1 Introduction

The HIPPARCOS satellite has provided high-precision parallaxes and proper motions of a relatively large number of Long-Period Variable stars (LPV) in the solar neighbourhood. In this paper and in the next ones (Barthès & Luri [1999]; Barthès et al. [1999], hereafter Papers II and III), which only concern those LPVs belonging to the Asymptotic Giant Branch (i.e. Mira, SRa and SRb stars), the HIPPARCOS data are exploited, together with radial velocities, K magnitudes, periods, as well as V-K and J-K colour indices, by using a specifically adapted maximum-likelihood method of luminosity calibration. We obtain model distributions of absolute magnitude, dereddened colours and period of several groups of stars. We derive de-biased relations between the period, the absolute magnitude and the colour indices (PLC relations). In this paper, the statistical model makes use of the V-K colour index. The J-K index is used only a posteriori in order to check the reliability of the results. In Paper II, the results will be confronted with theoretical models of LPV pulsation. In Paper III, a similar work will be performed with J-K included in the statistical model.

In the next section, the calibration method is presented. The data are detailed in Sect. 3. The results of the luminosity calibration are given in Sect. 4. Their consequences in terms of PLC, PL, PC and LC relations are given and commented in Sect. 5 (these relations concern the sample when they involve the J-K index, and the population in all other cases). Then, Sect. 6 summarizes and concludes this paper.