3 The statistical method

In this paper the Luri-Mennessier (LM) statistical method, described in Luri et al. (1996), is used to analyze the sample of LPV stars. The method has been specifically designed to exploit the HIPPARCOS data and thus is suitable for our purposes. This method has already given fruitful results, in particular for Barium stars (Mennessier et al. 1997c), Ap-Bp stars (Gomez et al. 1998) and the LMC distance modulus (Luri et al. 1998).

The use of appropriate statistical methods for the exploitation of the HIPPARCOS astrometric data is crucial in order to obtain correct results. Otherwise the values obtained can be affected by strong biases and the precision of the data will not be fully used. A discussion on the correct use of HIPPARCOS data and recommendations on analysis techniques can be found in Brown et al. (1997).

The LM method used in this paper is especially appropriate for use with the HIPPARCOS data. We refer to Luri et al. (1996) for a detailed description, and only briefly summarize here some of its main features.

First of all, the stellar population from which the sample is extracted is assumed to be composed of several distinct groups. These groups can differ in kinematics, luminosity or spatial distribution and its number is a priori not known. Therefore, using a sample extracted from this base population and taking into account the selection criteria used to create it, the LM method:

• determines the number of significant discriminating groups and the percentage of each of them in the population;

• produces, for each group, unbiased estimates of the parameters of the model used to describe it, i.e.:
  • a Schwarzschild ellipsoid velocity distribution characterized by ( $U_0, V_0, W_0, \sigma_U, \sigma_V, \sigma_W$). U₀, V₀ and W₀ point towards the galactic center, the galactic rotation, and the north galactic pole, respectively;
  • an exponential distribution of the number of stars in the direction perpendicular to the galactic plane, with a scale height Z₀;
  • a normal distribution of the absolute magnitude in the bandpass of the used observed magnitudes characterized by the parameters M₀ and $\sigma_M$.

The results of this first step for our LPVs sample are given in Sect.  4.

The minimum input data needed by the LM method are the measured positions, proper motions and apparent magnitudes of the stars, but it can also use the parallaxes and radial velocities if available. The method takes into account the selection effects of the sample, the observational errors, the galactic rotation and the interstellar absorption.

In a second step, once the groups are identified and parametrized, the method:

• computes for each star of the sample, the a posteriori probability that the star belongs to a given group;

• uses a Bayesian rule to assign each star to a group;

• uses a statistical estimator to obtain estimations of individual distances and luminosities for each star.

This second step is presented in Sect. 5 where the representativity of our sample with respect to the kinematic and photometric properties of the population is also discussed.