We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance $\Delta $; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value $f(t_i)$ of a regression function $f$ at a grid point $t_i$ (nonparametric GLM). When $f$ is in a H\"{o}lder ball with exponent $\beta >\frac 12 ,$ we establish global asymptotic equivalence to observations of a signal $\Gamma (f(t))$ in Gaussian white noise, where $\Gamma $ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process.

翻译：我们证明了一个非高斯非参数回归模型在渐近意义上等价于具有高斯噪声的回归模型。该近似是在Le Cam缺陷距离$\Delta$的意义上成立的；因此，对于所有有界损失统计决策问题，这两个模型是渐近等价的。我们的结果涉及一系列独立但非同分布的观测序列，其中每个分布属于同一实指数族。规范参数是回归函数$f$在网格点$t_i$处的值$f(t_i)$（非参数GLM）。当$f$位于指数为$\beta >\frac 12$的H\"{o}lder球中时，我们建立了该模型与高斯白噪声中观测信号$\Gamma (f(t))$的全局渐近等价性，其中$\Gamma$与指数族中的方差稳定变换相关。该结果是最近建立的独立同分布模型高斯近似在回归问题上的类比。证明基于部分和过程匈牙利构造的函数版本。