1 Correction: Journal of the Korean Statistical Society https://doi.org/10.1007/s42952-022-00187-z

In the appendix section, in “Proof of Theorem 1”, a part of equations was missing.

The correct equations are as follows.

For the second term on the right-hand side of (15), we can write

$$\begin{aligned} {\mathbf{S}}_{n}^{-1}({\varvec{\theta}}^{\mathrm{T}}{\varvec{x}}) \frac{1}{n} \sum \limits _{i=1}^{n} K_{h}(\varvec{\theta }^{\mathrm{T}}{\varvec{X}}_{i0}) \varphi '_{\gamma }(\epsilon _{i}) \begin{pmatrix} {\varvec{Z}}_{i} \\ {\varvec{Z}}_{i}\varvec{\theta }^{\mathrm{T}}{\varvec{X}}_{i0}/h \end{pmatrix} =\begin{pmatrix} {\varvec{r}}_{n,1}({\varvec{x}}) \\ {\varvec{r}}_{n,2}({\varvec{x}}) \end{pmatrix} +O_{p}(h\delta _{n}), \end{aligned}$$

where \({\varvec{r}}_{n,1}({\varvec{x}})=\big[n f_{\varvec{\theta }}({\varvec{\theta} }^{\mathrm{T}}{\varvec{x}}) {\mathbf{D}}_{\varvec{\theta }}({\varvec{\theta }}^{\mathrm{T}}{\varvec{x}}) E\{ \varphi _{\gamma }''(\epsilon ) \} \big]^{-1} \sum _{i=1}^{n} K_{h}({\varvec{\theta}}^{\mathrm{T}}{\varvec{X}}_{i0}) \varphi '_{\gamma }(\epsilon _{i}) {\varvec{Z}}_{i}\), \({\varvec{r}}_{n,2}({\varvec{x}})=\big [n f_{\varvec{\theta }}({\varvec{\theta }}^{\mathrm{T}}{\varvec{x}}) {\mathbf{D}}_{\varvec{\theta }}({\varvec{\theta }}^{\mathrm{T}}{\varvec{x}}) E\{ \varphi _{\gamma }''(\epsilon ) \} \big ]^{-1} \sum _{i=1}^{n} K_{h}({\varvec{\theta}}^{\mathrm{T}}{\varvec{X}}_{i0}) \varphi '_{\gamma }(\epsilon _{i}) \big ( {\varvec{\theta}}^{\mathrm{T}}{\varvec{X}}_{ix}/h \big ) {\varvec{Z}}_{i}.\)