific article; zbMATH DE number 1359508