By Kanti V. Mardia, Peter E. Jupp
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Extra resources for Directional Statistics (Wiley Series in Probability and Statistics)
L - ms(n-*/2c}]). 6) (Batxhelet, 1965, reproduced by permission, @) 1999 -41nt~ic,m Institute of Biological Sciences). For large K ? 21), B + < =3 K - y -to ( K - 2 ) , N(0,I). Hence for large K , M ( / ! L , K ) jK C - ' / * ( 8 -p) + X ( 0 , l), K, -# 30. 64). ricmornients. 30),w e have hf(/d, K ) N WN(/6,A(K,)), K -# OC. ioxi for large K, Kent (1978) hm shown that the approximation holds to a higher order in K . 24) p, A ( K ) )denotc the densities of the von Miscs where f v ~ ~ (p,. hc? approximating wrapped normal distribution IV,V[p, A ( K ) ) ,respectively.
It turns out that the appropriate way of constructing sumiary statistics for circular data is to regard points on the circle as unit vectors in the plane. and then to take polnr coordixiatw of the sample Inem of t h e e vectors. ive1y. The idea, of 'multiplying angles by p' (for p = 1,2?.. 4. e. the circle of unit radius centred at the origin). h direr:tions - as arigles a i d as unit complex numbers. Choose an initial direction and an orientation for the unit circle. d by a11 arigle 8 or equivalently by a unit complex number z.
1) Extcnding this notion, we define the pth trigonometric moment about the m u darectdora for p = 1,2, . 4) where 8, arid Rp denote the sample rnaari dirm%ionmid sample xxieim resiiltmt length of @ I , . . ,p&. 6) that mI = R. 1) play an important role in the theory of distributions on the circle. ,x,, on the line with sample mean f. The line can be wrapped onto the circle to give angles ?. Bi 3= 11 + mi mod 2n, i = 1,. . ,n. e3 + o(e5), 1 = 1 - ,e2 2! 1 + _e4 + o(es) 4! 10)of trigonometric moments of concentrated distributioris 0x1 the circle motivate the itsf!
Directional Statistics (Wiley Series in Probability and Statistics) by Kanti V. Mardia, Peter E. Jupp