minor

Author: mmkim1210

docs/src/advanced.md

@@ -35,7 +35,7 @@ Standard errors for our estimates are calculated using the Fisher information ma
 where ``\text{vech}\ \boldsymbol{\Gamma}_i`` creates an ``\frac{d(d+1)}{2} \times 1`` vector from ``\boldsymbol{\Gamma}_i`` by stacking its lower triangular part and ``\boldsymbol{U}_i = (\boldsymbol{I}_d \otimes \boldsymbol{K}_{nd} \otimes \boldsymbol{I}_n) (\boldsymbol{I}_{d^2} \otimes \text{vec}\ \boldsymbol{V}_i) \boldsymbol{D}_{d}``. Here, ``\boldsymbol{K}_{nd}`` is the ``nd \times nd`` [commutation matrix](https://en.wikipedia.org/wiki/Commutation_matrix) and ``\boldsymbol{D}_{d}`` the ``d^2 \times \frac{d(d+1)}{2}`` [duplication matrix](https://en.wikipedia.org/wiki/Duplication_and_elimination_matrices).
 
 # Special case: missing response
-In the setting of missing response, we let ``\boldsymbol{P}`` be the ``nd \times nd`` permutation matrix such that ``\boldsymbol{P} \cdot \text{vec}\ \boldsymbol{Y} = \begin{bmatrix} \boldsymbol{y}_{\text{obs}} \\ \boldsymbol{y}_{\text{mis}} \end{bmatrix}``, where ``\boldsymbol{y}_{\text{obs}}`` and ``\boldsymbol{y}_{\text{mis}}`` are vectors of observed and missing response values, respectively, in column-major order. If we also let ``\boldsymbol{P} \boldsymbol{\Omega} \boldsymbol{P}^T = \begin{bmatrix} \boldsymbol{\Omega}_{11} & \boldsymbol{\Omega}_{12} \\ \boldsymbol{\Omega}_{21} & \boldsymbol{\Omega}_{22} \end{bmatrix}`` and ``\boldsymbol{P} \cdot\text{vec}(\boldsymbol{X}\boldsymbol{B}) = \begin{bmatrix} \boldsymbol{\mu}_{1} \\ \boldsymbol{\mu}_{2} \end{bmatrix}`` such that conditional mean and variance are ``\boldsymbol{\mu}_2 + \boldsymbol{\Omega}_{21}\boldsymbol{\Omega}_{11}^{-1}(\boldsymbol{y}_{\text{obs}}-\boldsymbol{\mu}_1)`` and ``\boldsymbol{\Omega}_{22} - \boldsymbol{\Omega}_{21}\boldsymbol{\Omega}_{11}^{-1}\boldsymbol{\Omega}_{12}``, respectively, then the adjusted MM updates in each interation become
+In the setting of missing response, we let ``\boldsymbol{P}`` be the ``nd \times nd`` permutation matrix such that ``\boldsymbol{P} \cdot \text{vec}\ \boldsymbol{Y} = \begin{bmatrix} \boldsymbol{y}_{\text{obs}} \\ \boldsymbol{y}_{\text{mis}} \end{bmatrix}``, where ``\boldsymbol{y}_{\text{obs}}`` and ``\boldsymbol{y}_{\text{mis}}`` are vectors of observed and missing response values, respectively, in column-major order. If we also let ``\boldsymbol{P} \cdot\text{vec}(\boldsymbol{X}\boldsymbol{B}) = \begin{bmatrix} \boldsymbol{\mu}_{1} \\ \boldsymbol{\mu}_{2} \end{bmatrix}`` and ``\boldsymbol{P} \boldsymbol{\Omega} \boldsymbol{P}^T = \begin{bmatrix} \boldsymbol{\Omega}_{11} & \boldsymbol{\Omega}_{12} \\ \boldsymbol{\Omega}_{21} & \boldsymbol{\Omega}_{22} \end{bmatrix}`` such that conditional mean and variance of ``\boldsymbol{y}_{\text{mis}}`` are ``\boldsymbol{\mu}_2 + \boldsymbol{\Omega}_{21}\boldsymbol{\Omega}_{11}^{-1}(\boldsymbol{y}_{\text{obs}}-\boldsymbol{\mu}_1)`` and ``\boldsymbol{\Omega}_{22} - \boldsymbol{\Omega}_{21}\boldsymbol{\Omega}_{11}^{-1}\boldsymbol{\Omega}_{12}``, respectively, then the adjusted MM updates in each interation become
 ```math
 \begin{aligned}
 \text{vec}\ \boldsymbol{B}^{(t)} &= [(\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} (\boldsymbol{I}_d \otimes \boldsymbol{X})]^{-1} (\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} \text{vec}\ \boldsymbol{Z}^{(t)} \\