1. A Testimator Based Approach to Investigate the Non- Linearity of the LMC Cepheid Period-Luminosity Relation R. Stevens, A. Nanthakumar, S. M. Kanbur (Physics Department, SUNY Oswego) and C. Ngeow (Astronomy Department, UIUC) 209 th AAS Meeting, January 2007, Seattle, WA Abstract: The Cepheid Period-Luminosity (PL) relation is one the most fundamental relations in Astrophysics and is a crucial component in estimating the Hubble constant. Recent evidence has emerged that this relation, at least in the LMC, may be non-linear with two distinct slopes for short (P 10 days). Here we apply a new statistical test, the testimator method, for testing this non- linearity and apply it to the existing LMC data. Our result strongly suggests that the LMC PL relation is indeed non-linear. Introduction: The correlation of the period and the absolute brightness, the Period-Luminosity (PL) relation, for classical Cepheid variables is the key to the calibration of the extra-galactic distance scale and in turn can be used to estimate the Hubble constant. The most widely used PL relation is obtained from LMC Cepheids. This relation has been assumed to be linear for a long time. Recent evidence has emerged that the LMC Cepheid PL relation (at mean light) is non-linear: there are two relations with a break at/around 10 days. However, this non-linearity is difficult to visualize. This is demonstrated in Figure 1 using two simulated PL relations. This shows that direct testing for such a non-linearity is indeed subtle. Therefore, careful statistical tests are required to detect the non- linearity of the Cepheid PL relation, if present. A statistical test, the testimator method, is applied to the LMC data to detect the non-linearity of the PL relation. The Testimator Method and its Application to the PL Relation: The testimator method essentially breaks the data into small groups, ordered sequentially by period, and tests to see if the slope from one group is statistically similar to the average slope formed by all previous groups. The null hypothesis (Ho) is that these slopes are similar, while the alternate hypothesis (H A ) is they are statistically different. If the slopes are similar, the current group is incorporated into a revised estimate (the testimator) of the slope and the next group of data is studied. This procedure is illustrated in Figure 2. The testimator is such that: (a) it is an unbiased estimator under the null hypothesis; (b) its variance can be analytically proved to be smaller than the variance of the standard least squares estimator of the slope. Further, such a procedure can be used to bracket the period at which any slope change occurs. The testimator method correctly identifies the top panel in Figure 1 as being the PL relation with a break and was able to bracket the break around the period of 10 days. We applied this method to the V band LMC Cepheid data obtained by the OGLE project. Figure 1: Comparison of the simulated non-liner (A) and linear (B) PL relations. Figure 2: Illustration of the testimator method. The first two steps are shown here. The procedure are repeated until all the data is used or the null hypothesis is rejected. Results: The results of the testimator are shown in the Table: This dataset showed strong statistical evidence of a non-linearity of the LMC PL relation centered around a period of 10 days Log- Period range NDecision 0.4-0.5150Accept Ho 0.5-0.6150Accept Ho 0.6-0.7150Accept Ho 0.7-1.7191Reject Ho Acknowledgement: This research is supported in part by the AAS Small Research Grant.