Semiparametric jump-preserving estimation for single-index models

ABSTRACT

Estimation of the single-index model with a discontinuous unknown link function is considered in this paper. Existed refined minimum average variance estimation (rMAVE) method can estimate the single-index parameter and unknown link function simultaneously by minimising the average pointwise conditional variance, where the conditional variance can be estimated using the local linear fit method with centred kernel function. When there are jumps in the link function, big biases around jumps can appear. For this reason, we embed the jump-preserving technique in the rMAVE method, then propose an adaptive jump-preserving estimation procedure for the single-index model. Concretely speaking, the conditional variance is obtained by the one among local linear fits with centred, left-sided and right-sided kernel functions who has minimum weighted residual mean squares. The resulting estimators can preserve the jumps well and also give smooth estimates of the continuity parts. Asymptotic properties are established under some mild conditions. Simulations and real data analysis show the proposed method works well.

KEYWORDS:

• Single-index model
• jump-preserving technique
• minimum average variance estimation
• local linear kernel estimation

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is supported by National Natural Science Foundation of China (grant nos. 11471252 and 11571073), and Jiangsu Province Postdoctoral Research Funding Scheme (grant no. 1501021C).