Moslim et al. / Malaysian Journal of Fundamental and Applied Sciences Special Issue on Some Advances in Industrial and Applied Mathematics (2017) 390-393 390 On the approximation of the concentration parameter for von Mises distribution Nor Hafizah Moslim a, b , Yong Zulina Zubairi b,* , Abdul Ghapor Hussin c , Siti Fatimah Hassan b , Rossita Mohamad Yunus d a Faculty of Industrial Sciences & Technology, Universiti Malaysia Pahang, 26300 Gambang, Pahang b Centre for Foundation Studies in Science, University of Malaya, 50603 Kuala Lumpur c Faculty of Defence Sciences and Technology, National Defence University of Malaysia, Kem Sungai Besi, 57000 Kuala Lumpur d Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur * Corresponding author: yzulina@um.edu.my Article history Received 13 October 2017 Accepted 8 November 2017 Abstract The von Mises distribution is the ‘natural’ analogue on the circle of the Normal distribution on the real line and is widely used to describe circular variables. The distribution has two parameters, namely mean direction, and concentration parameter, κ. Solutions to the parameters, however, cannot be derived in the closed form. Noting the relationship of the κ to the size of sample, we examine the asymptotic normal behavior of the parameter. The simulation study is carried out and Kolmogorov- Smirnov test is used to test the goodness of fit for three level of significance values. The study suggests that as sample size and concentration parameter increase, the percentage of samples follow the normality assumption increase. Keywords: circular variable, concentration parameter, Monte Carlo, von Mises © 2017 Penerbit UTM Press. All rights reserved INTRODUCTION Statistical data can be classified according to their distributional topologies. A linear data set can be represented on a straight line and for circular data, they can be represented by the circumference of a circle. The data is commonly measured in the range of ( ] 0 ,360 ° ° degree or ( ] 0,2π radian. It is worthwhile to note that statistical theories for straight line and circle are very different from one to another because the circle is a closed curve but line is not. The application of directional statistics can be found in the area of meteorology such as wind direction. Circular data can be found in many fields (Mardia, 1972; Mardia et al., 2000; Amos, 1974). A circular random variable from a von Mises or Circular Normal distribution has a density function of () ( ) ( ) cos 0 1 , 0 2 2 f e I κ θ µ θ θ π π κ − = ≤ ≤ where 0 2 and 0 µ π κ ≤ ≤ ≥ are the parameters. ( ) 0 I κ is the modified Bessel function of order zero and can be defined as ( ) ( ) cos 2 0 0 1 2 I e d κ θ µ π κ θ π − = ∫ This distribution is the ‘natural’ analogue on the circle of the normal distribution on the real line and has few similar characteristics with the normal distribution (Dobson, 1978). It has been proved that the von Mises distribution can be approximated to the standard normal distribution for sufficiently large κ (Jammalamadaka et al., 2001; Fisher, 1993). As , κ →∞ ( ) ( ) 0,1 d N β κθ µ = − ⇒ Let ( ) β κθ µ = − . For large κ , β θ µ κ − = is small and from the Taylor series expansion 2 1 cos 2 θ θ − = , it gives ( cos ) cos β θ µ κ ⎛ ⎞ − = ⎜ ⎟ ⎝ ⎠ κ β 2 1 2 − = (1) Using the change of variable formula, ( ) ( ) ( ) 1 g f g θ β β β − ∂ = ∂ ( ) cos 0 1 2 e I β κ κ π κ κ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = (2) RESEARCH ARTICLE