In this paper we discuss and examine various issues concerning the recent findings that suggest that the observed period-luminosity (P-L) relation for the Large Magellanic Cloud (LMC) Cepheids is nonlinear. These include (1) visualizing the nonlinear P-L relation; (2) long-period Cepheids and sample selection; (3) outlier removal; (4) issues of extinction; (5) nonlinearity of the period-color (P-C) relation; (6) nonlinear P-L relations in different passbands; and (7) the universality of the P-L relation. Our results imply that a statistical test is needed to detect the nonlinear P-L relation. We then show that sample selection, number of long-period Cepheids in the sample, outlier removal, and extinction errors are unlikely to be responsible for the detection of the nonlinear P-L relation. We also argue for the existence of a nonlinear P-L relation from the perspective of the nonlinear P-C relation and the nonuniversality of the P-L relation. Combining the evidence and discussion from these aspects, we find that there is a strong indication that the observed LMC P-L relation is indeed nonlinear in the optical bands (however, the K-band LMC P-L relation is apparently linear). This could be due to internal physical reasons or to external hidden/additional factors. Compared to the nonlinear P-L relation, the systematic error in distance scale introduced from using the (incorrect) linear P-L relation is at most at the few percent level. While this is small compared to other systematic errors, it will be important in future efforts to produce a Cepheid distance scale accurate to 1% in order to remove degeneracies presented in CMB results.
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