Maximum likelihood trend estimation in exponential noise

This paper considers the problem of estimating a linear trend in noise, where the noise is modeled as independent and identically distributed (i.i.d.) random process with exponential distribution. The corresponding maximum likelihood parameter estimator of the trend and noise parameters is derived, and its performance is analyzed. It turns out that the resulting maximum likelihood estimator has to solve a linear programming problem with number of constraints equal to the number of received data. A recursive form of the maximum likelihood estimator, which makes it suitable for implementation in real-time systems, is then proposed. The memory requirements of the recursive algorithm are data dependent and are investigated by simulations using both computer-generated and recorded data sets.