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**Asymptotic nonequivalence of nonparametric experiments when the smoothness index is \(1/2\).**
*(English)*
Zbl 0932.62061

There have recently been several papers demonstrating the global asymptotic equivalence of certain nonparametric problems. See especially L. D. Brown and 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 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 \(\alpha>1/2\).

S. Efromovich and 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 \(\alpha=1/2\).

S. Efromovich and 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 \(\alpha=1/2\).

### MSC:

62G20 | Asymptotic properties of nonparametric inference |

62G08 | Nonparametric regression and quantile regression |

62M05 | Markov processes: estimation; hidden Markov models |

62G07 | Density estimation |

### References:

[1] | Brown, L. D. and Low, M. G. (1996). Asy mptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384-2398. · Zbl 0867.62022 · doi:10.1214/aos/1032181159 |

[2] | Brown, L. D. and Zhang, C.-H. (1996). Coupling inequalities for some random design matrices and asy mptotic equivalence of nonparametric regression and white noise. |

[3] | Efromovich, S. and Samarov, A. (1996). Asy mptotic equivalence of nonparametric regression and white noise has its limits. Statist. Probab. Lett. 28 143-145. · Zbl 0849.62023 · doi:10.1016/0167-7152(95)00109-3 |

[4] | Nussbaum, M. (1996). Asy mptotic equivalence of density estimation and white noise. Ann. Statist. 24 2399-2430. · Zbl 0867.62035 · doi:10.1214/aos/1032181160 |

[5] | Robbins, H. (1951). Asy mptotically subminimax solutions of compound statistical decision problems. Proc. Second Berkeley Sy mp. Math. Statist. Probab. 1 131-148. Univ. California Press, Berkeley. · Zbl 0044.14803 |

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