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Phase transitions in nonparametric regressions

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  • Zhu, Ying

Abstract

When the unknown regression function of a single variable is known to have derivatives up to the (γ+1)th order bounded in absolute values by a common constant everywhere or a.e. (i.e., (γ+1)th degree of smoothness), the minimax optimal rate of the mean integrated squared error (MISE) is stated as 1n2γ+22γ+3 in the literature. This paper shows that: (i) if n≤γ+12γ+3, the minimax optimal MISE rate is lognnlog(logn) and the optimal degree of smoothness to exploit is roughly maxlogn2loglogn,1; (ii) if n>γ+12γ+3, the minimax optimal MISE rate is 1n2γ+22γ+3 and the optimal degree of smoothness to exploit is γ+1.

Suggested Citation

  • Zhu, Ying, 2025. "Phase transitions in nonparametric regressions," Journal of Econometrics, Elsevier, vol. 252(PB).
    • Handle: RePEc:eee:econom:v:252:y:2025:i:pb:s0304407623003561
    DOI: 10.1016/j.jeconom.2023.105640
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    References listed on IDEAS

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