Angular power spectrum, bias from N weighted events

[ChristianS]

My general question is:
What is the angular power spectrum C_{l,N,ω} of N weighted (weight ω_i for event i) events from a full sky map with distribution C_l?

I'm interested in:

• Mean of C_{l,N,ω}: <C_{l,N,ω}>
  
• Variance of C_{l,N,ω}: Var(C_{l,N,ω})

The question is important, since we observe in reality only a certain number N of events of the true sky-distribution and this leads to a bias of the C_l s.
Due to energy dependent detector effects it is often important to weight each event i by the observed Energy ω_i(E). Maybe this problem is solved for the CMB-Powerspectrum, but I couldn't find anything :(.

For simplification I would like to start with the special case of a pure isotropic sky map. If we neglect the weights, we know from Poisson noise/shot noise (we observe N events at random positions):

1. <C_{l,N}>=4π/N
2. Var(C_{l,N})= (2/(2l+1)) (4π/N)^2

I would be very very thankful, if anybody could tell me, how this expression changes, if we weight each event i by the observed Energy ω_i(E)?

 

[Chalnoth]

What are typically used are maximum-likelihood methods. See here, for example:
https://arxiv.org/abs/astro-ph/0201438

Getting an accurate estimate of the CMB signal with realistic errors is a very complex subject overall, unfortunately. Especially difficult are estimating the foreground signals that must be subtracted (especially important for polarization measurements).

 

[ChristianS]

Thank you for the paper. Nevertheless I think that for the pure isotropic sky map it should be possible to derive an analytical expression, if the N events have instead of all weights ω_i=1 (for this case the above expression is true) different weights ω_i for each event i.
If e.g. N/4 events have weigt ω_i=1 and N*3/4 events have weight weigt ω_i=0, it should be <C_{l,N}>=4π/(N/4) and Var(C_{l,N})= (2/(2l+1)) (4π/(N/4))^2, since events with ω_i=0 shouldn't contribute to mean and variance.

I hope anybody can help me to find a gerneral expression for the case of a pure isotropic sky map.
There is no window function of the detector ... needed.