Appendix A: Object intrinsic weight

The simple weighting scheme considered in this paper [see Eq. (5)] is actually the basis of several similar schemes with slightly different properties. In this paper, for simplicity and clarity, we have confined the discussion to the simple case, which is already rich in peculiarities and unexpected results (for example, the behavior in the case of vanishing weights). In this Appendix, however, we will describe a slightly more complicated estimator often used in astronomy.

Suppose, as for the Eq. (5), that we can measure a given field $f(\vec\theta)$ at some positions $\vec\theta_n$ inside a set $\Omega$. Suppose also that we decide to use, for each object, a weight u_n > 0 which is independent of the position $\vec\theta_n$(the weight u_n could however depend on other intrinsic properties of the object, such as its luminosity or angular size). We then replace Eq. (5) with

$$\tilde f(\vec\theta) = \frac{\sum_{n=1}^N \hat f_n u_n w(\... ...heta_n)}{\sum_{n=1}^N u_n w(\vec\theta - \vec\theta_n)} \cdot$$ (A.1)

A typical situation for which this estimator is useful is when some error estimates $\{ \sigma_n \}$ are available for each object. Then, if we set $u_n = 1/\sigma_n^2$, we obtain an estimator which optimizes the signal-to-noise ratio.

Since the quantities $\{ u_n \}$ do not depend on the position, we can study the statistical properties of the estimator (A.1) provided that the probability distribution p_u(u) of u is available. In particular we obtain [cf. Eq. (8)]

 

$$\bigl\langle \tilde f(\vec\theta) \bigr\rangle \frac{N}{A^N} \int_\Omega \rm d^2 \theta_1 \int_0^\infty \rm du_1... ...c\theta - \vec\theta_1)}{\sum_{n=1}^N u_n w( \vec\theta - \vec\theta_n )} \cdot$$ (A.2)

Similar to Eq. (9), we define

$$y(\vec\theta) = \sum_{n=2}^N u_n w(\vec\theta - \vec\theta_n) = \sum_{n=2}^N v_n ,$$ (A.3)

with $v_n = u_n w(\vec\theta_n)$. Using y and $\bigl\{ v_n \bigr\}$and defining the probability distributions p_y(y) and p_v(v), we can write the analogous forms of Eqs. (10) and (11). In this way we obtain results similar to Eqs. (12-15) that we do not report here. Finally we have

  

$$\mathcal L[p_v](s) = \int_0^\infty \rm e^{-sv} p_v(v) \, \rm dv =... ...fty p_u(u) \, \rm du \int_\Omega \rm e^{-s u w(\vec\theta)} \, \rm d^2 \theta ,$$ (A.4)

$$\mathcal L[p_y](s) = \int_0^\infty \rm e^{-sy} p_y(y) \, \rm dy = \bigl[ V(s) \bigr]^{N-1} .$$ (A.5)

In other words, we basically recover Eqs. (16) and (17) with the significant difference that now the Laplace transform V(s) of p_v(v) is given by the superposition of functions like W(us) weighted with the probability distribution p_u(u). Note that all functions W(us) have the same shape but differ in the scaling of the independent variable. In the case where $p_u(u) = \delta(u - 1)$ is Dirac's delta distribution centered on unity we exactly reproduce Eqs. (16) and (17), as expected.

The continuous limit (cf. Sect. 3) does not present particular difficulties. The first significant change concerns Eq. (18), which now becomes

 

$$R(s) =\int_0^\infty p_u(u) \, \rm du \int_\Omega \left[ \rm e^{-s... ...c\theta)} - 1 \right] \, \rm d^2 \theta =\int_0^\infty p_u(u) Q(us) \, \rm du .$$ (A.6)

Correspondingly, $Y(s) = \exp \bigl[ \rho R(s) \bigr]$. Finally, writing the effective weight as $w_{\rm eff} = w C(w)$, we have

 

$$\int_0^\infty p_u(u) B(u w) \, \rm du ,$$ (A.7)

$$\mathcal L[\rho Y](w) .$$ (A.8)

The equations written here can be used to practically evaluate B(w)(and thus $w_{\rm eff}(\vec\theta)$), but also to derive the properties of the effective weight, as done in for the simple estimator (5). Here we do not carry out the calculations, since they are straightforward modifications of the calculations of Sect. 5; moreover the results obtained are basically identical to the ones reported in that section.