An adaptive lack of fit test for big data

New technological advancements combined with powerful computer hardware and high-speed network make big data available. The massive sample size of big data introduces unique computational challenges on scalability and storage of statistical methods. In this paper, we focus on the lack of fit test of parametric regression models under the framework of big data. We develop a computationally feasible testing approach via integrating the divide-and-conquer algorithm into a powerful nonparametric test statistic. Our theory results show that under mild conditions, the asymptotic null distribution of the proposed test is standard normal. Furthermore, the proposed test benefits from the use of data-driven bandwidth procedure and thus possesses certain adaptive property. Simulation studies show that the proposed method has satisfactory performances, and it is illustrated with an analysis of an airline data.