Large-Scale Structure of the Universe and Cosmological ...

6.11.2 Quadratic Estimators
In reality it is in general difficult to express explicitly the likelihood function in
terms of the parameters. In addition, even if we restrict to the case where the
parameters are given by the power spectrum as a function of scale as discussed
in the previous section, one must iterate numerically to obtain the ML estimates,
and their probability distribution also must be computed numerically
in order to provide error bars 84 . As a result, a useful approach is to seek an
optimal estimator, unbiased and having minimum variance, by restricting the
optimization to a subspace of estimators, as discussed in Sect. 6.9. Of course,
this method is not restricted to the assumption of Gaussianity, provided that
the variance is calculated including non-Gaussian contributions. It turns out
there is an elegant solution to the problem [293,296], which in its exact form
is unfortunately difficult to implement in practice, but it does illustrate the
connection to the ML estimate (505) in the Gaussian limit, and also provides
a generalization of the standard optimal weighting results, Eqs. (474,476) to
include non-Gaussian (and non-diagonal) elements of the covariance matrix.
Since the power spectrum is by definition a quadratic quantity in the overdensities,
it is natural to restrict the search to quadratic functions of the data.
In this framework, the unbiased estimator 85 of the power spectrum having
minimum variance reads [293,296]
ˆfα = F −1
αβ
∂Cij
[ ˜ C −1 ]ijkl(δkδl − ˆ Nkl), (507)
∂fβ
where the variance is given by Eq. (504) and the Fisher matrix by Eq. (506)
replacing 1
2 [C−1 ]ik[C −1 ]jl with [ ˜ C −1 ]ijkl, where
˜Cijkl = 〈(δiδj − ˆ Nij − ξij)(δkδl − ˆ Nkl − ξkl)〉 (508)
is the (shot noise subtracted) power spectrum covariance matrix. Here ˆ
Nij
denotes the ‘actual’ shot noise, meaning that the self-pairs contributions to
ξij are not included, see [296] for details. In the Gaussian limit, [ ˜ C −1 ]ijkl →
1
2 [C−1 ]ik[C −1 ]jl (symmetrized over indices k and l) and the minimum variance
estimator, Eq. (507), reduces to ML estimator, Eq. (505), assuming iteration to
convergence is carried out as discussed above. If the iteration is not done, the
estimator remains quadratic in the data, and it corresponds to using Eq. (505)
with a fixed prior; this should be already a good approximation to the full ML
estimator, otherwise it would indicate that the result depends sensitively on
the prior and thus there is not significant information coming from the data.
The use of such quadratic estimators in the Gaussian limit to measure the
galaxy power spectrum is discussed in detail in [648], see also [647,646,80].
84 However, see [81] for an analytic approximation in the case of the 2-D power
spectrum using an offset lognormal.
85 This is assuming that the mean density is perfectly known.
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