Abstract: This study discusses some of the approximation which can be used to obtain the maximum likelihood estimate of the concentration parameter of the von Mises distribution. This study shows that the efficient approximation may also be obtained by solving the equation for the ratio of the modified Bessel function of first kind of order one and first kind of order zero. The closed-form solution for parameter concentration is also given. We have found out that the new proposed method performs well especially for large κ.
INTRODUCTION
The von Mises distribution is denoted by
Suppose
There are several approximations available for A-1 (x) for all x in (0,1). For instance Amos (1974) proved that:
and hence A-1 (x) is approximately given by:
Mardia and Zemroch (1975) give a computer algorithm for calculating A-1 (x) together with the tables which are obtained iteratively. Meanwhile, by using the power series for the Bessel function I0 (x) and I1(x), Dobson (1978) has given the approximation of A-1 (x) as follows:
It is shown that this approximation gives less maximum relative error compared to Amos (1974). Further, an improved approximation for A-1 (x) was given by Best and Fisher (1981) which is
Its tabulated values can be found in Fisher (1993).
APPROXIMATION BASED ON THE MODIFIED BESSEL FUNCTION
By definition,
| (1) |
and from the power series for the Bessel function I0 (x) and I1 (x), we have
| (2) |
Solving (1) and (2), we obtain:
| (3) |
where, a3 = 8t-8, a2 = 4, a1 = 1 and a0 = 1.
By taking transformation of
| (4) |
Where:
and
Following Rades and Westergren (1988), the roots for Eq. (4) are of one real root and two complex roots and are given by
and |
(5) |
Where:
Hence, the approximation for
| (6) |
Here we show that this approximation gives more efficient estimation
for
SIMULATION RESULTS
Computer programs were written using SPLUS language to carry out the simulation study to assess the efficiency of the three different methods of approximating the concentration parameter κ. Circular samples of length n = 100 were generated from von Mises distribution with mean 0 and κ = 2, 4, 6, 8, 10, 12 14 and 16.
Let s be the number of simulations and the following computation were obtained from the simulation study.
| • | Mean |
| • | Estimated Bias = |
| • | Absolute Relative Estimated Bias (%) |
| • | Estimated Standard Error = |
| • | Estimated Root Mean Square Error (RMSE) = |
| • | Relative Efficiency = |
| Table 1: | Simulation results for various true value of parameter concentration |
The values of mean, estimated bias, absolute relative bias, estimated standard error and RMSE are computed for the Dobson`s method, Best and Fisher`s method and the new proposed method. The results are given in rows 1-15 of Table 1. We also compare the efficiency of Dobson and the new proposed method relative to the Best and Fisher method and the values are given in row 16-18 of (Table 1).
It appears from rows 1 to 3 that the mean estimate obtained from the new proposed method is very close to the true k value compared to the estimates obtained by Dobson and Best and Fisher approximation method. It can be seen clearer from rows 4 to 6 as the biases of estimates for new proposed method are closer to zero. Note also that biases are an increasing function of true κ. Further, it is also clear that, for κ≥6, the absolute relative estimated bias, estimated standard error and estimated root mean square error are the smallest for the new proposed method and followed by the Best and Fisher`s method as given in row 7-15. However, for smaller κ, Dobson`s method has the smallest estimated standard error and estimated root mean square error. To make a direct comparison between these approximation methods, the relative efficiency of Dobson`s method and new proposed method relative to Best and Fisher`s method were computed and results are given in row 16-18. Result in row 18 shows that new proposed method has the relative efficiency smaller than one for all values of true κ. These indicate that new proposed method is relatively better than Best and Fisher approximation method especially for small k. New proposed method is also found to be relatively better than the Dobson`s method except for small κ.
CONCLUSION
The objective of present study is to evaluate the performance of three different approximating methods of concentration parameter κ, namely, the Dobson`s method, Best and Fisher`s method and new proposed method through simulation study. Generally, it appears that, for large κ, the new proposed method have a better performance than the Dobson`s and Best and Fisher`s methods. For smaller κ, new proposed method has the least biases and absolute relative biases but Dobson`s method is more efficient with smaller values of estimated standard error and RMSE. Hence, it is shown that new proposed method is superior for large κ. However, the new proposed method and the Dobson`s method could be used for small κ.