Rice Distribution - Parameter Estimation (the Koay Inversion Technique)

Parameter Estimation (the Koay Inversion Technique)

There are three different methods for estimating the parameters of the Rice distribution, (1) method of moments, (2) method of maximum likelihood, and (3) method of least squares. In the first two methods the interest is in estimating the parameters of the distribution, ν and σ, from a sample of data. This can be done using the method of moments, e.g., the sample mean and the sample standard deviation. The sample mean is an estimate of μ₁' and the sample standard deviation is an estimate of μ₂1/2.

The following is an efficient method, known as the "Koay inversion technique". for solving the estimating equations, based on the sample mean and the sample standard deviation, simultaneously . This inversion technique is also known as the fixed point formula of SNR. Earlier works on the method of moments usually use a root-finding method to solve the problem, which is not efficient.

First, the ratio of the sample mean to the sample standard deviation is defined as r, i.e., . The fixed point formula of SNR is expressed as

where is the ratio of the parameters, i.e., and is given by:

where and are modified Bessel functions of the first kind.

Note that is a scaling factor of and is related to by:

To find the fixed point, of, an initial solution is selected, that is greater than the lower bound, which is and occurs when (Notice that this is the of a Rayleigh distribution). This provides a starting point for the iteration, which uses functional composition, and this continues until is less than some small positive value. Here, denotes the composition of the same function, -th times. In practice, we associate the final for some integer as the fixed point, i.e., .

Once the fixed point is found, the estimates and are found through the scaling function, as follows:

,

and

.

To speed up the iteration even more, one can use the Newton's method of root-finding. This particular approach is highly efficient.