Appendix: Proofs for theorems 1 and 2
Theorem 1 Proof. The risk function for \(\widetilde{\pmb {\beta }} _{N}^{F}\) under BLINEX loss is defined as:
$$\begin{aligned} \mathfrak {R}_\textit{BLINEX}\left( \widetilde{\pmb {\beta }}_{N}^{F},\pmb {\beta } \right)= & {} E\left[ \L _\textit{BLINEX}\left( \widetilde{\pmb {\beta }} _{N}^{F},\pmb {\beta }\right) \right] \\= & {} \frac{1}{\lambda }E\left[ 1-\frac{1}{1+b\left[ \exp \left( \mathbf {a} ^{\prime }\left( \widetilde{\pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) \right) -\mathbf {a}^{\prime }\left( \widetilde{\pmb {\beta }}_{N}^{F}- \pmb {\beta }\right) -1\right] }\right] \end{aligned}$$
By applying the binomial expansion \(\left( 1-M\right) ^{-1}=1+M+M^{2}+O\left( M^{3}\right) \)
where
$$\begin{aligned} M=b\left[ \mathbf {a}^{\prime }\left( \widetilde{\pmb {\beta }}_{N}^{F}- \pmb {\beta }\right) -\exp \left( \mathbf {a}^{\prime }\left( \widetilde{ \pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) \right) +1\right] \end{aligned}$$
The risk function reduces to
$$\begin{aligned} E\left[ \L _\textit{BLINEX}\left( \widetilde{\pmb {\beta }}_{N}^{F}, \pmb {\beta }\right) \right]= & {} \frac{1}{\lambda }E\left[ 1-\left( 1+M+M^{2}+{\ldots }\right) \right] \\= & {} -\frac{1}{\lambda }E\left[ M+M^{2}+{\ldots }\right] \\= & {} -\frac{1}{\lambda }\left[ E\left[ M\right] +E\left[ M^{2}\right] \right] +{\ldots } \end{aligned}$$
Firstly we consider \(E\left[ M\right] =bE\left[ \mathbf {a}^{\prime }\left( \widetilde{\pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) -\exp \left( \mathbf {a}^{\prime }\left( \widetilde{\pmb {\beta }}_{N}^{F}- \pmb {\beta }\right) \right) +1\right] \) and obtain
$$\begin{aligned} E\left[ M\right]= & {} bE\left[ \mathbf {a}^{\prime }\left( \widetilde{ \pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) \right] -bE\left[ \exp \left( \mathbf {a}^{\prime }\left( \widetilde{\pmb {\beta }}_{N}^{F}- \pmb {\beta }\right) \right) \right] +b\\ E\left[ M\right]= & {} bE\left[ \mathbf {a}^{\prime }\left( \widetilde{ \pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) \right] -b\exp \left( - \pmb {\beta }^{\prime }\mathbf {a}\right) E\left[ \exp \left( \mathbf {a} ^{\prime }\widetilde{\pmb {\beta }}_{N}^{F}\right) \right] +b \end{aligned}$$
Since \(\widetilde{\pmb {\beta }}_{N}^{F}\) has is asymptotically normally distributed, i.e. \(\widetilde{\pmb {\beta }}_{N}^{F}\sim N\left( {{\varvec{\beta }} ,{\varvec{\Sigma }} }_{\tilde{\pmb {\beta }}_{N}^{F}}\right) \) we evaluate the expectation in the limit as \(N\rightarrow \infty .\)
$$\begin{aligned} \underset{N\rightarrow \infty }{\lim }E\left( M\right)= & {} b\mathbf {a }^{\prime }\mathbf {0-}b\exp \left( -\pmb {\beta }^{\prime }\mathbf {a} \right) \exp \left( \pmb {\beta }^{\prime }\mathbf {a+}\frac{1}{2}\mathbf {a} ^{\prime }{\varvec{\Sigma }}_{\tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a} \right) +b\nonumber \\ \underset{N\rightarrow \infty }{\lim }E\left( M\right)= & {} -b\left( \exp \left( \frac{1}{2}\mathbf {a}^{\prime }{\varvec{\Sigma }}_{\tilde{{{\varvec{\beta }}}}_{N}^{F}}\mathbf {a}\right) -1\right) \end{aligned}$$
(11)
since \(E\left[ \exp \left( \mathbf {a}^{\prime }\widetilde{{\varvec{{\beta }} }}_{N}^{F}\right) \right] \) is the moment-generating function of the normal distribution.
Secondly we consider the calculation of \(E\left( M^{2}\right) \)
$$\begin{aligned}&=b^{2}E\left[ \left( \widetilde{\pmb {\beta }}_{N}^{F}-{\varvec{{\beta }}}\right) ^{\prime }\mathbf {aa}^{\prime }\left( \widetilde{{\varvec{{\beta }} }}_{N}^{F}-\pmb {\beta }\right) \right] \end{aligned}$$
(12)
$$\begin{aligned}&\quad +\,2b^{2}E\left[ \left( \widetilde{\pmb {\beta }}_{N}^{F}-{\varvec{{ \beta }}}\right) ^{\prime }\right] \mathbf {a} \end{aligned}$$
(13)
$$\begin{aligned}&\quad +\,b^{2}E\left[ \exp \left[ 2\mathbf {a}^{\prime }\left( \widetilde{ \pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) \right] \right] \end{aligned}$$
(14)
$$\begin{aligned}&\quad -\,2b^{2}E\left[ \mathbf {a}^{\prime }\left( \widetilde{{\varvec{{\beta }} }}_{N}^{F}-\pmb {\beta }\right) {\exp }\left[ \mathbf {a} ^{\prime }\left( \widetilde{\pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) \right] \right] \end{aligned}$$
(15)
$$\begin{aligned}&\quad -\,2b^{2}E\left[ \exp \left[ \mathbf {a}^{\prime }\left( \widetilde{ \pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) \right] \right] +b^{2} \end{aligned}$$
(16)
As in the previous simplification of \(E\left( M\right) \) we evaluate each of these expectations in the limit as \(N\rightarrow \infty .\)
Expression (12):
$$\begin{aligned}&b^{2}E\left[ \left( \widetilde{\pmb {\beta }}_{N}^{F}-{\varvec{{\beta }}}\right) ^{\prime }\mathbf {aa}^{\prime }\left( \widetilde{{\varvec{{\beta }}}}_{N}^{F}-\pmb {\beta }\right) \right] =b^{2}\mathbf {a}^{\prime }E \left[ \left( \widetilde{\pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) \left( \widetilde{\pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) ^{\prime } \right] \mathbf {a}\\&\quad =b^{2}\mathbf {a}^{\prime }\underset{N\rightarrow \infty }{\lim }E \left[ \left( \widetilde{\pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) \left( \widetilde{\pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) ^{\prime } \right] \mathbf {a}\\&\quad =b^{2}\mathbf {a}^{\prime }{\varvec{\Sigma }}_{\tilde{\pmb {\beta } }_{N}^{F}}\mathbf {a} \end{aligned}$$
Expression (13):
$$\begin{aligned} 2b^{2}E\left[ \left( \widetilde{\pmb {\beta }}_{N}^{F}-{{\varvec{\beta }} }\right) ^{\prime }\mathbf {a}\right] =\,2b^{2}\underset{N\rightarrow \infty }{\lim }E\left[ \left( \widetilde{\pmb {\beta }}_{N}^{F}-{\varvec{{\beta }}}\right) ^{\prime }\right] \mathbf {a=}\,2b^{2}\mathbf {0}^{\prime } \mathbf {a=}\,0 \end{aligned}$$
Expression (14):
$$\begin{aligned}&b^{2}E\left[ \exp \left[ 2\mathbf {a}^{\prime }\left( \widetilde{ \pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) \right] \right] =b^{2}\exp \left( -2\mathbf {a}^{\prime }\pmb {\beta }\right) \underset{N\rightarrow \infty }{\lim }E\left[ \exp \left( 2\mathbf {a}^{\prime }\widetilde{{\varvec{{\beta }} }}_{N}^{F}\right) \right] \\&\quad =b^{2}\exp \left( -2\pmb {\beta }^{\prime }\mathbf {a}\right) \exp \left( 2\pmb {\beta }^{\prime }\mathbf {a+}2\mathbf {a}^{\prime } {\varvec{\Sigma }}_{\tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a}\right) =b^{2}\exp \left[ 2\mathbf {a}^{\prime }{\varvec{\Sigma }}_{\tilde{{{\varvec{\beta }} }}_{N}^{F}}\mathbf {a}\right] \end{aligned}$$
Expression (15):
By defining \(\underset{N\rightarrow \infty }{\lim }M_{\left( \widetilde{\pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) }\left( \mathbf {a} \right) =\exp \left( \mathbf {a}^{\prime }\mathbf {0}+\frac{1}{2}\mathbf {a} ^{\prime }{\varvec{\Sigma }}_{\tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a} \right) \) and making use of Casella and Berger (2002, p62)
$$\begin{aligned}&-2b^{2}\mathbf {a}^{\prime }\underset{N\rightarrow \infty }{\lim }E \left[ \left( \widetilde{\pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) {\exp }\left[ \mathbf {a}^{\prime }\left( \widetilde{\pmb {\beta }} _{N}^{F}-\pmb {\beta }\right) \right] \right] \\&\quad =-2b^{2}\mathbf {a}^{\prime }\underset{N\rightarrow \infty }{\lim } \frac{\partial }{\partial \mathbf {a}}M_{\left( \widetilde{\pmb {\beta }} _{N}^{F}-\pmb {\beta }\right) }\left( \mathbf {a}\right) \\&\quad =-2b^{2}\mathbf {a}^{\prime }\frac{\partial }{\partial \mathbf {a}} \left( \exp \left( \mathbf {0}^{\prime }\mathbf {a}+\frac{1}{2}\mathbf {a} ^{\prime }{\varvec{\Sigma }}_{\tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a} \right) \right) \\&\quad =-2b^{2}\left( \mathbf {a}^{\prime }{\varvec{\Sigma }}_{\tilde{ \pmb {\beta }}_{N}^{F}}\mathbf {a}\right) \exp \left( \frac{1}{2}\mathbf {a} ^{\prime }{\varvec{\Sigma }}_{\tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a} \right) \end{aligned}$$
Expression (16):
$$\begin{aligned}&-2b^{2}\underset{N\rightarrow \infty }{\lim }E\left[ \exp \left[ \mathbf {a}^{\prime }\left( \widetilde{\pmb {\beta }}_{N}^{F}-{{\varvec{\beta }} }\right) \right] \right] +b^{2}\\&\quad =-2b^{2}\exp \left( -\mathbf {a}^{\prime } \pmb {\beta }\right) \underset{N\rightarrow \infty }{\lim }E\left[ \exp \left( \mathbf {a}^{\prime }\widetilde{\pmb {\beta }}_{N}^{F}\right) \right] +b^{2}\\&\quad =-2b^{2}\exp \left( \frac{1}{2}\mathbf {a}^{\prime }{\varvec{\Sigma }}_{\tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a}\right) +b^{2} \end{aligned}$$
Combining expressions (12)–(16) we obtain an result for \(\underset{N\rightarrow \infty }{\lim }E\left( M^{2}\right) \) given by
$$\begin{aligned}&\underset{N\rightarrow \infty }{\lim }E\left( M^{2}\right) =b^{2} \mathbf {a}^{\prime }{\varvec{\Sigma }}_{\tilde{\pmb {\beta }}_{N}^{F}} \mathbf {a+}b^{2}\exp \left[ 2\mathbf {a}^{\prime }{\varvec{\Sigma }}_{\tilde{ \pmb {\beta }}_{N}^{F}}\mathbf {a}\right] -2b^{2}\left( \mathbf {a}^{\prime } {\varvec{\Sigma }}_{\tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a}\right) \exp \left( \frac{1}{2}\mathbf {a}^{\prime }{\varvec{\Sigma }}_{\tilde{{\varvec{{\beta }} }}_{N}^{F}}\mathbf {a}\right) \nonumber \\&\quad -2b^{2}\exp \left( \frac{1}{2}\mathbf {a} ^{\prime }{\varvec{\Sigma }}_{\tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a} \right) +b^{2} \end{aligned}$$
(17)
Combining the final expressions of (11) and (17 ) completes the proof.
Theorem 2 Proof
In order to proof this theorem we need to show the following result:
From (8) we have
$$\begin{aligned} \hat{\pmb {\beta }}_{N}^\textit{FPT}= & {} \tilde{\pmb {\beta }}_{N}^{F}-\left( \tilde{\pmb {\beta }}_{N}^{F}-\pmb {\beta }_{0}\right) I_{NR}\left( F\right) \nonumber \\ \hat{\pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }= & {} \left( \tilde{{\varvec{{\beta }} }}_{N}^{F}-\pmb {\beta }\right) -\tilde{\pmb {\beta }} _{N}^{F}I_{NR}\left( F\right) +\pmb {\beta }_{0}I_{NR}\left( F\right) \nonumber \\= & {} \left( \tilde{\pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) -\left( \tilde{\pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) I_{NR}\left( F\right) +\left( \pmb {\beta }_{0}-\pmb {\beta }\right) I_{NR}\left( F\right) \end{aligned}$$
(18)
Proof
The risk function reduces to
$$\begin{aligned} E\left[ \L _\textit{BLINEX}\left( \hat{\pmb {\beta }}_{N}^\textit{FPT},{\varvec{{\beta }}}\right) \right]= & {} \frac{1}{\lambda }E\left[ 1-\left( 1+M+M^{2}+{\ldots }\right) \right] \nonumber \\= & {} -\frac{1}{\lambda }E\left[ M+M^{2}+{\ldots }\right] \nonumber \\= & {} -\frac{1}{\lambda }\left[ E\left[ M\right] +E\left[ M^{2}\right] \right] +{\ldots } \end{aligned}$$
(19)
Firstly considering \(E\left( M\right) \) we have
$$\begin{aligned} E\left( M\right)= & {} -bE\left[ \exp \left( \mathbf {a}^{\prime }\left( \hat{\pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) \right) - \mathbf {a}^{\prime }\left( \hat{\pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta } \right) -1\right] \\= & {} -\frac{b}{2}E\left[ \left( \mathbf {a}^{\prime }\left( \hat{\pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) \right) ^{2}\right] \end{aligned}$$
Considering \(E\left[ \left( \mathbf {a}^{\prime }\left( \hat{ \pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) \right) ^{2}\right] \) reduces to
$$\begin{aligned}&E\left[ \left( \mathbf {a}^{\prime }\left( \hat{\pmb {\beta }} _{N}^\textit{FPT}-\pmb {\beta }\right) \right) ^{\prime }\left( \mathbf {a} ^{\prime }\left( \hat{\pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) \right) \right] =\mathbf {a}^{\prime }E\left[ \left( \hat{\pmb {\beta }} _{N}^\textit{FPT}-\pmb {\beta }\right) \left( \hat{\pmb {\beta }}_{N}^\textit{FPT}- \pmb {\beta }\right) ^{\prime }\right] \mathbf {a}\\&\quad =\mathbf {a}^{\prime }E\left[ \left( \hat{\pmb {\beta }}_{N}^{F}- \pmb {\beta }\right) \left( \widetilde{\pmb {\beta }}_{N}^{F}-{\varvec{{\beta }}}\right) ^{\prime }-I_{NR}\left( F\right) \left( \widetilde{{{\varvec{\beta }} }}_{N}^{F}-\pmb {\beta }\right) \left( \widetilde{\pmb {\beta }} _{N}^{F}-\pmb {\beta }\right) ^{\prime }\right. \\&\qquad \left. +\,I_{NR}\left( F\right) \left( \pmb {\beta }_{0}-\pmb {\beta }\right) \left( \pmb {\beta }_{0}- \pmb {\beta }\right) ^{\prime }\right] \mathbf {a} \end{aligned}$$
Examining this expression in the limit as \(N\rightarrow \infty \) we obtain an expression for \(E\left( M\right) \):
$$\begin{aligned}&=\frac{-b}{2}\mathbf {a}^{\prime }\underset{N\rightarrow \infty }{ \lim }E\left[ \left( \widetilde{\pmb {\beta }}_{N}^{F}-\pmb {\beta } \right) \left( \widetilde{\pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) ^{\prime }-I_{NR}\left( F\right) \left( \widetilde{\pmb {\beta }}_{N}^{F}\right. \right. \nonumber \\&\quad \left. \left. -\, \pmb {\beta }\right) \left( \widetilde{\pmb {\beta }}_{N}^{F}-{\varvec{{\beta }}}\right) ^{\prime }+I_{NR}\left( F\right) \left( \pmb {\beta }_{0}- \pmb {\beta }\right) \left( \pmb {\beta }_{0}-\pmb {\beta }\right) ^{\prime }\right] \mathbf {a}\nonumber \\&\quad \underset{N\rightarrow \infty }{\lim }E\left( M\right) =\frac{-b}{ 2}\left( \mathbf {a}^{\prime }{\varvec{\Sigma }}_{\tilde{\pmb {\beta }} _{N}^{F}}\mathbf {a}-\mathbf {a}^{\prime }{\varvec{\Sigma }}_{\tilde{{\varvec{{\beta }} }}_{N}^{F}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2,\frac{\varDelta ^{2}}{2}}^{2}\right) \right] \right. \nonumber \\&\quad \left. +\,\mathbf {a}^{\prime }{\varvec{\Sigma }}_{{\varvec{{\beta }} }_{0}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2,\frac{\varDelta ^{2}}{2} }^{2}\right) \right] \right) \end{aligned}$$
(20)
Secondly, considering \(E\left( M^{2}\right) \) we obtain
$$\begin{aligned} E\left( M^{2}\right)= & {} b^{2}E\left[ \left( \mathbf {a}^{\prime }\left( \hat{\pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) -\exp \left( \mathbf {a}^{\prime }\left( \hat{\pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta } \right) \right) +1\right) ^{\prime }\left( \mathbf {a}^{\prime }\left( \hat{ \pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) \right. \right. \nonumber \\&\quad \left. \left. -\exp \left( \mathbf {a} ^{\prime }\left( \hat{\pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) \right) +1\right) \right] \nonumber \\= & {} b^{2}E\left[ \left( \hat{\pmb {\beta }}_{N}^\textit{FPT}-{\varvec{{\beta }}}\right) ^{\prime }\mathbf {aa}^{\prime }\left( \hat{\pmb {\beta }} _{N}^\textit{FPT}-\pmb {\beta }\right) \right] \end{aligned}$$
(21)
$$\begin{aligned}&+\,2b^{2}E\left[ \mathbf {a}^{\prime }\left( \hat{\pmb {\beta }} _{N}^\textit{FPT}-\pmb {\beta }\right) \right] \end{aligned}$$
(22)
$$\begin{aligned}&+\,b^{2}E\left[ \exp \left( 2\mathbf {a}^{\prime }\left( \hat{ \pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) \right) \right] \end{aligned}$$
(23)
$$\begin{aligned}&-\,2b^{2}E\left[ \mathbf {a}^{\prime }\left( \hat{\pmb {\beta }} _{N}^\textit{FPT}-\pmb {\beta }\right) \exp \left( \mathbf {a}^{\prime }\left( \hat{\pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) \right) \right] \end{aligned}$$
(24)
$$\begin{aligned}&-\,2b^{2}E\left[ \exp \left( \mathbf {a}^{\prime }\left( \hat{ \pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) \right) \right] +b^{2} \end{aligned}$$
(25)
Each one of these expectations will again be evaluated in the limit as \(N\rightarrow \infty .\)
Expression (21):
As in the case of expression (20) we obtain:
$$\begin{aligned}&b^{2}E\left[ \left( \hat{\pmb {\beta }}_{N}^\textit{FPT}-{{\varvec{\beta }} }\right) ^{\prime }\mathbf {aa}^{\prime }\left( \hat{\pmb {\beta }} _{N}^\textit{FPT}-\pmb {\beta }\right) \right] =b^{2}\left[ \mathbf {a}^{\prime } {\varvec{\Sigma }}_{\tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a}-\mathbf {a} ^{\prime }{\varvec{\Sigma }}_{\tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2,\frac{\varDelta ^{2}}{2}}^{2}\right) \right] \right. \\&\quad \left. +\,\mathbf { a}^{\prime }{\varvec{\Sigma }}_{\pmb {\beta }_{0}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2,\frac{\varDelta ^{2}}{2}}^{2}\right) \right] \right] \end{aligned}$$
Expression (22):
$$\begin{aligned}&2b^{2}E\left[ \mathbf {a}^{\prime }\left( \hat{\pmb {\beta }} _{N}^\textit{FPT}-\pmb {\beta }\right) \right] =2b^{2}E\left[ \mathbf {a}^{\prime }\left( \left( \tilde{\pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) -\left( \tilde{\pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) I_{NR}\left( F\right) \right. \right. \\&\qquad \left. \left. +\,\left( \pmb {\beta }_{0}-\pmb {\beta }\right) I_{NR}\left( F\right) \right) \right] \\&\quad =2b^{2}\mathbf {a}^{\prime }E\left[ \left( \tilde{\pmb {\beta }} _{N}^{F}-\pmb {\beta }\right) \right] -2b^{2}\mathbf {a}^{\prime }E\left[ \left( \tilde{\pmb {\beta }}_{N}^{F}-\pmb {\beta }\right) I_{NR}\left( F\right) \right] \\&\qquad +\,2b^{2}\mathbf {a}^{\prime }E\left[ \left( \pmb {\beta } _{0}-\pmb {\beta }\right) I_{NR}\left( F\right) \right] \end{aligned}$$
By evaluating all these expectations in the limit as \(N\rightarrow \infty \) and making use of Judge and Bock,1978, p321 we obtain
$$\begin{aligned} 2b^{2}\mathbf {a}^{\prime }\mathbf {0-}2b^{2}\mathbf {a}^{\prime } \mathbf {0}E\left[ \varphi \left( \chi _{p+2}^{2},\frac{\varDelta ^{2}}{2} \right) \right] +2b^{2}\mathbf {a}^{\prime }\mathbf {0}E\left[ \varphi \left( \chi _{p+2}^{2},\frac{\varDelta ^{2}}{2}\right) \right] =0 \end{aligned}$$
Expression (23):
$$\begin{aligned}&b^{2}E\left[ \exp \left( 2\mathbf {a}^{\prime }\left( \hat{{\varvec{{\beta }}}}_{N}^\textit{FPT}-\pmb {\beta }\right) \right) \right] =b^{2}E\left[ 1+2 \mathbf {a}^{\prime }\left( \hat{\pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta } \right) \right. \\&\quad \left. +\frac{1}{2!}\left( 2\mathbf {a}^{\prime }\left( \hat{\pmb {\beta }} _{N}^\textit{FPT}-\pmb {\beta }\right) \right) ^{2}+O\left( x^{3}\right) \right] \\&\approx b^{2}\left[ 1+2\mathbf {a}^{\prime }E\left( \hat{{\varvec{{\beta }} }}_{N}^\textit{FPT}-\pmb {\beta }\right) +2\mathbf {a}^{\prime }E\left[ \left( \hat{\pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) \left( \hat{ \pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) ^{\prime }\right] \mathbf {a }\right] \end{aligned}$$
From expression (22) we know that \(E\left( \hat{{\varvec{{\beta }}}}_{N}^\textit{FPT}-\pmb {\beta }\right) \) reduces to zero in the limit as \( N\rightarrow \infty \) therefore we obtain
$$\begin{aligned}&b^{2}+2b^{2}\mathbf {a}^{\prime }E\left[ \left( \hat{{\varvec{{\beta }}}}_{N}^\textit{FPT}-\pmb {\beta }\right) \left( \hat{\pmb {\beta }}_{N}^\textit{FPT}- \pmb {\beta }\right) ^{\prime }\right] \mathbf {a}\\&\quad =b^{2}+2b^{2}\left( \mathbf {a}^{\prime }{\varvec{\Sigma }}_{\tilde{ \pmb {\beta }}_{N}^{F}}\mathbf {a}-\mathbf {a}^{\prime }{\varvec{\Sigma }}_{ \tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2, \frac{\varDelta ^{2}}{2}}^{2}\right) \right] +\mathbf {a}^{\prime }{\varvec{{\Sigma }}}_{\pmb {\beta }_{0}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2, \frac{\varDelta ^{2}}{2}}^{2}\right) \right] \right) \end{aligned}$$
Expression (24):
$$\begin{aligned}&-2b^{2}E\left[ \mathbf {a}^{\prime }\left( \hat{\pmb {\beta }} _{N}^\textit{FPT}-\pmb {\beta }\right) \exp \left( \mathbf {a}^{\prime }\left( \hat{\pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) \right) \right] \\&\quad =-2b^{2}E\left[ \mathbf {a}^{\prime }\left( \hat{\pmb {\beta }} _{N}^\textit{FPT}-\pmb {\beta }\right) \left( 1+\mathbf {a}^{\prime }\left( \hat{ \pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) \right. \right. \\&\qquad \left. \left. +\,\frac{1}{2!}\left( \mathbf {a}^{\prime }\left( \hat{\pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta } \right) \right) ^{2}+O\left( x^{3}\right) \right) \right] \\&\quad \approx -\,2b^{2}E\left[ \mathbf {a}^{\prime }\left( \hat{{\varvec{{\beta }}}}_{N}^\textit{FPT}-\pmb {\beta }\right) \right] \\&\qquad -\,2b^{2}\mathbf {a}^{\prime }E\left[ \left( \hat{\pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) \left( \hat{\pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) ^{\prime } \right] \mathbf {a-}b^{2}E\left[ \left( \mathbf {a}^{\prime }\left( \hat{ \pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) \right) ^{3}\right] \end{aligned}$$
By making use of the results in expression (21) and (22) we obtain
$$\begin{aligned}&-2b^{2}\left( \mathbf {a}^{\prime }{\varvec{\Sigma }}_{\tilde{ \pmb {\beta }}_{N}^{F}}\mathbf {a}-\mathbf {a}^{\prime }{\varvec{\Sigma }}_{ \tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2, \frac{\varDelta ^{2}}{2}}^{2}\right) \right] +\mathbf {a}^{\prime }{\varvec{{\Sigma }}}_{\pmb {\beta }_{0}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2, \frac{\varDelta ^{2}}{2}}^{2}\right) \right] \right) \\&\quad -b^{2}E\left[ \left( \mathbf {a}^{\prime }\left( \hat{\pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta } \right) \right) ^{3}\right] \end{aligned}$$
Using the Stein identity, Saleh (2006, p. 3) and Kleyn (2014) we find that \(E\left[ \left( \mathbf {a}^{\prime }\left( \hat{\pmb {\beta }} _{N}^\textit{FPT}-\pmb {\beta }\right) \right) ^{3}\right] =0\) therefore expression (24) reduces to
$$\begin{aligned} -2b^{2}\left( \mathbf {a}^{\prime }{\varvec{\Sigma }}_{\tilde{ \pmb {\beta }}_{N}^{F}}\mathbf {a}-\mathbf {a}^{\prime }{\varvec{\Sigma }}_{ \tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2, \frac{\varDelta ^{2}}{2}}^{2}\right) \right] +\mathbf {a}^{\prime }{\varvec{{\Sigma }} }_{\pmb {\beta }_{0}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2, \frac{\varDelta ^{2}}{2}}^{2}\right) \right] \right) \end{aligned}$$
Expression (25):
$$\begin{aligned}&-2b^{2}E\left[ \exp \left( \mathbf {a}^{\prime }\left( \hat{ \pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) \right) \right] +b^{2}\\&\quad =-2b^{2}E\left[ 1+\mathbf {a}^{\prime }\left( \hat{\pmb {\beta }} _{N}^\textit{FPT}-\pmb {\beta }\right) +\frac{1}{2!}\left( \mathbf {a}^{\prime }\left( \hat{\pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) \right) ^{2}+O\left( x^{3}\right) \right] +b^{2}\\&\quad \approx -2b^{2}-2b^{2}\mathbf {a}^{\prime }E\left( \hat{{\varvec{{\beta }}}}_{N}^\textit{FPT}-\pmb {\beta }\right) -b^{2}\mathbf {a}^{\prime }E\left[ \left( \hat{\pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) \left( \hat{ \pmb {\beta }}_{N}^\textit{FPT}-\pmb {\beta }\right) ^{\prime }\right] \mathbf { a+}b^{2}\\&\quad =-2b^{2}-2b^{2}\mathbf {a}^{\prime }\mathbf {0-}b^{2}\left( \mathbf { a}^{\prime }{\varvec{\Sigma }}_{\tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a}- \mathbf {a}^{\prime }{\varvec{\Sigma }}_{\tilde{\pmb {\beta }}_{N}^{F}} \mathbf {a}E\left[ \varphi \left( \chi _{p+2,\frac{\varDelta ^{2}}{2} }^{2}\right) \right] \right. \\&\qquad \left. +\,\mathbf {a}^{\prime }{\varvec{\Sigma }}_{\pmb {\beta } _{0}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2,\frac{\varDelta ^{2}}{2} }^{2}\right) \right] \right) +b^{2}\\&\quad =-b^{2}\mathbf {-}b^{2}\left( \mathbf {a}^{\prime }{\varvec{\Sigma }} _{\tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a}-\mathbf {a}^{\prime }{\varvec{{\Sigma }} }_{\tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2,\frac{\varDelta ^{2}}{2}}^{2}\right) \right] +\,\mathbf {a}^{\prime } {\varvec{\Sigma }}_{\pmb {\beta }_{0}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2,\frac{\varDelta ^{2}}{2}}^{2}\right) \right] \right) \end{aligned}$$
By combining expressions (21)–(25) we obtain a final expression for \(E\left( M^{2}\right) \) in the limit as \(N\rightarrow \infty \):
$$\begin{aligned}&b^{2}\left[ \mathbf {a}^{\prime }{\varvec{\Sigma }}_{\tilde{{\varvec{{\beta }} }}_{N}^{F}}\mathbf {a}-\mathbf {a}^{\prime }{\varvec{\Sigma }}_{\tilde{ \pmb {\beta }}_{N}^{F}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2,\frac{ \varDelta ^{2}}{2}}^{2}\right) \right] +\mathbf {a}^{\prime }{\varvec{\Sigma }}_{ \pmb {\beta }_{0}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2,\frac{\varDelta ^{2}}{2}}^{2}\right) \right] \right] \\&\quad +\,b^{2}+2b^{2}\left( \mathbf {a}^{\prime }{\varvec{\Sigma }}_{\tilde{ \pmb {\beta }}_{N}^{F}}\mathbf {a}-\mathbf {a}^{\prime }{\varvec{\Sigma }}_{ \tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2, \frac{\varDelta ^{2}}{2}}^{2}\right) \right] +\,\mathbf {a}^{\prime }{\varvec{{\Sigma }}}_{\pmb {\beta }_{0}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2, \frac{\varDelta ^{2}}{2}}^{2}\right) \right] \right) \\&\quad -\,2b^{2}\left( \mathbf {a}^{\prime }{\varvec{\Sigma }}_{\tilde{ \pmb {\beta }}_{N}^{F}}\mathbf {a}-\mathbf {a}^{\prime }{\varvec{\Sigma }}_{ \tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2, \frac{\varDelta ^{2}}{2}}^{2}\right) \right] +\,\mathbf {a}^{\prime }{\varvec{{\Sigma }}}_{\pmb {\beta }_{0}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2, \frac{\varDelta ^{2}}{2}}^{2}\right) \right] \right) -b^{2}\\&\quad {-}\,b^{2}\left( \mathbf {a}^{\prime }{\varvec{\Sigma }}_{ \tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a}-\mathbf {a}^{\prime }{\varvec{{\Sigma }} }_{\tilde{\pmb {\beta }}_{N}^{F}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2,\frac{\varDelta ^{2}}{2}}^{2}\right) \right] +\mathbf {a}^{\prime } {\varvec{\Sigma }}_{\pmb {\beta }_{0}}\mathbf {a}E\left[ \varphi \left( \chi _{p+2,\frac{\varDelta ^{2}}{2}}^{2}\right) \right] \right) \end{aligned}$$
Therefore
$$\begin{aligned} \underset{N\rightarrow \infty }{\lim }E\left( M^{2}\right) =0 \end{aligned}$$
(26)
By substituting (20) and (26) into (19) completes the proof. \(\square \)
