The commonly made assumption of Gaussian noise is an approximation to reality. In this paper, influence function, an analysis tool in robust statistics, is used to formulate a recursive solution for the filtering of the ARMAX process with generalized t-distribution noise. By being a superset encompassing Gaussian, uniform, t, and double exponential distributions, generalized t-distribution has the flexibility of characterizing noise with Gaussian or non-Gaussian statistical properties. The filter is formulated as a maximum likelihood problem, but instead of solving the optimization problem numerically, influence function approximation is used to obtain a recursive solution to reduce the computational load and facilitate real-time implementation. The influence function equations derived are also useful in determining the variance of the filter and the impact of outliers.