Recent studies, using OGLE data for LMC Cepheids in the optical, strongly suggest that the period-luminosity (PL) relation for the Large Magellanic Cloud (LMC) Cepheids shows a break or non-linearity at a period of 10 d. In this paper we apply statistical tests, the chisquared test and the F-test, to the Cepheid data from the MACHO project to test for a non-linearity of the V- and R-band PL relations at 10 d, and extend these tests to the near-infrared (JHK-band) PL relations with 2MASS data. We correct the extinction for these data by applying an extinction map towards the LMC. The statistical test we use, the F-test, is able to take account of small numbers of data points and the nature of that data on either side of the period cut at 10 d. With our data, the results we obtained imply that the VRJH-band PL relations are non-linear around a period of 10 d, while the K-band PL relation is (marginally) consistent with a single-line regression. The choice of a period of 10 d, around which this non-linearity occurs, is consistent with the results obtained when this 'break' period is estimated from the data. We show that robust parametric (including least-squares, least absolute deviation, robust regression) and non-parametric regression methods, which restrict the influence of outliers, produce similar results. Long-period Cepheids are supplemented from the literature to increase our sample size. The photometry of these long-period Cepheids is compared with our data and no trend with period is found. Our main results remain unchanged when we supplement our data set with these long-period Cepheids. By examining our data at maximum light, we also suggest arguments as to why errors in reddening are unlikely to be responsible for our results. The non-linearity of the mean V-band PL relation as seen in both of the OGLE and the MACHO data, using different extinction maps, suggests that this non-linearity is real.