Supplementary Material for “ Correlation Curve Estimation for Multiplicative Distortion Measurement Errors Data ”

1.School of Economics and the Wang Yanan Institute for Studies in Economics, Xiamen University, Xiamen, China. 2. School of Statistics and Mathematics, Central University of Finance and Economics, Beijing, 100081, China. 3. Department of Biostatistics, School of Public Health, University of Texas at Houston, Houston, TX, 77225, USA. 4. College of Mathematics and Statistics, Institute of Statistical Sciences, Shenzhen University, Shenzhen-Hong Kong Joint Research Center for Applied Statistical Sciences, Shenzhen University, Shenzhen, 518060, China.

1. PROOFS OF PROPOSITIONS AND THEOREM 1
Proof. Lemma 1 can be immediately proven using the result obtained in Mack and Silverman (1982), see also in Lemma 7.1 of Fan and Huang (2005).

Proof of Propositions 1-2
In the following, Step 1-Step 3 give the detailed proof of asymptotic expression ofĝ(x) − g (x).
Using Lemma 1, we have Using the condition (C3), we have Using (A.2), Lemma 1 and the U-statistic's property (Serfling;1980) we have , then we have Similar to (A.5)-(A.6), the third term in (A.7) is of order O P (h 4 1 ). The U-statistic method in Serfling (1980) entails that the second term of (A.7) is of order O P (n −1/2 h 2 1 ). The second order of the projection of U-statistic entails that the first term is ) .
Using asymptotic expression (A.70) and the equation , we complete the proof of Theorem 1.