An approach for designing reduced order state estimators for discrete time systems driven by completely unknown exogenous disturbances is provided here. Necessary, as well as necessary and sufficient, conditions for existence of the discrete estimator are given, along with an easy to follow and implement approach for construction of the estimator. Such estimators have been proved useful in a variety of robust control applications, as well as in designing control systems with monitoring and fault detection capabilities. Here, the application of the estimator for actuator fault detection in robotic applications is explored. Also, although the estimator design is carried out in a deterministic setting, the presence of measurement noise is explored and it is shown that the estimator and fault detection strategy can still be very successful even in the presence of random noise.