Appendix A: Populations, sampling and biases

A striking feature of Figs. 5 through 7 is that many stars are located outside, most often above the projected $2\sigma$ contour of the fitted distribution of the corresponding group, whereas one would expect most of them to be located inside or symmetrically around it.

First, one must remember that, in the PL and LC diagrams, the projected $2\sigma$ contour (i.e. the projection of the $2\sigma$ contour of the mean PLC plane) is not the $2\sigma$ contour of the projected distribution. This explains why, in the Luminosity-Colour diagram (Fig. 7), the width of each sequence of sample stars is much larger than the corresponding "ellipse''.

On the other hand, the offset of the sample distributions with respect to the model ones is due to the fact that the sample selection is based on the apparent magnitude (see Sect. 3.1). This is analogous to the Malmquist bias. To better understand the phenomenon in our case, a closer look at this well known bias may be helpful:

Malmquist ([1936]) studied the bias in the mean absolute magnitude that is derived from a sample of stars with the following characteristics:

• The base population from which the sample is extracted has (a) a homogeneous spatial distribution and (b) a Gaussian distribution of absolute magnitudes $G(M_{0},\sigma _{M})$;
• The sample is selected within the base population by means of a limit-magnitude criterium: $m\leq m_{\rm lim}$.

Under these conditions, Malmquist ([1936]) proved that the mean absolute magnitude of the sample <M> differs from the mean absolute magnitude of the base population M₀ according to: $<M>~=M_{0}-1.38\; \sigma _{M}^{2}$.

In other words, the stars in the sample are, on average, brighter that the base population. The reason for this is that, because of the apparent magnitude limit, brighter stars are over-represented in the sample: as they can be seen at longer distances, more of them are included.

In our case, the effects are more complicated (inhomogeneous spatial distribution, PLC relations and complex selection function) and also stronger. The PLC relations have, in some cases, large slopes and thus the groups may contain stars of very different absolute magnitudes. Like in the case of the Malmquist bias, brighter stars are favoured and thus over-represented in our sample. This favours, in turn, stars with long periods and large colour indices but also, at a given period and colour, stars located on the "bright'' side of the main PLC plane.

Figure A1: Simulated sample of Group 1 stars with no magnitude limit

Figure A2: Same simulation when applying the magnitude-based selection function

This effect can be illustrated by Monte-Carlo simulations. Let us first simulate a sample of Group 1 stars with no magnitude limit. As can be seen in Fig. A1, most of these stars are located, in the $\{\log P,M_{K}\}$ plane, inside the projected $2\sigma$ contour of the distribution used for the simulation. However, when the selection function (the one whose parameters were calculated in Sect. 4) is applied, the $\{\log P,M_{K}\}$ distribution of the sample drastically changes, as shown in Fig. A2: in this case, a majority of the stars are brighter than the projected $2\sigma$ contour.

This example clearly shows that the suprising peculiarities of Figs. 5 to 7 are nothing but natural. Moreover it shows that, in the most general case, a "naive'' fit to the absolute magnitude distribution of a sample is not at all representative of the base population. Fortunately, this bias is probably negligible in the case of Magellanic Clouds studies, since all their LPVs may be considered as located at the same distance with a reasonable approximation, and their amplitudes are small in the near-infrared bands where they are observed.