This paper discusses the limitations of statistical theories of elasticity when applied to polymer melts. Several workers have used such theories to predict the magnitude of die swell, but have largely failed to recognize the fact that the classical relationships for elasticity do not hold at large deformations. The relationships between stress and strain are derived for tensile and shear deformations, based on the nonGaussian statistical theory, which is valid for the large deformations encountered in polymer melt flow. Results of die swell experiments are presented, which suggest that the theory can successfully be applied to polymer melts despite the major assumptions which its use entails.