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- I need help about the understanding of all the steps in a demonstration of the optimal variance by inverse-weighted variance average.
I have a problem of understanding in the following demo :
In a cosmology context with 2 probes (spectroscopic and photometric), let notice ##a_{\ell m, s p}## the spectroscopic and ##a_{\ell m, p h}## the photometric coefficients of the decomposition in spherical harmonics of the distributions of each population. In the absence of any Poisson noise we have:
##\dfrac{a_{\ell m, s p}^{2}}{a_{\ell m, p h}^{2}}=\left(\dfrac{b_{s p}}{b_{p h}}\right)^{2}\quad(1)##
Now let assume the spectroscopic sample is a Poisson realization of density ##N_{s p}## (the galaxy density of the spectroscopic sample) and that we have an unbiased estimator ##\hat{a}_{\ell m, s p}## of ##a_{\ell m, s p}##. We then have the average :
##
\left\langle\dfrac{\hat{a}_{\ell m, s p}^{2}}{a_{\ell m, p h}^{2}}\right\rangle=\dfrac{\left\langle\hat{a}_{\ell m, s p}^{2}\right\rangle}{a_{\ell m, p h}^{2}}=\left(\dfrac{b_{s p}}{b_{p h}}\right)^{2}\quad(2)
##
with its variance :
##
\dfrac{2}{f_{s k y} a_{\ell m, p h}^{4} N_{s p}^{2}}
##
We can therefore build an estimator ##\hat{O}## of ##\left(\dfrac{b_{s p}}{b_{p h}}\right)^{2}## by taking the optimal (inverse-variance weighted) average over all ##\ell## and ##m## :
##\hat{O}=\dfrac{\sum_{\ell=\ell_{\min }}^{\ell_{\max }} \sum_{m=-\ell}^{\ell} a_{\ell m, p h}^{2} \hat{a}_{\ell m, s p}^{2}}{\sum_{\ell=\ell_{\min }}^{\ell_{\max }} \sum_{m=-\ell}^{\ell} a_{\ell m, p h}^{4}}\quad(3)##
the variance of which being:
##\sigma_{\hat{o}}^{2}=\left(\sum_\limits{\ell=\ell_{\min }}^{\ell_{\max }}(2 \ell+1) C_{\ell, p h}^{2}\right)^{-1} \dfrac{2}{f_{s k y} N_{sp}^{2}}\quad(4)##
I don't understanding the passing between eq(3) and eq(4). Indeed, I can't make appear from eq(3) the existing factor ##2\ell+1## in eq(4). The goal is to prove the relation eq(4).
If someone could help me to detail the different necessary steps to obtain eq(4), this would be fine.
I recall that in general, ##C_{\ell}=\dfrac{1}{2\ell+1}\sum_{m=-\ell}^{+\ell} a_{\ell m}^{2}##
Best regards
In a cosmology context with 2 probes (spectroscopic and photometric), let notice ##a_{\ell m, s p}## the spectroscopic and ##a_{\ell m, p h}## the photometric coefficients of the decomposition in spherical harmonics of the distributions of each population. In the absence of any Poisson noise we have:
##\dfrac{a_{\ell m, s p}^{2}}{a_{\ell m, p h}^{2}}=\left(\dfrac{b_{s p}}{b_{p h}}\right)^{2}\quad(1)##
Now let assume the spectroscopic sample is a Poisson realization of density ##N_{s p}## (the galaxy density of the spectroscopic sample) and that we have an unbiased estimator ##\hat{a}_{\ell m, s p}## of ##a_{\ell m, s p}##. We then have the average :
##
\left\langle\dfrac{\hat{a}_{\ell m, s p}^{2}}{a_{\ell m, p h}^{2}}\right\rangle=\dfrac{\left\langle\hat{a}_{\ell m, s p}^{2}\right\rangle}{a_{\ell m, p h}^{2}}=\left(\dfrac{b_{s p}}{b_{p h}}\right)^{2}\quad(2)
##
with its variance :
##
\dfrac{2}{f_{s k y} a_{\ell m, p h}^{4} N_{s p}^{2}}
##
We can therefore build an estimator ##\hat{O}## of ##\left(\dfrac{b_{s p}}{b_{p h}}\right)^{2}## by taking the optimal (inverse-variance weighted) average over all ##\ell## and ##m## :
##\hat{O}=\dfrac{\sum_{\ell=\ell_{\min }}^{\ell_{\max }} \sum_{m=-\ell}^{\ell} a_{\ell m, p h}^{2} \hat{a}_{\ell m, s p}^{2}}{\sum_{\ell=\ell_{\min }}^{\ell_{\max }} \sum_{m=-\ell}^{\ell} a_{\ell m, p h}^{4}}\quad(3)##
the variance of which being:
##\sigma_{\hat{o}}^{2}=\left(\sum_\limits{\ell=\ell_{\min }}^{\ell_{\max }}(2 \ell+1) C_{\ell, p h}^{2}\right)^{-1} \dfrac{2}{f_{s k y} N_{sp}^{2}}\quad(4)##
I don't understanding the passing between eq(3) and eq(4). Indeed, I can't make appear from eq(3) the existing factor ##2\ell+1## in eq(4). The goal is to prove the relation eq(4).
If someone could help me to detail the different necessary steps to obtain eq(4), this would be fine.
I recall that in general, ##C_{\ell}=\dfrac{1}{2\ell+1}\sum_{m=-\ell}^{+\ell} a_{\ell m}^{2}##
Best regards
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