Accuracy and efficiency of alarm systems are of paramount importance in safe operations of industrial processes. Accuracy is measured by false and missed alarm rates (probabilities); while efficiency relates to the detection delay and complexity of the technique used. Moving average filters are often employed in industry for improved alarm accuracy. Can one do better than moving average filters? The following two problems are studied in this paper: First, given both normal and abnormal statistic distributions, how to design an optimal alarm filter (of fixed complexity) for best alarm accuracy, minimizing a weighted sum of false and missed alarm rates? Second, in what cases are moving average filters optimal? For the first problem, design of optimal linear FIR alarm filters is studied, and a numerical optimization based procedure is proposed. For the second problem, a sufficient condition is given under which the moving average filters are optimal.