Abstract
In this paper, we propose an estimator for $g(x)$ under the model $Y_{i}=g(Z_{i}),\ i=1,2,...,n$ where $Z_{i},\ i=1,2,...$ are random variables with known distribution but unknown observed values, $Y_{i},\ i=1,2,...$ are observed data and $g(x)$ is an unknown strictly monotonically increasing function (we call $g(x)$ transformation function). We prove the almost sure convergence of the estimator and construct confidence intervals and bands when $Z_{i},i=1,2,...$ are i.i.d data based on their asymptotic distribution. Corresponding case when $Z_{i}$ being linear process is handled by resampling method. We also design the hypothesis test regarding whether $g(x)$ equals an expected transformation function or not. The finite sample performance is evaluated by applying the method to simulated data and an urban waste water treatment plant’s dataset.
Citation
Yunyi Zhang. Jiazheng Liu. Zexin Pan. Dimitris N. Politis. "Estimating transformation function." Electron. J. Statist. 13 (2) 3095 - 3119, 2019. https://doi.org/10.1214/19-EJS1603