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We present a detailed statistical analysis of possible non-linearities in thePeriod-Luminosity (P-L), Period-Wesenheit (P-W) and Period-Color (P-C)relations for Cepheid variables in the LMC at optical ($VI$) and near-infrared($JHK_{s}$) wavelengths. We test for the presence of possible non-linearitiesand determine their statistical significance by applying a variety of robuststatistical tests ($F$-test, Random-Walk, Testimator and the Davies test) tooptical data from OGLE III and near-infrared data from LMCNISS. Forfundamental-mode Cepheids, we find that the optical P-L, P-W and P-C relationsare non-linear at 10 days. The near-infrared P-L and the $W^H_{V,I}$ relationsare non-linear around 18 days; this break is attributed to a distinct variationin mean Fourier amplitude parameters near this period for longer wavelengths ascompared to optical bands. The near-infrared P-W relations are also non-linearexcept for the $W_{H,K_s}$ relation. For first-overtone mode Cepheids, asignificant change in the slope of P-L, P-W and P-C relations is found around2.5 days only at optical wavelengths. We determine a global slope of$\textrm{-}3.212\pm0.013$ for the $W^H_{V,I}$ relation by combining our LMCdata with observations of Cepheids in Supernovae host galaxies \citep{riess11}.We find this slope to be consistent with the corresponding LMC relation atshort periods, and significantly different to the long-period value. We do notfind any significant difference in the slope of the global-fit solution using alinear or non-linear LMC P-L relation as calibrator, but the linear versionprovides a $2\times$ better constraint on the slope and metallicitycoefficient.

Large Magellanic Cloud Near-Infrared Synoptic Survey. III. A Statistical Study of Non-Linearity in the Leavitt Laws