This paper studies the inference problem of index coefficient in single-index models under massive dataset. Analysis of massive dataset is challenging owing to formidable computational costs or memory requirements. A natural method is the averaging divide-and-conquer approach, which splits data into several blocks, obtains the estimators for each block and then aggregates the estimators via averaging. However, there is a restriction on the number of blocks. To overcome this limitation, this paper proposed a computationally efficient method, which only requires an initial estimator and then successively refines the estimator via multiple rounds of aggregations. The proposed estimator achieves the optimal convergence rate without any restriction on the number of blocks. We present both theoretical analysis and experiments to explore the property of the proposed method.

References

• Chen, X., Liu, W., & Zhang, Y. (2019). Quantile regression under memory constraint. The Annals of Statistics, 47(6), 3244–3273. https://doi.org/10.1214/18-AOS1777 
• Chen, X. Y., & Xie, M. G. (2014). A split-and-conquer approach for analysis of extraordinarily large data. Statistica Sinica, 24(4), 1655–1684. https://doi.org/10.5705/ss.2013.088 
• Christou, E., & Akritas, M. (2016). Single index quantile regression for heteroscedastic data. Journal of Multivariate Analysis, 150, 169–182. https://doi.org/10.1016/j.jmva.2016.05.010 
• Delecroix, M., Hristache, M., & Patilea, V. (2006). On semiparametric M-estimation in single-index regression. Journal of Statistical Planning and Inference, 136(3), 730–769. https://doi.org/10.1016/j.jspi.2004.09.006 
• Ichimura, H. (1993). Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. Journal of Econometrics, 58(1-2), 71–120. https://doi.org/10.1016/0304-4076(93)90114-K 
• Jiang, R., Guo, M. F., & Liu, X. (2020). Composite quasi-likelihood for single-index models with massive datasets. Communications in Statistics – Simulation and Computation, 51(9), 5024–5040. https://doi.org/10.1080/03610918.2020.1753074 
• Jiang, R., Qian, W. M., & Zhou, Z. G. (2016). Weighted composite quantile regression for single-index models. Journal of Multivariate Analysis, 148, 34–48. https://doi.org/10.1016/j.jmva.2016.02.015 
• Jiang, R., Zhou, Z. G., Qian, W. M., & Chen, Y. (2013). Two step composite quantile regression for single-index models. Computational Statistics & Data Analysis, 64, 180–191. https://doi.org/10.1016/j.csda.2013.03.014 
• Jordan, M., Lee, J., & Yang, Y. (2019). Communication-efficient distributed statistical learning. Journal of the American Statistical Association, 14(526), 668–681. https://doi.org/10.1080/01621459.2018.1429274
• Lin, N., & Xi, R. (2011). Aggregated estimating equation estimation. Statistics and Its Interface, 4(1), 73–83. https://doi.org/10.4310/SII.2011.v4.n1.a8 
• Schifano, E., Wu, J., Wang, C., Yan, J., & Chen, M. H. (2016). Online updating of statistical inference in the big data setting. Technometrics, 58(3), 393–403. https://doi.org/10.1080/00401706.2016.1142900 
• Tang, Y., Wang, H., & Liang, H. (2018). Composite estimation for single-index models with responses subject to detection limits. Scandinavian Journal of Statistics, 45(3), 444–464. https://doi.org/10.1111/sjos.v45.3 
• Wang, J. L., Xue, L., Zhu, L., & Chong, Y. (2010). Estimation for a partial-linear single-index model. The Annals of Statistics, 38(1), 246–274. https://doi.org/10.1214/09-AOS712
• Wu, T., Yu, K., & Yu, Y. (2010). Single-index quantile regression. Journal of Multivariate Analysis, 101(7), 1607–1621. https://doi.org/10.1016/j.jmva.2010.02.003 
• Xia, Y., Tong, H., Li, W., & Zhu, L. X. (2002). An adaptive estimation of dimension reduction space. Journal of the Royal Statistical Society Series B, 64(3), 363–410. https://doi.org/10.1111/rssb.2002.64.issue-3 
• Zhu, L., & Xue, L. (2006). Empirical likelihood confidence regions in a partially linear single-index model. Journal of the Royal Statistical Society: Series B, 68(3), 549–570. https://doi.org/10.1111/rssb.2006.68.issue-3

To cite this article: Rong Jiang & Yexun Peng (2023) A short note on fitting a singleindex model with massive data, Statistical Theory and Related Fields, 7:1, 49-60, DOI: 10.1080/24754269.2022.2135807 To link to this article: https://doi.org/10.1080/24754269.2022.2135807