  We present a detailed statistical analysis of possible non-linearities in the
Period-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-linearities
and determine their statistical significance by applying a variety of robust
statistical tests ($F$-test, Random-Walk, Testimator and the Davies test) to
optical data from OGLE III and near-infrared data from LMCNISS. For
fundamental-mode Cepheids, we find that the optical P-L, P-W and P-C relations
are non-linear at 10 days. The near-infrared P-L and the $W^H_{V,I}$ relations
are non-linear around 18 days; this break is attributed to a distinct variation
in mean Fourier amplitude parameters near this period for longer wavelengths as
compared to optical bands. The near-infrared P-W relations are also non-linear
except for the $W_{H,K_s}$ relation. For first-overtone mode Cepheids, a
significant change in the slope of P-L, P-W and P-C relations is found around
2.5 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 LMC
data with observations of Cepheids in Supernovae host galaxies \citep{riess11}.
We find this slope to be consistent with the corresponding LMC relation at
short periods, and significantly different to the long-period value. We do not
find any significant difference in the slope of the global-fit solution using a
linear or non-linear LMC P-L relation as calibrator, but the linear version
provides a $2\times$ better constraint on the slope and metallicity
coefficient.
