Performance Evaluation and Stochastic Optimization with Gradually Changing Non-Stationary Data

We propose estimators and analyze their large-sample distribution for two problems, data-oriented expected performance evaluation and data-oriented stochastic optimization, in presence of a specific class of non-stationarities in the observed data. The considered non-stationarities reflect nonparametric gradual distribution changes in a sequence of independently generated data, which can be viewed as relaxing the ``identically distribution" part in the independent and identically distributed (iid) data generation assumption. Such non-stationarities can pose challenges to standard estimators that would, in the i.i.d. settings, enjoy reliable estimator performance and provable large-sample behavior. In presence of the gradually changing non-stationarities, we demonstrate the effectiveness of a class of weighted estimators in terms of mean square error and show that the optimal weighted estimators intuitively assign relatively more weights to the more recently collected data and abandon data that were collected long ago. We prove central limit theorems for the proposed weighted estimators under one plausible non-stationary limiting regime, so that asymptotically valid confidence intervals can be constructed. We then prove central limit theorems for a weighted sample average approximation approach that is designed to solve data-oriented stochastic optimization problems under gradually changing non-stationarity