Compliant offshore structures are subject to two distinct types of excitation, namely Gaussian first order forces occurring at the dominant wind and wave frequencies, and non-Gaussian low frequency second order wave forces acting at the surge, sway and yaw natural frequencies. Although the response to the low frequencysecond order forces tends to dominate the response in the horizontal plane, the response to the first order forces can significantly influence the extreme response of the vessel. Since the design of an offshore mooring system is based on the extreme response of the vessel, it is essential that the combined first and second order wave and wind induced response is considered. The discretised Kac-Siegert technique is used in the present work to analyse the combined first and second order wave and wind induced response of a linearly moored vessel, where the sea-state may be uni-directional or short-crested. A closed-form series solution for the probability density function (pdf) of the combined response is derived in terms of the eigenvalues and eigenvectors of the Kac-Siegert matrix. The first term of the resulting series corresponds to the assumption that the first and second order responses are statistically independent, while subsequent terms depend upon the joint cumulants of the response. The accuracy of the method is investigated by comparison with time domain simulation. An analytic series solution for the mean up-crossing rate is developed in which the first term of this series corresponds to the assumption that the displacement and velocity responses are statistically independent, and further terms depend upon the joint displacement-and velocity cumulants. The accuracy of the method is assessed by comparison with numerical simulation, and the extreme values of the response are estimated using the Poisson assumption of independent crossings.