Appendix
1.1 Proof of Theorem 1
Equation (3.6) is summarized as
$$\begin{aligned} w^{*}\equiv N^{1/2}({\hat{\alpha }}_{\mathrm{W}} -\alpha _0) =N^{-1/2}\eta _0 +\sum _{i=1}^3 {N^{-(i-1)/2}} {{\varvec{\uplambda }}}^{(i)} {}^{\prime }\mathbf{m}^{(i)}+O_p (N^{-3/2}), \end{aligned}$$
(8.1)
where \(\eta _0 =O(1), {{\varvec{\uplambda }}}^{(i)}=O(1)\) and \(\mathbf{m}^{(i)}=O_p (1) (i=1,2,3)\), which are
$$\begin{aligned} \eta _0&= -({{\varvec{\Lambda }}}^{-1}\mathbf{q}_0^*)_{(\alpha )}, {{\varvec{\uplambda }}}^{(1)}=\frac{\partial \alpha _0 }{\partial {{\varvec{\uppi }}}_{\mathrm{T}} }, {{\varvec{\uplambda }}}^{(2)}=\frac{1}{2}\left\{ {\left( {\frac{\partial }{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}} \right) ^{\langle 2\rangle }\alpha _0} \right\} , \nonumber \\ {{\varvec{\uplambda }}}^{(3)}&= \left[ {\frac{1}{6}\left\{ {\left( {\frac{\partial }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}} \right) ^{\langle 3\rangle }\alpha _0 } \right\} ,\frac{\partial \alpha _{\Delta \mathrm{W}} }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}}\right] ^{{\prime }}, \\ \mathbf{m}^{(1)}&= \mathbf{u}, \mathbf{m}^{(2)}=\mathbf{u}^{\langle 2\rangle }, \mathbf{m}^{(3)}=(\mathbf{u}^{\langle 3\rangle \prime },\mathbf{u}^{\prime })^{\prime }.\nonumber \end{aligned}$$
(8.2)
In (8.1), the differences between \(w^{*}\) for \({\hat{\alpha }}_{\mathrm{W}}\) and \(w^{*}\) for \({\hat{\alpha }}_{\mathrm{ML}}\), say \(w_{\mathrm{ML}}^*\), are found in \(\eta _0\) and \(\partial \alpha _{\Delta \mathrm{W}} /\partial {{\varvec{\uppi }}}_{\mathrm{T}}\) of \({{\varvec{\uplambda }}}^{(3)}\) in (8.2), which are null in the case of \(w_{\mathrm{ML}}^*\).
Note that the fixed term \(N^{-1/2}\eta _0\) in (8.1) is irrelevant to all the cumulants of \(w^{*}\) except the first one. Define \(\kappa _i (\cdot )\) as the \(i\)th cumulant of the argument. Then, from (8.1) and (8.2), it follows that
$$\begin{aligned} \kappa _1 (w^{*})&= N^{-1/2}\eta _0 +N^{-1/2}\beta _{\mathrm{ML}1} +O(N^{-3/2}), \nonumber \\ \kappa _2 (w^{*})&= \beta _{\mathrm{ML}2} +N^{-1}\left( {\beta _{\mathrm{ML}\Delta 2} +2\frac{\partial \alpha _0 }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}{{\varvec{\Omega }}}_{\mathrm{T}} \frac{\partial \alpha _{\Delta \mathrm{W}} }{\partial {{\varvec{\uppi }}}_{\mathrm{T}} }} \right) +O(N^{-2}), \\ \kappa _3 (w^{*})&= N^{-1/2}\beta _{\mathrm{ML}3} +O(N^{-3/2}),\nonumber \end{aligned}$$
(8.3)
and
$$\begin{aligned} \kappa _4 (w^{*})&= N^{-1}\left[ {\beta _{\mathrm{ML}4} +4\text{ E }\left\{ {\left( {\frac{\partial \alpha _0 }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}\mathbf{u}} \right) ^{3}\frac{\partial \alpha _{\Delta \mathrm{W}} }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}\mathbf{u}} \right\} -6\beta _{\mathrm{ML}2} (\beta _{\mathrm{W}\Delta 2} -\beta _{\mathrm{ML}\Delta 2} )} \right] \\&+O(N^{-2}) = N^{-1}\left( \beta _{\mathrm{ML}4} +4\times 3\beta _{\mathrm{ML}2} \frac{\partial \alpha _0 }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}{{\varvec{\Omega }}}_{\mathrm{T}} \frac{\partial \alpha _{\Delta \mathrm{W}} }{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}\right. \\&\left. -6\times 2\beta _{\mathrm{ML}2} \frac{\partial \alpha _0 }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}{{\varvec{\Omega }}}_{\mathrm{T}} \frac{\partial \alpha _{\Delta \mathrm{W}}}{\partial {{\varvec{\uppi }}}_{\mathrm{T}}} \right) +O(N^{-2}) =N^{-1}\beta _{\mathrm{ML}4} +O(N^{-2}), \end{aligned}$$
which gives (3.7), where the equality \(\beta _{\mathrm{W}3} =\beta _{\mathrm{ML}3}\) stems from the fact that the third asymptotic cumulant is given by \(N^{-(i-1)/2}{{\varvec{\uplambda }}}^{(i)} {}^{\prime }\mathbf{m}^{(i)} (i=1,2)\), which are common to \(w^{*}\) and \(w_{\mathrm{ML}}^*\).
1.2 Proof of Theorem 2
From (8.1) and (4.3) of Lemma 1, \(t_V\) is expanded as
$$\begin{aligned} t_V&= (N^{-1/2}\eta _0 +{{\varvec{\uplambda }}}^{(1)} {}^{\prime }\mathbf{m}^{(1)}+N^{-1/2}{{\varvec{\uplambda }}}^{(2)} {}^{\prime }\mathbf{m}^{(2)}+N^{-1}{{\varvec{\uplambda }}}^{(3)} {}^{\prime }\mathbf{m}^{(3)}) \nonumber \\&\ \times (v_{0\alpha \alpha }^{-1/2} + N^{-1}\eta _{V0} + \ N^{-1/2}{{\varvec{\uplambda }}}_{V}^{(1)} {}^{\prime }\mathbf{m}^{(1)}+N^{-1}{{\varvec{\uplambda }}}_{V}^{(2)} {}^{\prime }\mathbf{m}^{(2)})+O_p (N^{-3/2}) \nonumber \\&= N^{-1/2}\eta _0 v_{0\alpha \alpha }^{-1/2} +{{\varvec{\uplambda }}}^{(1)} {}^{\prime }v_{0\alpha \alpha }^{-1/2} \mathbf{m}^{(1)} \nonumber \\&\ +N^{-1/2}({{\varvec{\uplambda }}}^{(2)} {}^{\prime }v_{0\alpha \alpha }^{-1/2} +{{\varvec{\uplambda }}}^{(1)} {}^{\prime }\otimes {{\varvec{\uplambda }}}_{V}^{(1)} {}^{\prime })\mathbf{m}^{(1)\langle 2\rangle } (\text{ recall } \ \mathbf{m}^{(2)}=\mathbf{m}^{(1)\langle 2\rangle }) \nonumber \\&\ + N^{-1}({{\varvec{\uplambda }}}^{(3)} {}^{\prime }v_{0\alpha \alpha }^{-1/2}, \eta _0 {{\varvec{\uplambda }}}_{V}^{(1)} {}^{\prime }+\eta _{V0} {{\varvec{\uplambda }}}^{(1)} {}^{\prime }, {{\varvec{\uplambda }}}^{(2)} {}^{\prime }\otimes {{\varvec{\uplambda }}}_{V}^{(1)} {}^{\prime }+{{\varvec{\uplambda }}}^{(1)} {}^{\prime }\otimes {{\varvec{\uplambda }}}_{V}^{(2)} {}^{\prime }) \nonumber \\&\ \times (\mathbf{m}^{(3)} {}^{\prime }, \mathbf{m}^{(1)} {}^{\prime }, \mathbf{m}^{(1)\langle 3\rangle } {}^{\prime })^{\prime }+O_p (N^{-3/2}) \nonumber \\&= N^{-1/2}\eta _0 v_{0\alpha \alpha }^{-1/2} +{{\varvec{\uplambda }}}^{(1)} {}^{\prime }v_{0\alpha \alpha }^{-1/2} \mathbf{m}^{(1)} \nonumber \\&\ +N^{-1/2}({{\varvec{\uplambda }}}^{(2)} {}^{\prime }v_{0\alpha \alpha }^{-1/2} +{{\varvec{\uplambda }}}^{(1)} {}^{\prime }\otimes {{\varvec{\uplambda }}}_{V}^{(1)} {}^{\prime })\mathbf{m}^{(1)\langle 2\rangle } \nonumber \\&\ +N^{-1}\left[ {\left\{ {\frac{1}{6}\left( {\frac{\partial }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}} \right) ^{\langle 3\rangle }\alpha _0 } \right\} } v_{0\alpha \alpha }^{-1/2} +{{\varvec{\uplambda }}}^{(2)} {}^{\prime }\otimes {{\varvec{\uplambda }}}_{ V}^{(1)} {}^{\prime }+{{\varvec{\uplambda }}}^{(1)} {}^{\prime }\otimes {{\varvec{\uplambda }}}_{V}^{(2)} {}^{\prime }, \right. \nonumber \\&\left. {\frac{\partial \alpha _{\Delta \mathrm{W}} }{\partial {{\varvec{\uppi }}}_{\mathrm{T}} {}^{\prime }}v_{0\alpha \alpha }^{-1/2} +\eta _0 {{\varvec{\uplambda }}}_{V}^{(1)} {}^{\prime }+\eta _{V0} {{\varvec{\uplambda }}}^{(1)} {}^{\prime }} \right] (\mathbf{m}^{(1)\langle 3\rangle } {}^{\prime }, \mathbf{m}^{(1)} {}^{\prime })^{\prime }+O_p (N^{-3/2})\nonumber \\&\equiv N^{-1/2}\eta _0 v_{0\alpha \alpha }^{-1/2} +\sum _{j=1}^3 {N^{-(j-1)/2}} {{\varvec{\uplambda }}}^{(Vj)} {}^{\prime }\mathbf{m}^{(j)}+O_p (N^{-3/2}). \end{aligned}$$
(8.4)
From (8.4), as in Theorem 1, we note that \({{\varvec{\uplambda }}}^{(Vj)} {}^{\prime }\mathbf{m}^{(j)} (j=1,2)\) are common to \(t_V\) for \({\hat{\alpha }}_{\mathrm{W}}\) and to \(t_V\) for \({\hat{\alpha }}_{\mathrm{ML}}\). Then,
$$\begin{aligned} \kappa _1 (t_V)&= N^{-1/2}\eta _0 v_{0\alpha \alpha }^{-1/2} +N^{-1/2}{\beta _{\mathrm{ML}1V}}^{\prime }+O(N^{-3/2}) \nonumber \\&= N^{-1/2}\left\{ -({{\varvec{\Lambda }}}^{-1}\mathbf{q}_0^*)_{(\alpha )} v_{0\alpha \alpha }^{-1/2} +{\beta _{\mathrm{ML}1V}}^{\prime }\right\} +O(N^{-3/2}), \nonumber \\ \kappa _2 (t_V)&= v_{0\alpha \alpha }^{-1} \beta _{\mathrm{ML}2} +N^{-1}\left[ {{\beta _{\mathrm{ML}\Delta 2V}}^{\prime }+2v_{0\alpha \alpha }^{-1} \frac{\partial \alpha _0 }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}{{\varvec{\Omega }}}_{\mathrm{T}} \frac{\partial \alpha _{\Delta \mathrm{W}}}{\partial {{\varvec{\uppi }}}_{\mathrm{T}} }} \right. \nonumber \\&\left. +2v_{0\alpha \alpha }^{-1/2} (\eta _0 {{\varvec{\uplambda }}}_{V}^{(1)} {}^{\prime }{{\varvec{\Omega }}}_{\mathrm{T}} {{\varvec{\uplambda }}}^{(1)}+\eta _{V0} \beta _{\mathrm{ML}2} )\right] +O(N^{-2}), \nonumber \\ \kappa _3 (t_V)&= N^{-1/2}{\beta _{\mathrm{ML}3V}}^{\prime } +O(N^{-3/2}), \text{ and } \nonumber \\ \kappa _4 (t_V)&= N^{-1}{\beta _{\mathrm{ML}4V}}^{\prime }+O(N^{-2}), \end{aligned}$$
(8.5)
which gives (4.4).
1.3 Proof of Theorem 4
Expand \(N^{1/2}({\hat{\alpha }}_{\mathrm{B}\text{- }\mathrm{C\, W}} -\alpha _0)\) as
$$\begin{aligned} N^{1/2}({\hat{\alpha }}_{\mathrm{B}\text{- }\mathrm{C\, W}} -\alpha _0)&= {{\varvec{\uplambda }}}^{(1)} {}^{\prime }\mathbf{m}^{(1)}+N^{-1/2}{{\varvec{\uplambda }}}^{(2)} {}^{\prime }\mathbf{m}^{(2)}\nonumber \\&\ -\frac{N^{-1/2}}{2}\text{ tr }\left( {\frac{\partial ^{2}\alpha _0 }{\partial {{\varvec{\uppi }}}_{\mathrm{T}} {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}{{\varvec{\Omega }}}_{\mathrm{T}}} \right) +N^{-1}{{\varvec{\uplambda }}}_{\mathrm{B}\text{- }\mathrm{C\, W}}^{(3)} {}^{\prime }\mathbf{m}^{(3)}\nonumber \\&\ +O_p (N^{-3/2}), \end{aligned}$$
(8.6)
where
$$\begin{aligned}&{{\varvec{\uplambda }}}_{\mathrm{B}\text{- }\mathrm{C}}^{(3)} \!=\! \left[ {\frac{1}{6}\left\{ {\left( {\frac{\partial }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}} \right) ^{\langle 2 \rangle }\alpha _0 } \right\} , \frac{\partial \alpha _{\Delta \mathrm{W}} }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}\!-\!\frac{\partial \beta _{\mathrm{W}1}}{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}} \right] ^{{\prime }} \text{ and } \end{aligned}$$
(8.7)
$$\begin{aligned}&\left. -\frac{\partial \beta _{\mathrm{W}1} }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }} \!=\! \frac{\partial ({\hat{{\varvec{\Lambda }}}}_{\mathrm{W}}^{-1} {\hat{\mathbf{q}}}_{\mathrm{W}}^*)_{(\alpha )} }{\partial \mathbf{p}^{\prime }}\right| _{\mathop {{\hat{{\varvec{\upalpha }}}}_{\mathrm{W}} \!=\!{{\varvec{\upalpha }}}_0}\limits _{\mathbf{p}\!=\!{{\varvec{\uppi }}}_{\mathrm{T}}}} \left. -\frac{1}{2}\frac{\partial }{\partial \mathbf{p}^{\prime }}\text{ tr }\left( {\frac{\partial ^{2}{\hat{\alpha }}_{\mathrm{W}} }{\partial \mathbf{p}\partial \mathbf{p}^{\prime }}{\hat{{\varvec{\Omega }}}}_{\mathrm{T}}} \right) \right| _{\mathop {{\hat{{\varvec{\upalpha }}}}_{\mathrm{W}} \!=\!{{\varvec{\upalpha }}}_0}\limits _{\mathbf{p}={{\varvec{\uppi }}}_{\mathrm{T}}}} \nonumber \\&=\left\{ -\left( \left. {{\varvec{\Lambda }}}^{-1}\frac{\partial {\hat{{\varvec{\Lambda }}}}}{\partial p_1}\right| _{\mathop {{\hat{{\varvec{\upalpha }}}}_{\mathrm{W}} ={{\varvec{\upalpha }}}_0}\limits _{\mathbf{p}={{\varvec{\uppi }}}_{\mathrm{T}}}} {{\varvec{\Lambda }}}^{-1}\mathbf{q}_0^*,\ldots , {{{\varvec{\Lambda }}}^{-1}\left. \frac{\partial {\hat{{\varvec{\Lambda }}}}}{\partial p_K}\right| _{\mathop {{\hat{{\varvec{\upalpha }}}}_{\mathrm{W}} ={{\varvec{\upalpha }}}_0}\limits _{\mathbf{p}={{\varvec{\uppi }}}_{\mathrm{T}}}} {{\varvec{\Lambda }}}^{-1}\mathbf{q}_0^*} \right) \right. \nonumber \\&\quad \left. +{{\varvec{\Lambda }}}^{-1}\left. \frac{\partial {\hat{\mathbf{q}}}_{\mathrm{W}}^*}{\partial \mathbf{p}{\prime }}\right| _{\mathop {{\hat{{\varvec{\upalpha }}}}_{\mathrm{W}} ={{\varvec{\upalpha }}}_0}\limits _{\mathbf{p}={{\varvec{\uppi }}}_{\mathrm{T}}}} \right\} _{(\alpha )} -\frac{1}{2}\frac{\partial }{\partial \mathbf{p}^{\prime }}\text{ tr }\left. \left( {\frac{\partial ^{2}{\hat{\alpha }}_{\mathrm{W}} }{\partial \mathbf{p}\partial \mathbf{p}^{\prime }}{\hat{{\varvec{\Omega }}}}_{\mathrm{T}} } \right) \right| _{\mathop {{\hat{{\varvec{\upalpha }}}}_{\mathrm{W}} ={{\varvec{\upalpha }}}_0}\limits _{\mathbf{p}={{\varvec{\uppi }}}_{\mathrm{T}}}} \nonumber \\&=-\frac{\partial \alpha _{\Delta \mathrm{W}} }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}-\frac{1}{2}\frac{\partial }{\partial {{{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}\text{ tr }\left( {\frac{\partial ^{2}\alpha _0 }{\partial {{\varvec{\uppi }}}_{\mathrm{T}} {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}{{\varvec{\Omega }}}_{\mathrm{T}} } \right) , \end{aligned}$$
(8.8)
(see (3.4) with (3.2) and (3.3)), which gives
$$\begin{aligned} \frac{\partial \alpha _{\Delta \mathrm{W}} }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}-\frac{\partial \beta _{\mathrm{W}1} }{\partial {{{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}&= -\frac{1}{2}\left\{ {\text{ tr }\left( {\frac{\partial ^{3}\alpha _0 }{\partial {{\varvec{\uppi }}}_{\mathrm{T}} {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }\partial \pi _{\mathrm{T}1} }{{\varvec{\Omega }}}_{\mathrm{T}}} \right) ,\ldots ,\text{ tr }\left( {\frac{\partial ^{3}\alpha _0}{\partial {{\varvec{\uppi }}}_{\mathrm{T}} {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }\partial \pi _{\mathrm{T}K} }{{\varvec{\Omega }}}_{\mathrm{T}} } \right) } \right\} \nonumber \\&\quad -\frac{1}{2}\left\{ {\text{ tr }\left( {\frac{\partial ^{2}\alpha _0 }{\partial {{\varvec{\uppi }}}_{\mathrm{T}} {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}\frac{\partial {{\varvec{\Omega }}}_{\mathrm{T}} }{\partial \pi _{\mathrm{T}1} }} \right) ,\ldots ,\text{ tr }\left( {\frac{\partial ^{2}\alpha _0 }{\partial {{\varvec{\uppi }}}_{\mathrm{T}} {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}\frac{\partial {{\varvec{\Omega }}}_{\mathrm{T}} }{\partial \pi _{\mathrm{T}K} }} \right) } \right\} \nonumber \\&= -\frac{\partial \beta _{\mathrm{ML}1}}{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}. \end{aligned}$$
(8.9)
From the last result, \({{\varvec{\uplambda }}}_{\mathrm{B}\text{- }\mathrm{C\, W}}^{(3)}\) becomes equal to \({{\varvec{\uplambda }}}_{\mathrm{B}\text{- }\mathrm{C\, ML}}^{(3)}\) of \({\hat{\alpha }}_{\mathrm{B}\text{- }\mathrm{C\, ML}}\) (the bias-corrected \({\hat{\alpha }}_{\mathrm{ML}})\), which gives (5.6).
1.4 Proof of Theorem 5
Replace \(\eta _0\) and \(\partial \alpha _{\Delta \mathrm{W}} /{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }\) of \({{\varvec{\uplambda }}}^{(3)}\) in (8.2) with \(-\frac{1}{2}\text{ tr }\left( {\frac{\partial ^{2}\alpha _0}{\partial {{\varvec{\uppi }}}_{\mathrm{T}} {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}{{\varvec{\Omega }}}_{\mathrm{T}}} \right) (\equiv \eta _{\mathrm{B}\text{- }\mathrm{C}\, 0} \equiv ({{\varvec{\upeta }}}_{\mathrm{B}\text{- }\mathrm{C}\, 0} )_{(\alpha )})\) in (8.6) and \(-\frac{\partial \beta _{\mathrm{ML}1} }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}\) of \({{\varvec{\uplambda }}}_{\mathrm{B}\text{- }\mathrm{C}}^{(3)}\) (see also (8.9)), respectively. Similarly, \(\eta _{V0}\) is to be replaced by \(\eta _{\mathrm{B}\text{- }\mathrm{C}\, V0}\), which is defined as \(\eta _{V0}\), where \({{\varvec{\upeta }}}_{\mathrm{B}\text{- }\mathrm{C}\, 0}\) is used instead of \({{\varvec{\upeta }}}_0\) (for the actual expressions of \(\eta _{\mathrm{B}\text{- }\mathrm{C}\, V0}\), see Ogasawara 2014, Section (e)). From the \(t_{\mathrm{B}\text{- }\mathrm{C}\, V}\) version of the expectation of \(t_V\) (see (8.4))
$$\begin{aligned} \text{ E }(t_{\mathrm{B}\text{- }\mathrm{C}\, V} )\!&= \!N^{-1/2}\left\{ {\!-\!\frac{1}{2}\text{ tr }\left( {\frac{\partial ^{2}\alpha _0}{\partial {{\varvec{\uppi }}}_{\mathrm{T}} {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}{{\varvec{\Omega }}}_{\mathrm{T}} } \right) } \right\} v_{0\alpha \alpha }^{-1/2} \!+\!{{\varvec{\uplambda }}}^{(1)} {}^{\prime }v_{0\alpha \alpha }^{-1/2} \text{ E }(\mathbf{m}^{(1)}) \nonumber \\&\!+\!N^{-1/2}({{\varvec{\uplambda }}}^{(2)} {}^{\prime }v_{0\alpha \alpha }^{-1/2} \!+\!{{\varvec{\uplambda }}}^{(1)} {}^{\prime }\otimes {{\varvec{\uplambda }}}_{V}^{(1)} {}^{\prime })\text{ E }(\mathbf{m}^{(1)\langle 2 \rangle })\!+\!O(N^{-3/2}) \nonumber \\ \!&= \!N^{-1/2}({{\varvec{\uplambda }}}^{(1)} {}^{\prime }\otimes {{\varvec{\uplambda }}}_{V}^{(1)} {}^{\prime })\text{ vec }({{\varvec{\Omega }}}_{\mathrm{T}})\!+\!O(N^{-3/2})\nonumber \\ \!&= \!N^{-1/2}{{\varvec{\uplambda }}}^{(1)} {}^{\prime }{{\varvec{\Omega }}}_{\mathrm{T}} {{\varvec{\uplambda }}}_{V}^{(1)} \!+\!O(N^{-3/2})\equiv N^{-1/2}{\beta _{\mathrm{B}\text{- }\mathrm{C\, W}1V}}^{\prime }\nonumber \\&+O(N^{-3/2}),\quad \quad \end{aligned}$$
(8.10)
where \(\text{ vec }(\cdot )\) is the vectorizing operator stacking columns of a matrix sequentially. Noting that \({{\varvec{\upeta }}}_{\mathrm{B}\text{- }\mathrm{C}\, 0}\) and, consequently, \(\eta _{\mathrm{B}\text{- }\mathrm{C}\, V0}\) are common to those by ML and BM, from (8.10) and the similar results in Theorem 2 (see (4.4)), the remaining asymptotic cumulants of \(t_{\mathrm{B}\text{- }\mathrm{C}\, V}\) are given as in (5.8).
1.5 Asymptotic cumulants of the bias-prevented estimator
Define \({\hat{\alpha }}_{\mathrm{B}\text{- }\mathrm{P}}\) that satisfies the condition:
$$\begin{aligned} \frac{\partial \hat{{l}}_{\mathrm{B}\text{- }\mathrm{P}} }{\partial {\hat{{\varvec{\upalpha }}}}_{\mathrm{B}\text{- }\mathrm{P}} }\equiv \frac{\partial \hat{{l}}_{\mathrm{ML}}}{\partial {\hat{{\varvec{\upalpha }}}}_{\mathrm{B}\text{- }\mathrm{P}} }+N^{-1}{\hat{\mathbf{q}}}_{\mathrm{B}\text{- }\mathrm{P}}^*=\mathbf{0}, \end{aligned}$$
(8.11)
where
$$\begin{aligned} \frac{\partial \hat{{l}}_{\mathrm{ML}}}{\partial {\hat{{\varvec{\upalpha }}}}_{\mathrm{B}\text{- }\mathrm{P}}}&= \sum _{k=1}^K {\frac{p_k }{\hat{{\pi }}_k}} \frac{\partial \pi _k }{\partial {\varvec{\upalpha }}}|_{{\varvec{\upalpha }}={\hat{{\varvec{\upalpha }}}}_{\mathrm{B}\text{- }\mathrm{P}}}, \hat{{\pi }}_k =\pi _k ({\hat{{\varvec{\upalpha }}}}_{\mathrm{B}\text{- }\mathrm{P}} ), {\hat{\mathbf{q}}}_{\mathrm{B}\text{- }\mathrm{P}}^*=\mathbf{q}^{*}({\hat{{\varvec{\upalpha }}}}_{\mathrm{B}\text{- }\mathrm{P}}, \mathbf{p})={\hat{{\varvec{\Lambda }}}}_{\mathrm{B}\text{- }\mathrm{P}} {\hat{{\varvec{\upbeta }}}}_{\mathrm{B}\text{- }\mathrm{P} 1}, \nonumber \\ {\hat{{\varvec{\Lambda }}}}_{\mathrm{B}\text{- }\mathrm{P}}&= \left. \frac{\partial ^{2}l_{\mathrm{ML}} }{\partial {\varvec{\upalpha }} {\partial {\varvec{\upalpha }}}^{\prime }}\right| _{\mathop {{\varvec{\upalpha }}={\hat{{\varvec{\upalpha }}}}_{\mathrm{B}\text{- }\mathrm{P}}}\limits _{\mathbf{p}=\mathbf{p}}} \text{ and } ({\hat{{\varvec{\upbeta }}}}_{\mathrm{B}\text{- }\mathrm{P}1})_{(\alpha )} ={\hat{\beta }}_{\mathrm{B}\text{- }\mathrm{P}1} =\frac{1}{2}\text{ tr }\left( {\frac{\partial ^{2}{\hat{\alpha }}_{\mathrm{B}\text{- }\mathrm{P}} }{\partial \mathbf{p} \partial \mathbf{p}^{\prime }}|_{\mathbf{p}=\mathbf{p}} {\hat{{\varvec{\Omega }}}}_{\mathrm{T}} } \right) .\nonumber \\ \end{aligned}$$
(8.12)
Theorem 6
Let \(w_{\mathrm{B}\text{- }\mathrm{P}}^*\equiv N^{1/2}({\hat{\alpha }}_{\mathrm{B}\text{- }\mathrm{P}} -\alpha _0)\) Then, the cumulants of \(w_{\mathrm{B}\text{- }\mathrm{P}}^*\) up to the fourth order are written as
$$\begin{aligned} \kappa _1 (w_{\mathrm{B}\text{- }\mathrm{P}}^*)&= O(N^{-3/2}) (\beta _{\mathrm{B}\text{- }\mathrm{P}1} =0), \nonumber \\ \kappa _2 (w_{\mathrm{B}\text{- }\mathrm{P}}^*)&= \beta _{\mathrm{ML}2} +N^{-1}\left( {\beta _{\mathrm{ML}\Delta 2} +2\frac{\partial \alpha _0 }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}{{\varvec{\Omega }}}_{\mathrm{T}} \frac{\partial \alpha _{\Delta \mathrm{B}\text{- }\mathrm{P}} }{\partial {{\varvec{\uppi }}}_{\mathrm{T}} }} \right) +O(N^{-2}) \nonumber \\&\left( {\beta _{\mathrm{B}\text{- }\mathrm{P} 2} =\beta _{\mathrm{ML}2} , \beta _{\mathrm{B}\text{- }\mathrm{P} \Delta 2} =\beta _{\mathrm{ML}\Delta 2} +2\frac{\partial \alpha _0 }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}{{\varvec{\Omega }}}_{\mathrm{T}} \frac{\partial \alpha _{\Delta \mathrm{B}\text{- }\mathrm{P}} }{\partial {{\varvec{\uppi }}}_{\mathrm{T}} }}\right) , \nonumber \\ \kappa _3 (w_{\mathrm{B}\text{- }\mathrm{P}}^*)&= N^{-1/2}\beta _{\mathrm{ML}3} +O(N^{-3/2}) (\beta _{\mathrm{B}\text{- }\mathrm{P}3} =\beta _{\mathrm{ML}3}) \nonumber \\ and\, \kappa _4 (w_{\mathrm{B}\text{- }\mathrm{P}}^*)&= N^{-1}\beta _{\mathrm{ML}4} +O(N^{-2}) (\beta _{\mathrm{B}\text{- }\mathrm{P} 4} =\beta _{\mathrm{ML}4} ). \end{aligned}$$
(8.13)
Proof
Define \({{\varvec{\upalpha }}}_{\mathrm{B}\text{- }\mathrm{P}}\) as a special case of \({{\varvec{\upalpha }}}_{\mathrm{W}}\) given by (2.5), and \(\mathbf{q}_{\mathrm{B}\text{- }\mathrm{P}0}^*={{\varvec{\Lambda }}} {{\varvec{\upbeta }}}_{\mathrm{ML}1}\) as a special case of \(\mathbf{q}_0^*\) (see (2.6)). Then,
$$\begin{aligned} {{\varvec{\upalpha }}}_{\mathrm{B}\text{- }\mathrm{P}}&= {{\varvec{\upalpha }}}_0 -N^{-1}\left( {\frac{\partial ^{2}l_{\mathrm{ML}} }{\partial {{\varvec{\upalpha }}}_0 {\partial {{\varvec{\upalpha }}}_0}^{\prime }}} \right) ^{-1}\mathbf{q}_{\mathrm{B}\text{- }\mathrm{P}0}^*+O(N^{-2})\nonumber \\&= {{\varvec{\upalpha }}}_0 - N^{-1}{{\varvec{\Lambda }}}^{-1}{{\varvec{\Lambda }} {\varvec{\upbeta }}}_{\mathrm{ML}1} +O(N^{-2})={{\varvec{\upalpha }}}_0 -N^{-1} {{\varvec{\upbeta }}}_{\mathrm{ML}1} +O(N^{-2}).\nonumber \\ \end{aligned}$$
(8.14)
Similarly, define \(\eta _{\mathrm{B}\text{- }\mathrm{P}\, 0}\) as a special case of \(\eta _{0}\) in (3.7):
$$\begin{aligned} \eta _{\mathrm{B}\text{- }\mathrm{P}\, 0} = -({{\varvec{\Lambda }}}^{-1}\mathbf{q}_{\mathrm{B}\text{- }\mathrm{P}\, 0}^*)_{(\alpha )} =-({{\varvec{\upbeta }}}_{\mathrm{ML}1} )_{(\alpha )} =-\beta _{\mathrm{ML}1}. \end{aligned}$$
(8.15)
Then, as before,
$$\begin{aligned} w_{\mathrm{B}\text{- }\mathrm{P}}^*&= N^{1/2}({\hat{\alpha }}_{\mathrm{B}\text{- }\mathrm{P}} -\alpha _0)=-N^{-1/2}\beta _{\mathrm{ML}1} +{{\varvec{\uplambda }}}^{(1)} {}^{\prime }\mathbf{m}^{(1)}+N^{-1/2}{{\varvec{\uplambda }}}^{(2)} {}^{\prime }\mathbf{m}^{(2)} \nonumber \\&+N^{-1}{{\varvec{\uplambda }}}_{\mathrm{B}\text{- }\mathrm{P}}^{(3)} {}^{\prime }\mathbf{m}^{(3)}+O_p (N^{-3/2}), \end{aligned}$$
(8.16)
where \({{\varvec{\uplambda }}}_{\mathrm{B}\text{- }\mathrm{P}}^{(3)} =\left[ {\left\{ {\frac{1}{6}\left( {\frac{\partial }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}} \right) ^{\langle 3 \rangle }\alpha _0 } \right\} , \frac{\partial \alpha _{\Delta \mathrm{B}\text{- }\mathrm{P}} }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}} \right] ^{{\prime }}\) with \(\frac{\partial \alpha _{\Delta \mathrm{B}\text{- }\mathrm{P}} }{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}\) being defined below.
Note that for the Bayes estimator \({\hat{\alpha }}_{\mathrm{W}}, \mathbf{q}^{*}=\mathbf{q}^{*}({\varvec{\upalpha }})\) is a function of \({\varvec{\upalpha }}\), while for \({\hat{\alpha }}_{\mathrm{B}\text{- }\mathrm{P}},\, \mathbf{q}_{\mathrm{B}\text{- }\mathrm{P}}^*=\mathbf{q}^{*}({\varvec{\upalpha }},\mathbf{p})\) is a function of p as well as \({\varvec{\upalpha }}\). Then,
$$\begin{aligned} \frac{\partial {{\varvec{\upalpha }}}_{\mathrm{B}\text{- }\mathrm{P}} }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}&= -\left( {\frac{\partial ^{2}l_{\mathrm{B}\text{- }\mathrm{P}} }{\partial {{\varvec{\upalpha }}}_{\mathrm{B}\text{- }\mathrm{P}}\partial {{\varvec{\upalpha }}}_{\mathrm{B}\text{- }\mathrm{P}} {\prime }}} \right) ^{-1}\frac{\partial ^{2}l_{\mathrm{B}\text{- }\mathrm{P}} }{\partial {{\varvec{\upalpha }}}_{\mathrm{B}\text{- }\mathrm{P}} {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }} \nonumber \\&= -\left( {\frac{\partial ^{2}l_{\mathrm{ML}} }{\partial {{\varvec{\upalpha }}}_{\mathrm{B}\text{- }\mathrm{P}} {\partial {{\varvec{\upalpha }}}_{\mathrm{B}\text{- }\mathrm{P}}}^{\prime }}+N^{-1}\frac{\partial \mathbf{q}_{\mathrm{B}\text{- }\mathrm{P}}^*}{{\partial {{\varvec{\upalpha }}}_{\mathrm{B}\text{- }\mathrm{P}}}^{\prime }}} \right) ^{-1}\left( {\frac{\partial ^{2}l_{\mathrm{ML}}}{\partial {{\varvec{\upalpha }}}_{\mathrm{B}\text{- }\mathrm{P}} {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}+N^{-1}\frac{\partial \mathbf{q}_{\mathrm{B}\text{- }\mathrm{P}}^*}{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}} \right) ,\nonumber \\ \end{aligned}$$
(8.17)
where the term \(N^{-1}\partial \mathbf{q}_{\mathrm{B}\text{- }\mathrm{P}}^*/{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }\) has been added (see (3.2)). In (8.17),
$$\begin{aligned} \frac{\partial ^{2}l_{\mathrm{ML}}}{\partial {{\varvec{\upalpha }}}_{\mathrm{B}\text{- }\mathrm{P}} {\partial {{\varvec{\upalpha }}}_{\mathrm{B}\text{- }\mathrm{P}}}^{\prime }}\!&= \!\frac{\partial ^{2}l_{\mathrm{ML}} }{\partial {{\varvec{\upalpha }}}_0 {\partial {{\varvec{\upalpha }}}_0}^{\prime }}\!+\!\sum _{i=1}^q {\frac{\partial ^{3}l_{\mathrm{ML}} }{\partial {{\varvec{\upalpha }}}_0 {\partial {{\varvec{\upalpha }}}_0}^{\prime }\partial \alpha _{0i} }} (\alpha _{\mathrm{B}\text{- }\mathrm{P} i} \!-\!\alpha _{0i} )\!+\!O(N^{-2}) \nonumber \\ \!&= \!\frac{\partial ^{2}l_{\mathrm{ML}} }{\partial {{\varvec{\upalpha }}}_0 {\partial {{\varvec{\upalpha }}}_0}^{\prime }}\!-\!N^{-1}\sum _{i=1}^q {\frac{\partial ^{3}l_{\mathrm{ML}} }{\partial {{\varvec{\upalpha }}}_0 {\partial {{\varvec{\upalpha }}}_0}^{\prime }\partial \alpha _{0i} }} \left\{ {\left( {\frac{\partial ^{2}l_{\mathrm{ML}} }{\partial {{\varvec{\upalpha }}}_0 {\partial {{\varvec{\upalpha }}}_0}^{\prime }}} \right) ^{-1}\mathbf{q}_{\mathrm{B}\text{- }\mathrm{P}\, 0}^*} \right\} _i\!+\!O(N^{-2}) \nonumber \\ \!&= \!\frac{\partial ^{2}l_{\mathrm{ML}} }{\partial {{\varvec{\upalpha }}}_0 {\partial {{\varvec{\upalpha }}}_0}^{\prime }}\!-\!N^{-1}\sum _{i=1}^q {\frac{\partial ^{3}l_{\mathrm{ML}} }{\partial {{\varvec{\upalpha }}}_0 {\partial {{\varvec{\upalpha }}}_0}^{\prime }\partial \alpha _{0i} }} ({{\varvec{\upbeta }}}_{\mathrm{ML}1} )_i \!+\! O(N^{-2}), \nonumber \\ \frac{\partial \mathbf{q}_{\mathrm{B}\text{- }\mathrm{P}}^*}{{\partial {{\varvec{\upalpha }}}_{\mathrm{B}\text{- }\mathrm{P}}}^{\prime }}\!&= \!\frac{\partial \mathbf{q}_{\mathrm{B}\text{- }\mathrm{P}}^*}{{\partial {{\varvec{\upalpha }}}_0}^{\prime }}\!+\!O(N^{-1})\nonumber \\ \frac{\partial ^{2}l_{\mathrm{ML}} }{\partial {{\varvec{\upalpha }}}_{\mathrm{B}\text{- }\mathrm{P}} {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}\!&= \!\frac{\partial ^{2}l_{\mathrm{ML}} }{\partial {{\varvec{\upalpha }}}_0 {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}+N^{-1}\sum _{i=1}^q {\frac{\partial ^{3}l_{\mathrm{ML}} }{\partial {{\varvec{\upalpha }}}_0 {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }\partial \alpha _{0i} }} (\alpha _{\mathrm{B}\text{- }\mathrm{P} i} \!-\!\alpha _{0i} )\!+\!O(N^{-2}) \nonumber \\ \!&= \!\frac{\partial ^{2}l_{\mathrm{ML}} }{\partial {{\varvec{\upalpha }}}_0 {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}\!-\!N^{-1}\sum _{i=1}^q {\frac{\partial ^{3}l_{\mathrm{ML}} }{\partial {{\varvec{\upalpha }}}_0 {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }\partial \alpha _{0i} }} ({{\varvec{\upbeta }}}_{\mathrm{ML}1} )_i \!+\!O(N^{-2}). \end{aligned}$$
(8.18)
From the above results, we have
$$\begin{aligned} \frac{\partial {{\varvec{\upalpha }}}_{\mathrm{B}\text{- }\mathrm{P}} }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}\!&= \!-{{\varvec{\Lambda }}}^{-1}\frac{\partial ^{2}l_{\mathrm{ML}} }{\partial {{\varvec{\upalpha }}}_0 {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}\!+\!N^{-1}\left[ {{{\varvec{\Lambda }}}^{-1}\left\{ {\!-\!\sum _{i=1}^q {\frac{\partial ^{3}l_{\mathrm{ML}} }{\partial {{\varvec{\upalpha }}}_0 {\partial {{\varvec{\upalpha }}}_0}^{\prime }\partial \alpha _{0i} }} ({{\varvec{\upbeta }}}_{\mathrm{ML}1} )_i \!+\!\frac{\partial \mathbf{q}_{\mathrm{B}\text{- }\mathrm{P}}^*}{{\partial {{\varvec{\upalpha }}}_0}^{\prime }}} \right\} } \right. \nonumber \\&\left. \!\times {{\varvec{\Lambda }}}^{-1}\frac{\partial ^{2}l_{\mathrm{ML}} }{\partial {{\varvec{\upalpha }}}_0 {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}\!+\!{{\varvec{\Lambda }}}^{-1} {\left\{ {\sum _{i=1}^q {\frac{\partial ^{3}l_{\mathrm{ML}}}{\partial {{\varvec{\upalpha }}}_0 {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }\partial \alpha _{0i} }} ({{\varvec{\upbeta }}}_{\mathrm{ML}1})_i \!-\!\frac{\partial \mathbf{q}_{\mathrm{B}\text{- }\mathrm{P}}^*}{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}} \right\} }\right] \nonumber \\&\!+O(N^{-2}) \equiv \frac{\partial {{\varvec{\upalpha }}}_0 }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}\!+\!N^{-1}\frac{\partial {{\varvec{\upalpha }}}_{\Delta \mathrm{B}\text{- }\mathrm{P}} }{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}+O(N^{-2}), \end{aligned}$$
(8.19)
$$\begin{aligned} \left( {\frac{\partial \mathbf{q}_{\mathrm{B}\text{- }\mathrm{P}}^*}{{\partial {{\varvec{\upalpha }}}_0}^{\prime }}} \right) _{ij}&= \frac{\partial ^{3}l_{\mathrm{ML}} }{\partial \alpha _{0i} {\partial {{\varvec{\upalpha }}}_0}^{\prime }\partial \alpha _{0j}}{{\varvec{\upbeta }}}_{\mathrm{ML}1}, \nonumber \\ \left( {\frac{\partial \mathbf{q}_{\mathrm{B}\text{- }\mathrm{P}}^*}{{\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}} \right) _{ik}&= \frac{\partial ^{2}l_{\mathrm{ML}} }{\partial \alpha _{0i} {\partial {{\varvec{\upalpha }}}_0}^{\prime }}\frac{\partial {\hat{{\varvec{\upbeta }}}}_{\mathrm{ML}1} }{\partial p_k }|_{\mathbf{p}={{\varvec{\uppi }}}_{\mathrm{T}} } +\frac{\partial ^{3}l_{\mathrm{ML}}}{\partial \alpha _{0i} {\partial {{\varvec{\upalpha }}}_0}^{\prime }\partial \pi _{\mathrm{T}k} }{{\varvec{\upbeta }}}_{\mathrm{ML}1} \nonumber \\&= \frac{1}{2}\sum _{j^{*}=1}^q {\frac{\partial ^{3}l_{\mathrm{ML}} }{\partial \alpha _{0i} \partial \alpha _{0j^{*}} }} \text{ tr }\left( {\frac{\partial ^{3}\alpha _{0j^{*}} }{\partial {{\varvec{\uppi }}}_{\mathrm{T}} {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }\partial \pi _{\mathrm{T}k} }{{\varvec{\Omega }}}_{\mathrm{T}} +\frac{\partial ^{2}\alpha _{0j^{*}} }{\partial {{\varvec{\uppi }}}_{\mathrm{T}} {\partial {{\varvec{\uppi }}}_{\mathrm{T}}}^{\prime }}\frac{\partial {{\varvec{\Omega }}}_{\mathrm{T}} }{\partial \pi _{\mathrm{T}k} }} \right) \nonumber \\&+\frac{\partial ^{3}l_{\mathrm{ML}} }{\partial \alpha _{0i} {\partial {{\varvec{\upalpha }}}_0}^{\prime }\partial \pi _{\mathrm{T}k} }{{\varvec{\upbeta }}}_{\mathrm{ML}1} (i,j=1,\ldots ,q; k=1,\ldots ,K). \end{aligned}$$
(8.20)
(8.16) with (8.20) gives (8.13). \(\square \)
