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The research was supported by the National Natural Science Foundation of China (Nos. 12271046 and 12201306)

the National Statistical Science Research Project of China (No. 2024LY088)

the Priority Academic Program Development of Jiangsu Higher Education Institutions (Statistics)

the Open Project of Joint Lab for Statistics and Finance of NAU (Nos. 2025JLSF306 and 2025JLSF320).

1. School of Statistics and Data Science, Nanjing Audit University, Nanjing, 211815, China

   Lili Yue & Xiong Cai

2. Joint Lab for Statistics and Finance, Nanjing Audit University, Nanjing, 211815, China

   Lili Yue & Xiong Cai

3. College of Business and Economics, California State University, Fullerton, 90802, USA

   Daoji Li

4. School of Statistics, Beijing Normal University, Beijing, 100875, China

   Gaorong Li

Correspondence to Gaorong Li.

The authors declare that they have no conflict of interest.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

In this Appendix, we shall give the proof of the Theorem 1. Review the definitions of \(\hat{f}_k\) and \(f_k\), we have

where \(\mathbb {P}_nX_{(k)}^2=n^{-1}\sum _{i=1}^nm_i^{-1}\sum _{j=1}^{m_i}X_{ijk}^2\) and \(\mathbb {P}_nX_{(k)}=n^{-1}\sum _{i=1}^nm_i^{-1}\sum _{j=1}^{m_i}X_{ijk}\). Let \(D_n=\mathbb {P}_nX_{(k)}^2-EX_{(k)}^2\). Then,

Similar with the discuss of He et al. (2013), \(|S_{k6}|=O(n^{-1/2}(\log n)^{1/2})=o(n^{-\xi })\), \(|S_{k9}|=O(n^{-1/2}(\log n)^{1/2})=o(n^{-\xi })\) using conditions (C2), (C4), (C5) and \(\textrm{max}_{i}\{m_i\}=m_a<\infty \). Then, for sufficiently large n, we have

For the second part of the inequality (Appendix A.1), we have

Let \(\epsilon _1=m_a\delta _1/n\). Combine Bernstein’s inequality, we have

where \(c_3\) is a positive constant.

By the conditions (C2)–(C5), the first part of the inequality (A.1) can be expressed as

the last inequality is derived using the \(\sqrt{n}\)-consistency of QIF estimator \(\hat{\beta }_k\), \(C_1\) and \(C_2\) are some positive constants. Note that,

where \(C_3\) is a positive constant. Let \(\epsilon =4m_a\delta _2/n\), and combine Bernstein’s inequality, we have

where \(c_4\) is a positive constant. Similarly, we have

where \(c_5\) and \(C_4\) are some positive constants.

Combine (A.2)–(A.4), we have

Let \(4m_a\delta _2/n=m_a\delta _1/n=Cn^{-\xi }\) for any given constant \(C>0\), that is, \(\delta _2=Cn^{1-\xi }/(4m_a)\) and \(\delta _1=Cn^{1-\xi }/m_a\). Then,

Similar with the Lemma 3.1 in He et al. (2013), there is a given constant \(c_1>0\) such that \(\textrm{min}_{k\in \mathcal{M}_\tau }\Vert f_k\Vert ^2\ge c_1n^{-\xi }/8\). Take the threshold value \(\nu _n=\delta n^{-\xi }\) with \(\delta <c_1/16\), we have

Then, the proof of Theorem 1 is finished. \(\square \)

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