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Astron. Astrophys. 362, 851-864 (2000)

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Appendix A

From Eq. (3), we write the probability density (hereafter PD) of [FORMULA] as:

[EQUATION]

where [FORMULA] and [FORMULA] are the PDs of [FORMULA] and [FORMULA], and [FORMULA] is the probability that, for a fixed output of [FORMULA], [FORMULA] is included between [FORMULA] and [FORMULA]. For brevity, we put [FORMULA].

By Eq. (A1) we obtain the following relations:

[EQUATION]

[EQUATION]

By using the normalization of [FORMULA] and the hypothesis [FORMULA], from (A2) we obtain Eq. (7). Eq. (8) follows from (A3) with the further hypothesis that [FORMULA] and [FORMULA] are not correlated (`hypothesis [FORMULA]').

If [FORMULA] does not hold, we obtain again Eq. (8) provided that the CM terms of [FORMULA] and [FORMULA] are substituted with their expected values with respect to the PDs of [FORMULA].

In a similar way, Eq. (9) can be proved.

Appendix B

Let's start by introducing the following notations:

[EQUATION]

The symbols [FORMULA], [FORMULA] and [FORMULA] will indicate the quantities obtained by substituting [FORMULA] in Eqs. (B3 - B5) with unbiased estimates [FORMULA].

From Eqs. (10 - 13) we have:

[EQUATION]

where [FORMULA] and [FORMULA] are defined by replacing [FORMULA] with [FORMULA] in Eqs. (10, 11), and o is a quantity that can be neglected when the sample size increases.

By using the relation

[EQUATION]

where [FORMULA] is a generic RV and [FORMULA] an unbiased estimate of [FORMULA], the following relations can be written:

[EQUATION]

Setting [FORMULA] and [FORMULA], we introduce the RVs [FORMULA], [FORMULA] and [FORMULA] by the following definitions:

[EQUATION]

Using Eqs. (B8 - B13) and setting [FORMULA], [FORMULA], after lengthy calculations, Eqs. (B6,B7) are rewritten:

[EQUATION]

From the above relations and Eq. (12), we also have:

[EQUATION]

where [FORMULA].

If [FORMULA] are normally distributed, estimates of the expected values in the previous equations are given by sample means. The CM terms of [FORMULA], [FORMULA] and [FORMULA] are thus obtained by the usual formulae:

[EQUATION]

where N is the sample size and [FORMULA], [FORMULA] and [FORMULA] are the RVs obtained by substituting in the expressions of [FORMULA], [FORMULA] and [FORMULA] the unknown quantities with their unbiased estimates.

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© European Southern Observatory (ESO) 2000

Online publication: October 30, 2000
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