Estimating transformation function

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.