Geomon

Combining unbiased estimators

Let ${{\hat {\phi }}_{1}}$ and ${{\hat {\phi }}_{2}}$ be unbiased estimators of $\phi \in {{\mathbb {R} }^{k}}$ with non-singular variances ${{V}_{1}}$ and ${{V}_{2}}$ respectively.

Then the minimum variance linear unbiased estimator of $\phi$ is obtained by combining ${{\hat {\phi }}_{1}}$ and ${{\hat {\phi }}_{2}}$ using weights that are proportional to the inverses of their variances. The result can be expressed in a variety of ways:

${\begin{aligned}{\hat {\phi }}&={{\left(V_{1}^{-1}+V_{2}^{-1}\right)}^{-1}}\left(V_{1}^{-1}{{\hat {\phi }}_{1}}+V_{2}^{-1}{{\hat {\phi }}_{2}}\right)\\&={{\left(V_{1}^{-1}+V_{2}^{-1}\right)}^{-1}}\left(V_{1}^{-1}{{\hat {\phi }}_{1}}+V_{2}^{-1}{{\hat {\phi }}_{2}}\right)+&\left[{{\left(V_{1}^{-1}+V_{2}^{-1}\right)}^{-1}}V_{2}^{-1}{{\hat {\phi }}_{1}}-{{\left(V_{1}^{-1}+V_{2}^{-1}\right)}^{-1}}V_{2}^{-1}{{\hat {\phi }}_{1}}\right]\\&={{\hat {\phi }}_{1}}+{{\left(V_{1}^{-1}+V_{2}^{-1}\right)}^{-1}}V_{2}^{-1}\left({{\hat {\phi }}_{2}}-{{\hat {\phi }}_{1}}\right)\\&={{\hat {\phi }}_{1}}+{{\left(I+{{V}_{2}}V_{1}^{-1}\right)}^{-1}}\left({{\hat {\phi }}_{2}}-{{\hat {\phi }}_{1}}\right)\\&={{\left(I+{{V}_{1}}V_{2}^{-1}\right)}^{-1}}\left({{\hat {\phi }}_{1}}+{{V}_{1}}V_{2}^{-1}{{\hat {\phi }}_{2}}\right)\end{aligned}}$ The proof is an application of the principle of Generalized Least-Squares. The problem can be formulated as a GLS problem by considering that: $\left[{\begin{matrix}{{\hat {\phi }}_{1}}\\{{\hat {\phi }}_{2}}\\\end{matrix}}\right]=\left[{\begin{matrix}I\\I\\\end{matrix}}\right]\phi +\left[{\begin{matrix}{{\varepsilon }_{1}}\\{{\varepsilon }_{1}}\\\end{matrix}}\right]$ with $\operatorname {Var} \left(\left[{\begin{matrix}{{\varepsilon }_{1}}\\{{\varepsilon }_{1}}\\\end{matrix}}\right]\right)=\left[{\begin{matrix}{{V}_{1}}&0\\0&{{V}_{2}}\\\end{matrix}}\right]$

Applying the GLS formula yields: ${\begin{aligned}{\hat {\phi }}&={{\left({{\left[{\begin{matrix}I\\I\\\end{matrix}}\right]}^{\prime }}{{\left[{\begin{matrix}{{V}_{1}}&0\\0&{{V}_{2}}\\\end{matrix}}\right]}^{-1}}\left[{\begin{matrix}I\\I\\\end{matrix}}\right]\right)}^{-1}}{{\left[{\begin{matrix}I\\I\\\end{matrix}}\right]}^{\prime }}{{\left[{\begin{matrix}{{V}_{1}}&0\\0&{{V}_{2}}\\\end{matrix}}\right]}^{-1}}\left[{\begin{matrix}{{\hat {\phi }}_{1}}\\{{\hat {\phi }}_{2}}\\\end{matrix}}\right]\\&={{\left(V_{1}^{-1}+V_{2}^{-1}\right)}^{-1}}\left(V_{1}^{-1}{{\hat {\phi }}_{1}}+V_{2}^{-1}{{\hat {\phi }}_{2}}\right)\end{aligned}}$

Help:Math

Expected value of SSH

Consider one-way MANOVA with $G$ groups, each with $n_{g}$ observations. Let $N=\sum _{g=1}^{G}n_{g}\!$ and let

$D={\begin{bmatrix}1_{n_{1}}&\cdots &0\\\vdots &\ddots &\vdots \\0&\cdots &1_{n_{g}}\end{bmatrix}}$

be the design matrix.

Let $Q$ be the $N\times N$ residual projection matrix defined by

$Q=I-1_{N}(1_{N}'1_{N})^{-1}1_{N}'=I-{\frac {1}{N}}U$

Analyzing SSH

We can find expressions for SSH in terms of the data and find expected values for SSH under a fixed effects or under a random effects model.

The following formula is used repeatedly to find the expected value of a quadratic form. If $Y$ is a random vector with $\operatorname {E} (Y)=\mu$ and $\operatorname {Var} (Y)=\Psi \!$, and $Q\!$ is symmetric, then

$\operatorname {E} (Y'QY)=\mu 'Q\mu +\operatorname {tr} (Q\Psi )\!$

We can model:

$\mathbf {Y} =D\mathbf {\mu } +\epsilon \!$

where

$\mu \sim N(1\psi ,\phi ^{2}I)\!$

and

$\epsilon \sim N(0,\sigma ^{2}I)\!$

and $\mu$ is independent of $\epsilon$.

Thus

$\operatorname {E} (Y)=1\psi$ and $\operatorname {Var} (Y)=\phi ^{2}DD'+\sigma ^{2}I\!$

Consequently

$\operatorname {E} (SSTO)\!$	$=$	$\operatorname {E} (Y'QY)=\psi 1'Q1\psi +\operatorname {tr} \left[(\phi ^{2}DD'+\sigma ^{2}I)(I-{\frac {1}{N}}U)\right]$
	$=$	$0+\operatorname {tr} \left[\phi ^{2}DD'-{\frac {\phi ^{2}}{N}}DD'U+\sigma ^{2}Q\right]$
	$=$	$\phi ^{2}N-{\frac {\phi ^{2}}{N}}\operatorname {tr} (DD'U)+\sigma ^{2}\operatorname {tr} (Q)$
	$=$	$\phi ^{2}N-{\frac {\phi ^{2}}{N}}\operatorname {tr} (1'DD'1)+\sigma ^{2}(N-1)$
	$=$	$\phi ^{2}N-{\frac {\phi ^{2}}{N}}\sum _{g=1}^{G}n_{g}^{2}+\sigma ^{2}(N-1)$
	$=$	$\phi ^{2}(N-{\tilde {n}})+\sigma ^{2}(N-1)$

where ${\tilde {n}}=\sum _{g=1}^{G}{\frac {n_{g}}{N}}n_{g}$ is the group-size weighted mean of group sizes. With equal groups ${\tilde {n}}=N/G$ and

$E(SSTO)=\phi ^{2}N{\frac {G-1}{G}}+\sigma ^{2}(N-1)=\phi ^{2}(N-n)+\sigma ^{2}(N-1)$

Thus

$E(SSH)\!$	=	$E(SSTO)-E(SSE)\!$
	=	$\phi ^{2}(N-{\tilde {n}})+\sigma ^{2}(N-1)-\sigma ^{2}(N-G)$
	=	$\phi ^{2}(N-{\tilde {n}})+\sigma ^{2}(G-1)$

Multivariate response

If we are sampling from a p-variate distribution in which

$\mathbf {Y} _{ig}\sim {\mbox{i.i.d.}}N(\mathbf {\mu } _{g},\Sigma )$

and

$\mathbf {\mu } _{1},...,\mathbf {\mu } _{G}\sim {\mbox{ i.i.d. }}N(\mathbf {\psi } ,\Phi ),{\mbox{ independently of }}\mathbf {Y} _{ig}$

then the analogous results are:

$E(SSE)=(N-G)\Sigma$

and

$E(SSH)=(N-{\overset {\sim }{n}})\Phi +(G-1)\Sigma$

Note that

$Var({\bar {\mathbf {Y} }}_{\cdot g})=\Phi +{\frac {1}{n_{g}}}\Sigma$

and that the group-size weighted average of these variances is:

$\sum _{g=1}^{G}{\frac {n_{g}}{N}}Var({\bar {\mathbf {Y} }}_{\cdot g})=\sum _{g=1}^{G}{\frac {n_{g}}{N}}\left[\Phi +{\frac {1}{n_{g}}}\Sigma \right]=\Phi +{\frac {G}{N}}\Sigma$

The expectation of combinations of $SSH$ and $SSE$ of the form $k_{H}SSH+k_{E}SSE$:

$k_{H}\!$	$k_{E}\!$	$E(k_{H}SSH+k_{E}SSE)\!$
1	0	$(N-{\tilde {n}})\Phi +(G-1)\Sigma$
0	1	$(N-G)\Sigma$
${\frac {1}{N-{\overset {\sim }{n}}}}$	0	$\Phi +{\frac {G-1}{N-{\overset {\sim }{n}}}}\Sigma$
${\frac {G}{N(G-1)}}$	0	$\Phi +{\frac {G}{N}}\Sigma ,{\mbox{ with equal groups}}$
${\frac {G}{N(G-1)}}$	$-{\frac {G}{N(N-G)}}$	$\Phi ,{\mbox{ with equal groups}}$