Further remarks on constrained over-parameterized linear models

Abstract

This paper considers comparison problems for dispersion matrices of predictors under two competing linear models with having the same restrictions on their joint unknown parameters. One of the competing model is a constrained linear model (CLM) and the other one is a constrained over-parameterized model (COLM), obtained by adding new regressors to the CLM. After converting explicitly CLM and its COLM to their implicitly constrained forms, analytical expressions and properties of the best linear unbiased predictors (BLUPs) are given via using some quadratic matrix optimization methods under the models. In particular, the authors provide some equalities and inequalities for dispersion matrices of BLUPs under the models by using some formulas of ranks and inertias of block matrices. Comparison results on the dispersion matrices are also derived for special cases.

Appendix A

Formulas of inertia and rank of block matrices, used in the proofs of the results in the paper, are given in the following two lemmas; see, Tian (2010).

Lemma A.1

Let ${\textbf{C}}_{1}$, ${\textbf{C}}_{2}$ $\in {\mathbb {R}}^{k\times n}$, or, let ${\textbf{C}}_{1}={\textbf{C}}_{1}'$, ${\textbf{C}}_{2}={\textbf{C}}_{2}'$ $\in {\mathbb {R}}^{k\times k}$. Then,

$$\begin{aligned}{} & {} {\varvec{r}}({\textbf{C}}_{1}-{\textbf{C}}_{2})=0 \Leftrightarrow {\textbf{C}}_{1}={\textbf{C}}_{2}. \end{aligned}$$

(A1)

$$\begin{aligned}{} & {} {\varvec{i}}_{-}({\textbf{C}}_{1}-{\textbf{C}}_{2})=k \Leftrightarrow {\textbf{C}}_{1} \prec {\textbf{C}}_{2} \hbox { and } {\varvec{i}}_{+}({\textbf{C}}_{1}-{\textbf{C}}_{2})=k \Leftrightarrow {\textbf{C}}_{1} \succ {\textbf{C}}_{2}. \end{aligned}$$

(A2)

$$\begin{aligned}{} & {} {\varvec{i}}_{-}({\textbf{C}}_{1}-{\textbf{C}}_{2})=0 \Leftrightarrow {\textbf{C}}_{1}\succcurlyeq {\textbf{C}}_{2} \hbox { and } {\varvec{i}}_{+}({\textbf{C}}_{1}-{\textbf{C}}_{2})=0 \Leftrightarrow {\textbf{C}}_{1}\preccurlyeq {\textbf{C}}_{2}. \end{aligned}$$

(A3)

Lemma A.2

Let ${\textbf{C}}={\textbf{C}}' \in {\mathbb {R}}^{k\times k}$, ${\textbf{D}}={\textbf{D}}' \in {\mathbb {R}}^{n\times n}$, ${\textbf{P}}\in {\mathbb {R}}^{k\times n}$, and $t \in {\mathbb {R}}$. Then,

$$\begin{aligned}{} & {} {\varvec{r}}({\textbf{C}})={\varvec{i}}_{+}({\textbf{C}})+{\varvec{i}}_{-}({\textbf{C}}). \end{aligned}$$

(A4)

$$\begin{aligned}{} & {} {\varvec{i}}_{\pm }(t{\textbf{C}})={\varvec{i}}_{\pm }({\textbf{C}}) \hbox { if } t>0 \; \hbox { and } \; {\varvec{i}}_{\pm }(t{\textbf{C}})={\varvec{i}}_{\mp }({\textbf{C}}) \hbox { if } t<0. \end{aligned}$$

(A5)

$$\begin{aligned}{} & {} {\varvec{i}}_{\pm }\begin{bmatrix}{\textbf{C}}&{} {\textbf{P}}\\ {\textbf{P}}' &{} {\textbf{D}}\end{bmatrix}={\varvec{i}}_{\pm }\begin{bmatrix}{\textbf{C}}&{} -{\textbf{P}}\\ -{\textbf{P}}' &{} {\textbf{D}}\end{bmatrix}={\varvec{i}}_{\mp }\begin{bmatrix}-{\textbf{C}}&{} {\textbf{P}}\\ {\textbf{P}}' &{} -{\textbf{D}}\end{bmatrix}. \end{aligned}$$

(A6)

$$\begin{aligned}{} & {} {\varvec{i}}_{\pm }\begin{bmatrix}{\textbf{C}}&{} {\textbf{0}}\\ {\textbf{0}}&{} {\textbf{D}}\end{bmatrix}={\varvec{i}}_{\pm }({\textbf{C}})+{\varvec{i}}_{\pm }({\textbf{D}}) \; \hbox { and } \; {\varvec{i}}_{+}\begin{bmatrix}{\textbf{0}}&{} {\textbf{P}}\\ {\textbf{P}}' &{} {\textbf{0}}\end{bmatrix}={\varvec{i}}_{-}\begin{bmatrix}{\textbf{0}}&{} {\textbf{P}}\\ {\textbf{P}}' &{} {\textbf{0}}\end{bmatrix}={\varvec{r}}({\textbf{P}}). \end{aligned}$$

(A7)

$$\begin{aligned}{} & {} {\varvec{i}}_{\pm }\begin{bmatrix}{\textbf{C}}&{} {\textbf{P}}\\ {\textbf{P}}' &{} {\textbf{0}}\end{bmatrix}={\varvec{r}}({\textbf{P}})+{\varvec{i}}_{\pm }({\textbf{E}}_{{\textbf{P}}}{\textbf{C}}{\textbf{E}}_{{\textbf{P}}}). \end{aligned}$$

(A8)

$$\begin{aligned}{} & {} {\varvec{i}}_{+}\begin{bmatrix}{\textbf{C}}&{} {\textbf{P}}\\ {\textbf{P}}' &{} {\textbf{0}}\end{bmatrix}={\varvec{r}}\begin{bmatrix}{\textbf{C}},&{\textbf{P}}\end{bmatrix} \;\; \hbox { and } \;\; {\varvec{i}}_{-}\begin{bmatrix}{\textbf{C}}&{} {\textbf{P}}\\ {\textbf{P}}' &{} {\textbf{0}}\end{bmatrix}={\varvec{r}}({\textbf{P}}) \;\; \hbox { if } \;\; {\textbf{C}}\succcurlyeq {\textbf{0}}. \end{aligned}$$

(A9)

$$\begin{aligned}{} & {} {\varvec{i}}_{\pm }\begin{bmatrix}{\textbf{C}}&{} {\textbf{P}}\\ {\textbf{P}}' &{} {\textbf{D}}\end{bmatrix}={\varvec{i}}_{\pm }({\textbf{C}})+ {\varvec{i}}_{\pm }({\textbf{D}}-{\textbf{P}}'{\textbf{C}}^{+}{\textbf{P}}) \;\; \hbox { if } \;\; {\mathscr {C}}({\textbf{P}}) \subseteq {\mathscr {C}}({\textbf{C}}). \end{aligned}$$

(A10)

The constrained quadratic matrix-valued function optimization problem related to minimization problems on dispersion matrices of predictors is given in the following lemma; see Tian (2012).

Lemma A.3

Let ${\textbf{K}}\in {\mathbb {R}}^{n\times p}$ and ${\textbf{D}}\in {\mathbb {R}}^{m\times p}$ be given matrices, and let ${\textbf{Q}}\in {\mathbb {R}}^{n\times n}$ be a symmetric positive semi-definite matrix. Assume that there exists ${\textbf{X}}_{0} \in {\mathbb {R}}^{m\times n}$ such that ${\textbf{X}}_{0}{\textbf{K}}={\textbf{D}}$. Then the maximal positive inertia of ${\textbf{X}}_{0}{\textbf{Q}}{\textbf{X}}_{0}'-{\textbf{X}}{\textbf{Q}}{\textbf{X}}'$ subject to all solutions of ${\textbf{X}}{\textbf{K}}={\textbf{D}}$ is

$$\begin{aligned} \max _{{\textbf{X}}{\textbf{K}}={\textbf{D}}} {\varvec{i}}_{+}({\textbf{X}}_{0}{\textbf{Q}}{\textbf{X}}_{0}'-{\textbf{X}}{\textbf{Q}}{\textbf{X}}') ={\varvec{r}}\begin{bmatrix} {\textbf{X}}_{0}{\textbf{Q}}\\ {\textbf{K}}' \end{bmatrix}-{\varvec{r}}({\textbf{K}}) ={\varvec{r}}({\textbf{X}}_{0}{\textbf{Q}}{\textbf{K}}^{\perp }). \end{aligned}$$

Hence there exists solution ${\textbf{X}}_{0}$ of ${\textbf{X}}_{0}{\textbf{K}}={\textbf{D}}$ such that ${\textbf{X}}_{0}{\textbf{Q}}{\textbf{X}}_{0}'\preccurlyeq {\textbf{X}}{\textbf{Q}}{\textbf{X}}'$ holds for all solutions of ${\textbf{X}}{\textbf{K}}={\textbf{D}}$ $\Leftrightarrow$ ${\textbf{X}}_{0}$ satisfies both ${\textbf{X}}_{0}{\textbf{K}}={\textbf{D}}$ and ${\textbf{X}}_{0}{\textbf{Q}}{\textbf{K}}^{\perp }={\textbf{0}}$.

Proof of Lemma 1

Let ${\textbf{T}}{{\widehat{{\textbf{y}}}}}$ be an unbiased linear predictor for ${\textbf{n}}$ in ${{\widehat{{\mathcal {N}}}}}$. Then,

$$\begin{aligned}&{{\,\textrm{E}\,}}({\textbf{T}}{{\widehat{{\textbf{y}}}}}-{\textbf{n}}) ={\textbf{0}}\Leftrightarrow {\textbf{T}}\begin{bmatrix}{{\widehat{{\textbf{X}}}}},&{{\widehat{{\textbf{A}}}}}\end{bmatrix}={\widehat{{\textbf{N}}}}, \; \hbox { i.e., } \;\begin{bmatrix}{\textbf{T}},&-{\textbf{I}}_{t}\end{bmatrix}\begin{bmatrix}{{\widehat{{\textbf{X}}}}} &{} {{\widehat{{\textbf{A}}}}}\\ {\textbf{N}}&{} {\textbf{0}}\end{bmatrix}&={\textbf{0}}, \end{aligned}$$

(A11)

$$\begin{aligned}&{{\,\textrm{D}\,}}({\textbf{T}}{{\widehat{{\textbf{y}}}}}-{\textbf{n}}) =\varvec{\sigma }^2({\textbf{T}}-{\textbf{R}}){{\widehat{\varvec{\Gamma }}}}({\textbf{T}}-{\textbf{R}})' \nonumber \\&\quad =\varvec{\sigma }^2\begin{bmatrix}{\textbf{T}},&-{\textbf{I}}_{t}\end{bmatrix}\begin{bmatrix}{\textbf{I}}_{n+m}\\ {\textbf{R}}\end{bmatrix}{{\widehat{\varvec{\Gamma }}}}\begin{bmatrix}{\textbf{I}}_{n+m}\\ {\textbf{R}}\end{bmatrix}'\begin{bmatrix}{\textbf{T}},&-{\textbf{I}}_{t}\end{bmatrix}':={\varvec{f}}({\textbf{T}}). \end{aligned}$$

(A12)

The similar expressions as in (A11) and (A12) can be written for the other unbiased predictor ${\textbf{S}}{{\widehat{{\textbf{y}}}}}$ by setting ${\textbf{S}}$ instead of ${\textbf{T}}$. Finding solution ${\textbf{T}}$ of the consistent equation ${\textbf{T}}\begin{bmatrix}{{\widehat{{\textbf{X}}}}},&{{\widehat{{\textbf{A}}}}}\end{bmatrix}={\widehat{{\textbf{N}}}}$ such that

$$\begin{aligned} \begin{aligned} {\varvec{f}}({\textbf{T}}) \preccurlyeq {\varvec{f}}({\textbf{S}}) \; \hbox { s.t. } \; {\textbf{S}}\begin{bmatrix}{{\widehat{{\textbf{X}}}}},&{{\widehat{{\textbf{A}}}}}\end{bmatrix}={\widehat{{\textbf{N}}}} \end{aligned} \end{aligned}$$

(A13)

corresponds to find the ${{\,\textrm{BLUP}\,}}$ of ${\textbf{n}}$ under ${{\widehat{{\mathcal {N}}}}}$. (A13) is a standard constrained quadratic matrix-valued function optimization problem in the L${\ddot{o}}$wner partial ordering as given in Lemma A.3. Applying this lemma to (A13), (9) is obtained and also from Lemma A.3, we obtain the fundamental equation of ${{\,\textrm{BLUP}\,}}$ of ${\textbf{n}}$ in (10). It is well-known that the general solution of (10) can be written as in (11). Item (1) follows from (11). The expressions in item (2) are well-known results; see also (Tian 2013, Lemma 2.1(a)). The equalities in item (3) is seen from (7) and (11). $\square$

Proof of Theorem 1

Let ${\textbf{G}}=\begin{bmatrix}{{\widehat{{\textbf{X}}}}},&{{\widehat{{\textbf{A}}}}} \end{bmatrix}$ and ${\textbf{H}}={{\,\textrm{D}\,}}[{{\textbf{n}}-{{\,\textrm{BLUP}\,}}_{{{\widehat{{\mathcal {M}}}}}}({\textbf{n}})}]$. Using (13) and (A10),

$$\begin{aligned}&{\varvec{i}}_{\pm }({\textbf{H}}-{{\,\textrm{D}\,}}[{{\textbf{n}}-{{\,\textrm{BLUP}\,}}_{{{\widehat{{\mathcal {N}}}}}}({\textbf{n}})}])\nonumber \\&\quad ={\varvec{i}}_{\pm }\left( {\textbf{H}}-\left( \begin{bmatrix}{\widehat{{\textbf{N}}}},&{\textbf{R}}{{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp }\end{bmatrix}{\textbf{J}}_{n}^{+} -{\textbf{R}}\right) {{\widehat{\varvec{\Gamma }}}}{{\widehat{\varvec{\Gamma }}}}^{+}{{\widehat{\varvec{\Gamma }}}}\left( \begin{bmatrix}{\widehat{{\textbf{N}}}},&{\textbf{R}}{{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp }\end{bmatrix}{\textbf{J}}_{n}^{+} -{\textbf{R}}\right) '\right) \nonumber \\&\quad ={\varvec{i}}_{\pm }\begin{bmatrix} {{\widehat{\varvec{\Gamma }}}}&{{\widehat{\varvec{\Gamma }}}}\left( \begin{bmatrix}{\widehat{{\textbf{N}}}}, &{} {\textbf{R}}{{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp }\end{bmatrix}{\textbf{J}}_{n}^{+} -{\textbf{R}}\right) ' \\ \left( \begin{bmatrix}{\widehat{{\textbf{N}}}}, &{} {\textbf{R}}{{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp }\end{bmatrix}{\textbf{J}}_{n}^{+} -{\textbf{R}}\right) {{\widehat{\varvec{\Gamma }}}}&{\textbf{H}}\end{bmatrix}-{\varvec{i}}_{\pm }({{\widehat{\varvec{\Gamma }}}})\nonumber \\&\quad ={\varvec{i}}_{\pm }\left( \begin{bmatrix} {{\widehat{\varvec{\Gamma }}}} &{} -{{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' \\ -{\textbf{R}}{{\widehat{\varvec{\Gamma }}}} &{} {\textbf{H}}\end{bmatrix} +\begin{bmatrix} {{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}\\ {\textbf{0}}&{} \begin{bmatrix}{\widehat{{\textbf{N}}}}, &{} {\textbf{R}}{{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp }\end{bmatrix} \end{bmatrix} \begin{bmatrix} {\textbf{0}}&{} {\textbf{J}}_{n} \\ {\textbf{J}}_{n}' &{} {\textbf{0}}\end{bmatrix}^{+} \begin{bmatrix} {{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}\\ {\textbf{0}}&{} \begin{bmatrix}{\widehat{{\textbf{N}}}}, &{} {\textbf{R}}{{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp }\end{bmatrix}' \end{bmatrix}\right) \nonumber \\&\qquad -{\varvec{i}}_{\pm }({{\widehat{\varvec{\Gamma }}}}) \end{aligned}$$

(A14)

is obtained, where ${\textbf{J}}_{n}=\begin{bmatrix}{\textbf{G}},&{{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp }\end{bmatrix}$. It is seen from (12) and Definition 1 that ${\mathscr {C}}({{\widehat{\varvec{\Gamma }}}})\subseteq {\mathscr {C}}({\textbf{J}}_{n})$ and ${\mathscr {C}}\left( \begin{bmatrix}{\widehat{{\textbf{N}}}},&{\textbf{R}}{{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp }\end{bmatrix}'\right) \subseteq {\mathscr {C}}({\textbf{J}}_{n}')$ hold. Then, applying (A10) to (A14), using (A5), (A7), and (12) with the elementary block matrix operations, and setting ${\textbf{J}}_{n}$ into (A14), (A14) is equivalently written as follows

$$\begin{aligned}&{\varvec{i}}_{\pm }\begin{bmatrix} {\textbf{0}}&{} -{\textbf{G}}&{} -{{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp } &{} {{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}\\ -{\textbf{G}}' &{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}&{} {\widehat{{\textbf{N}}}}'\\ -{\textbf{G}}^{\perp }{{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{G}}^{\perp }{{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' \\ {{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}&{} {\textbf{0}}&{} {{\widehat{\varvec{\Gamma }}}} &{} -{{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' \\ {\textbf{0}}&{} {\widehat{{\textbf{N}}}} &{} {\textbf{R}}{{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp } &{} -{\textbf{R}}{{\widehat{\varvec{\Gamma }}}} &{} {\textbf{H}}\end{bmatrix}- {\varvec{i}}_{\mp }\begin{bmatrix} {\textbf{0}}&{} {\textbf{G}}&{} {{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp } \\ {\textbf{G}}' &{} {\textbf{0}}&{} {\textbf{0}}\\ {\textbf{G}}^{\perp }{{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}&{} {\textbf{0}}\end{bmatrix} -{\varvec{i}}_{\pm }({{\widehat{\varvec{\Gamma }}}})\nonumber \\&\quad ={\varvec{i}}_{\pm }\begin{bmatrix} -{{\widehat{\varvec{\Gamma }}}} &{} -{\textbf{G}}&{} -{{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp } &{} {{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' \\ -{\textbf{G}}' &{} {\textbf{0}}&{} {\textbf{0}}&{} {\widehat{{\textbf{N}}}}'\\ -{\textbf{G}}^{\perp }{{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{G}}^{\perp }{{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' \\ {\textbf{R}}{{\widehat{\varvec{\Gamma }}}} &{} {\widehat{{\textbf{N}}}} &{} {\textbf{R}}{{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp } &{} {\textbf{H}}-{\textbf{R}}{{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' \end{bmatrix}-{\varvec{r}}\begin{bmatrix} {\textbf{G}},&{{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp } \end{bmatrix}\nonumber \\&\quad ={\varvec{i}}_{\mp }\begin{bmatrix} {{\widehat{\varvec{\Gamma }}}} &{} {\textbf{G}}&{} {{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' \\ {\textbf{G}}' &{} {\textbf{0}}&{} {\widehat{{\textbf{N}}}}' \\ {\textbf{R}}{{\widehat{\varvec{\Gamma }}}} &{} {\widehat{{\textbf{N}}}} &{} -{\textbf{H}}+{\textbf{R}}{{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' \end{bmatrix}+ {\varvec{i}}_{\pm }({\textbf{G}}^{\perp }{{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp })-{\varvec{r}}\begin{bmatrix} {\textbf{G}},&{{\widehat{\varvec{\Gamma }}}} \end{bmatrix}\nonumber \\&\quad ={\varvec{i}}_{\mp }\left( \begin{bmatrix} {{\widehat{\varvec{\Gamma }}}} &{} {{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' &{} {\textbf{G}}\\ {\textbf{R}}{{\widehat{\varvec{\Gamma }}}} &{} {\textbf{R}}{{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' &{} {\widehat{{\textbf{N}}}} \\ {\textbf{G}}' &{} {\widehat{{\textbf{N}}}}' &{} {\textbf{0}}\end{bmatrix}-\begin{bmatrix} {\textbf{0}}\\ {\textbf{I}}_{t} \\ {\textbf{0}}\end{bmatrix}{\textbf{H}}\begin{bmatrix} {\textbf{0}}\\ {\textbf{I}}_{t} \\ {\textbf{0}}\end{bmatrix}'\right) + {\varvec{i}}_{\pm }({\textbf{G}}^{\perp }{{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp })-{\varvec{r}}\begin{bmatrix} {\textbf{G}},&{{\widehat{\varvec{\Gamma }}}} \end{bmatrix}. \end{aligned}$$

(A15)

We can reapply (A10) to (A15) after substituting ${\textbf{H}}={{\,\textrm{D}\,}}[{{\textbf{n}}-{{\,\textrm{BLUP}\,}}_{{{\widehat{{\mathcal {M}}}}}}({\textbf{n}})}]$ in (14). Then, by using a similar way to obtaining (A14), (A15) is equivalently written as

$$\begin{aligned}&{\varvec{i}}_{\mp }\left( \begin{bmatrix} {{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}&{} -{{\widehat{\varvec{\Gamma }}}}{\textbf{R}}'&{} {\textbf{0}}\\ {\textbf{0}}&{} {{\widehat{\varvec{\Gamma }}}} &{} {{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' &{} {\textbf{G}}\\ -{\textbf{R}}{{\widehat{\varvec{\Gamma }}}} &{} {\textbf{R}}{{\widehat{\varvec{\Gamma }}}} &{} {\textbf{R}}{{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' &{} {\widehat{{\textbf{N}}}} \\ {\textbf{0}}&{} {\textbf{G}}' &{} {\widehat{{\textbf{N}}}}' &{} {\textbf{0}}\end{bmatrix} +\begin{bmatrix} {{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}\\ {\textbf{0}}&{} {\textbf{0}}\\ {\textbf{0}}&{} \begin{bmatrix}{\textbf{N}}, &{} {\textbf{R}}{{\widehat{\varvec{\Gamma }}}}{{\widehat{{\textbf{X}}}}}^{\perp }\end{bmatrix} \\ {\textbf{0}}&{} {\textbf{0}}\end{bmatrix} \begin{bmatrix} {\textbf{0}}&{} {\textbf{J}}_{m} \\ {\textbf{J}}_{m}' &{} {\textbf{0}}\end{bmatrix}^{+} \right. \nonumber \\&\quad \left. \times \begin{bmatrix} {{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}\\ {\textbf{0}}&{} {\textbf{0}}&{} \begin{bmatrix}{\textbf{N}}, &{} {\textbf{R}}{{\widehat{\varvec{\Gamma }}}}{{\widehat{{\textbf{X}}}}}^{\perp }\end{bmatrix}'&{\textbf{0}}\end{bmatrix}\right) + {\varvec{i}}_{\pm }({\textbf{G}}^{\perp }{{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp })-{\varvec{r}}\begin{bmatrix} {\textbf{G}},&{{\widehat{\varvec{\Gamma }}}} \end{bmatrix}-{\varvec{i}}_{\mp }({{\widehat{\varvec{\Gamma }}}}), \end{aligned}$$

(A16)

where ${\textbf{J}}_{m}=\begin{bmatrix}{{\widehat{{\textbf{X}}}}},&{{\widehat{\varvec{\Gamma }}}}{{\widehat{{\textbf{X}}}}}^{\perp }\end{bmatrix}$. We can apply (A10) to (A16) since ${\mathscr {C}}({{\widehat{\varvec{\Gamma }}}})\subseteq {\mathscr {C}}({\textbf{J}}_{m})$ and ${\mathscr {C}}(\begin{bmatrix}{\textbf{N}},&{\textbf{R}}{{\widehat{\varvec{\Gamma }}}}{{\widehat{{\textbf{X}}}}}^{\perp }\end{bmatrix}')\subseteq {\mathscr {C}}({\textbf{J}}_{m}')$ hold. Then, by setting ${\textbf{J}}_{m}$, (A16) is equivalently written as

$$\begin{aligned}&{\varvec{i}}_{\mp }\begin{bmatrix} {\textbf{0}}&{} -{{\widehat{{\textbf{X}}}}} &{} -{{\widehat{\varvec{\Gamma }}}}{{\widehat{{\textbf{X}}}}}^{\perp } &{} {{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}\\ -{{\widehat{{\textbf{X}}}}}' &{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{N}}' &{} {\textbf{0}}\\ -{{\widehat{{\textbf{X}}}}}^{\perp }{{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}&{} {{\widehat{{\textbf{X}}}}}^{\perp }{{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' &{} {\textbf{0}}\\ {{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}&{} {\textbf{0}}&{} {{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}&{} -{{\widehat{\varvec{\Gamma }}}}{\textbf{R}}'&{} {\textbf{0}}\\ {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}&{} {{\widehat{\varvec{\Gamma }}}} &{} {{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' &{} {\textbf{G}}\\ {\textbf{0}}&{} {\textbf{N}}&{} {\textbf{R}}{{\widehat{\varvec{\Gamma }}}}{{\widehat{{\textbf{X}}}}}^{\perp } &{} -{\textbf{R}}{{\widehat{\varvec{\Gamma }}}} &{} {\textbf{R}}{{\widehat{\varvec{\Gamma }}}} &{} {\textbf{R}}{{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' &{} {\widehat{{\textbf{N}}}}\\ {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{G}}' &{} {\widehat{{\textbf{N}}}}' &{} {\textbf{0}}\end{bmatrix}- {\varvec{i}}_{\pm }\begin{bmatrix} {\textbf{0}}&{} {{\widehat{{\textbf{X}}}}} &{} {{\widehat{\varvec{\Gamma }}}}{{\widehat{{\textbf{X}}}}}^{\perp } \\ {{\widehat{{\textbf{X}}}}}' &{} {\textbf{0}}&{} {\textbf{0}}\\ {{\widehat{{\textbf{X}}}}}^{\perp }{{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}&{} {\textbf{0}}\end{bmatrix} \nonumber \\&\quad + {\varvec{i}}_{\pm }({\textbf{G}}^{\perp }{{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp })-{\varvec{r}}\begin{bmatrix} {\textbf{G}},&{{\widehat{\varvec{\Gamma }}}} \end{bmatrix}-{\varvec{i}}_{\mp }({{\widehat{\varvec{\Gamma }}}}). \end{aligned}$$

(A17)

Using (A5)–(A8), and some congruence operations, (A17) is equivalently written as

$$\begin{aligned}&{\varvec{i}}_{\mp }\begin{bmatrix} -{{\widehat{\varvec{\Gamma }}}} &{} -{{\widehat{{\textbf{X}}}}} &{} -{{\widehat{\varvec{\Gamma }}}}{{\widehat{{\textbf{X}}}}}^{\perp } &{} {\textbf{0}}&{} {{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' &{} {\textbf{0}}\\ -{{\widehat{{\textbf{X}}}}}' &{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{N}}' &{} {\textbf{0}}\\ -{{\widehat{{\textbf{X}}}}}^{\perp }{{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}&{} {{\widehat{{\textbf{X}}}}}^{\perp }{{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' &{} {\textbf{0}}\\ {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}&{} {{\widehat{\varvec{\Gamma }}}} &{} {{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' &{} {\textbf{G}}\\ {\textbf{R}}{{\widehat{\varvec{\Gamma }}}} &{} {\textbf{N}}&{} {\textbf{R}}{{\widehat{\varvec{\Gamma }}}}{{\widehat{{\textbf{X}}}}}^{\perp } &{} {\textbf{R}}{{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}&{} {\widehat{{\textbf{N}}}} \\ {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{G}}' &{} {\widehat{{\textbf{N}}}}' &{} {\textbf{0}}\end{bmatrix}- {\varvec{r}}\begin{bmatrix} {{\widehat{{\textbf{X}}}}},&{{\widehat{\varvec{\Gamma }}}}{{\widehat{{\textbf{X}}}}}^{\perp } \end{bmatrix} + {\varvec{i}}_{\pm }({\textbf{G}}^{\perp }{{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp }) \nonumber \\&\qquad -{\varvec{r}}\begin{bmatrix} {\textbf{G}},&{{\widehat{\varvec{\Gamma }}}} \end{bmatrix} \nonumber \\&\quad ={\varvec{i}}_{\mp }\begin{bmatrix} -{{\widehat{\varvec{\Gamma }}}} &{} -{{\widehat{{\textbf{X}}}}} &{} {\textbf{0}}&{} {{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' &{} {\textbf{0}}\\ -{{\widehat{{\textbf{X}}}}}' &{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{N}}' &{} {\textbf{0}}\\ {\textbf{0}}&{} {\textbf{0}}&{} {{\widehat{\varvec{\Gamma }}}} &{} {{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' &{} {\textbf{G}}\\ {\textbf{R}}{{\widehat{\varvec{\Gamma }}}} &{} {\textbf{N}}&{} {\textbf{R}}{{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}&{} {\widehat{{\textbf{N}}}} \\ {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{G}}' &{} {\widehat{{\textbf{N}}}}' &{} {\textbf{0}}\end{bmatrix}+{\varvec{i}}_{\mp }({{\widehat{{\textbf{X}}}}}^{\perp }{{\widehat{\varvec{\Gamma }}}}{{\widehat{{\textbf{X}}}}}^{\perp })- {\varvec{r}}\begin{bmatrix} {{\widehat{{\textbf{X}}}}},&{{\widehat{\varvec{\Gamma }}}} \end{bmatrix} + {\varvec{i}}_{\pm }({\textbf{G}}^{\perp }{{\widehat{\varvec{\Gamma }}}}{\textbf{G}}^{\perp })\nonumber \\&\qquad -{\varvec{r}}\begin{bmatrix} {\textbf{G}},&{{\widehat{\varvec{\Gamma }}}} \end{bmatrix} \nonumber \\&\quad ={\varvec{i}}_{\pm }\begin{bmatrix} {{\widehat{\varvec{\Gamma }}}} &{} {{\widehat{\varvec{\Gamma }}}} &{} {{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' &{} {\textbf{0}}&{} {{\widehat{{\textbf{X}}}}} \\ {{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{G}}&{} {\textbf{0}}\\ {\textbf{R}}{{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}&{} {\textbf{0}}&{} {\widehat{{\textbf{N}}}} &{} {\textbf{0}}\\ {\textbf{0}}&{} {\textbf{G}}' &{} {\widehat{{\textbf{N}}}}' &{} {\textbf{0}}&{} {\textbf{0}}\\ {{\widehat{{\textbf{X}}}}}' &{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}\end{bmatrix} +{\varvec{i}}_{\mp }\begin{bmatrix} {{\widehat{\varvec{\Gamma }}}} &{} {{\widehat{{\textbf{X}}}}}\\ {{\widehat{{\textbf{X}}}}}' &{} {\textbf{0}}\end{bmatrix} -{\varvec{r}}({{\widehat{{\textbf{X}}}}}) -{\varvec{r}}\begin{bmatrix} {{\widehat{{\textbf{X}}}}},&{{\widehat{\varvec{\Gamma }}}} \end{bmatrix} +{\varvec{i}}_{\pm }\begin{bmatrix} {{\widehat{\varvec{\Gamma }}}} &{} {\textbf{G}}\\ {\textbf{G}}' &{} {\textbf{0}}\end{bmatrix}\nonumber \\&\qquad -{\varvec{r}}({\textbf{G}})-{\varvec{r}}\begin{bmatrix} {\textbf{G}},&{{\widehat{\varvec{\Gamma }}}} \end{bmatrix}\end{aligned}$$

(A18)

$$\begin{aligned}&\quad ={\varvec{i}}_{\pm }\begin{bmatrix} {{\widehat{\varvec{\Gamma }}}} &{} {{\widehat{\varvec{\Gamma }}}} &{} {{\widehat{\varvec{\Gamma }}}}{\textbf{R}}' &{} {\textbf{0}}&{} {\textbf{0}}&{} {{\widehat{{\textbf{X}}}}} \\ {{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}&{} {\textbf{0}}&{} {{\widehat{{\textbf{X}}}}} &{} {{\widehat{{\textbf{A}}}}} &{} {\textbf{0}}\\ {\textbf{R}}{{\widehat{\varvec{\Gamma }}}} &{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{N}}&{} {\textbf{0}}&{} {\textbf{0}}\\ {\textbf{0}}&{} {{\widehat{{\textbf{X}}}}}' &{} {\textbf{N}}' &{} {\textbf{0}}&{} {\textbf{0}}&{}{\textbf{0}}\\ {\textbf{0}}&{} {{\widehat{{\textbf{A}}}}}' &{}{\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}&{}{\textbf{0}}\\ {{\widehat{{\textbf{X}}}}}' &{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}&{} {\textbf{0}}\end{bmatrix}+{\varvec{i}}_{\mp }\begin{bmatrix} {{\widehat{\varvec{\Gamma }}}} &{} {{\widehat{{\textbf{X}}}}}\\ {{\widehat{{\textbf{X}}}}}' &{} {\textbf{0}}\end{bmatrix} -{\varvec{r}}({{\widehat{{\textbf{X}}}}}) -{\varvec{r}}\begin{bmatrix} {{\widehat{{\textbf{X}}}}},&{{\widehat{\varvec{\Gamma }}}} \end{bmatrix} +{\varvec{i}}_{\pm }\begin{bmatrix} {{\widehat{\varvec{\Gamma }}}} &{} {{\widehat{{\textbf{X}}}}} &{} {{\widehat{{\textbf{A}}}}}\\ {{\widehat{{\textbf{X}}}}}' &{} {\textbf{0}}&{} {\textbf{0}}\\ {{\widehat{{\textbf{A}}}}}' &{} {\textbf{0}}&{} {\textbf{0}}\end{bmatrix}\nonumber \\&\qquad -{\varvec{r}}\begin{bmatrix} {{\widehat{{\textbf{X}}}}},&{{\widehat{{\textbf{A}}}}} \end{bmatrix}-{\varvec{r}}\begin{bmatrix} {{\widehat{{\textbf{X}}}}},&{{\widehat{{\textbf{A}}}}},&{{\widehat{\varvec{\Gamma }}}} \end{bmatrix}, \end{aligned}$$

(A19)

where (A19) is obtained by substituting ${\textbf{G}}$ in (A18). In consequence, by using (A9) and writing ${\textbf{S}}$ instead of the first matrix in (A19), we obtain the positive and negative inertia of difference ${{\,\textrm{D}\,}}[{{\textbf{n}}-{{\,\textrm{BLUP}\,}}_{{{\widehat{{\mathcal {M}}}}}}({\textbf{n}})}]-{{\,\textrm{D}\,}}[{{\textbf{n}}-{{\,\textrm{BLUP}\,}}_{{{\widehat{{\mathcal {N}}}}}}({\textbf{n}})}]$ as follows:

$$\begin{aligned} \begin{aligned} {\varvec{i}}_{+}({\textbf{S}})={\varvec{r}}\begin{bmatrix} {{\widehat{{\textbf{X}}}}},&{{\widehat{\varvec{\Gamma }}}} \end{bmatrix}-{\varvec{r}}\begin{bmatrix} {{\widehat{{\textbf{X}}}}},&{{\widehat{{\textbf{A}}}}} \end{bmatrix} \;\hbox { and } \; {\varvec{i}}_{-}({\textbf{S}})={\varvec{r}}\begin{bmatrix} {{\widehat{{\textbf{X}}}}},&{{\widehat{{\textbf{A}}}}},&{{\widehat{\varvec{\Gamma }}}} \end{bmatrix}-{\varvec{r}}({{\widehat{{\textbf{X}}}}}), \end{aligned} \end{aligned}$$

(A20)

respectively. The rank of ${{\,\textrm{D}\,}}[{{\textbf{n}}-{{\,\textrm{BLUP}\,}}_{{{\widehat{{\mathcal {M}}}}}}({\textbf{n}})}]-{{\,\textrm{D}\,}}[{{\textbf{n}}-{{\,\textrm{BLUP}\,}}_{{{\widehat{{\mathcal {N}}}}}}({\textbf{n}})}]$ is written as

$$\begin{aligned} \begin{aligned} {\varvec{r}}({\textbf{S}})={\varvec{r}}\begin{bmatrix} {{\widehat{{\textbf{X}}}}},&{{\widehat{\varvec{\Gamma }}}} \end{bmatrix}-{\varvec{r}}\begin{bmatrix} {{\widehat{{\textbf{X}}}}},&{{\widehat{{\textbf{A}}}}} \end{bmatrix}-{\varvec{r}}\begin{bmatrix} {{\widehat{{\textbf{X}}}}},&{{\widehat{{\textbf{A}}}}},&{{\widehat{\varvec{\Gamma }}}} \end{bmatrix}-{\varvec{r}}({{\widehat{{\textbf{X}}}}}) \end{aligned} \end{aligned}$$

(A21)

from (A4), by adding the expressions in (A20). Applying (A1)–(A3) to (A20) and (A21) yields items (1)–(5). $\square$