  In this paper, instead of the usual Gaussian noise assumption,
$t$-distribution noise is assumed. A Maximum Likelihood Estimator using the
most recent N measurements is proposed for the Auto-Regressive-Moving-Average
with eXogenous input (ARMAX) process with this assumption. The proposed
estimator is robust to outliers because the `thick tail' of the t-distribution
reduces the effect of large errors in the likelihood function. Instead of
solving the resulting nonlinear estimator numerically, the Influence Function
is used to formulate a computationally efficient recursive solution, which
reduces to the traditional Moving Horizon Estimator when the noise is Gaussian.
The formula for the variance of the estimate is derived. This formula shows
explicitly how the variance of the estimate is affected by the number of
measurements and noise variance. The simulation results show that the proposed
estimator has smaller variance and is more robust to outliers than the Moving
Window Least-Squares Estimator. For the same accuracy, the proposed estimator
is an order of magnitude faster than the particle filter.
