A generalized Cauchy process with a cubic nonlinear term (a nonlinear friction) is studied under the influence of independent multiplicative and additive Gaussian-white noises. Three methods of parameter estimation (i.e. the maximum likelihood, the moment and the log-amplitude moment) are presented in detail. The effect of nonlinearity-noise mterplay associated with the nonlinear friction under the influences of both multiplicative and additive noises are discussed in conjunction with fluctuation-dissipation theorem. The physical significance of nonlinear friction is demonstrated with the use of time series data in economics and fluid turbulence.