A test for the significance of the mean direction and the concentration parameter of a circular distribution.

W. S. Rhode
March, 1976

This note describes the development of the Rayleigh test for the significance of the mean direction in a cycle (period) histogram. It is important to know what confidence is to be placed in the mean and concentration parameter (sync coefficient). This is a problem in circular distributions. It is described in the Statistics of Directional Data by K.V. Mardia (AP, 1972).

We shall assume that the underlying population is von Mises,

where U0 is the mean direction and k is the concentration parameter and is a modified Bessel function of the first kind.

Tests of Uniformity

Let q 1,...q n be a random sample from a population with p.d.f. f(q We are interested in testing the null hypothesis

against the alternative

where g(q) has the given form but may contain unknown parameters. The parameters U0,k will be assumed unknown in M(U0,k ).

We want to obtain the likelihood ratio test for this situation. Consider a random sample X1,X2,.,Xn from a distribution having p.d.f. f(x,q ),q e W . The joint p.d.f. may be regarded as a function of q . When so regarded, it is called the likelihood function L of the random sample, and we write

 

We seek a function, say U(x1,.,xn) such that when q is replaced by

U(x1,.,xn) the likelihood function, L, is a maximum.

 

Therefore

=

 

 

The maximal likelihood statistics are and .

 

To determine and , often the derivative of the log L is used.

 

(1)

 

(2)

 

(3)

 

Let

 

therefore

 

(4)

 

this derives from eqn.(2)

 

® eqn.(4).

 

From eqn.(3)

 

or, A(k )=R

 

(4)

 

The likelihood ratio statistic is

 

 

 

(5)

 

now,

 

If q is distributed as M(0,k ) then

 

 

since and

therefore , using this in eqn. (5) implies that l is a monotonically

decreasing function of .

 

,therefore is a monotonically increasing function of .

 

The critical region (CR) l < K reduces to and

 

 

The pdf of R under H0 is

 

 

and the pdf of R under H1 is

 

 

Note that CR is that subset of a space which leads us to reject the H0 hypothesis.

The power function of a test of a statistical hypothesis H0 against an alternative

hypothesis H1 is that function which yields the probability that the sample point

falls in the critical region C of the test, i.e., a function which yields the

prob. of rejecting the hypothesis under consideration. The value of the power

function at a parameter point is called the power of the test.

 

Critical values* of the Rayleigh test of uniformity with the test-statistics

n a 0.10 0.05 0.025 0.01 0.001

5 0.677 0.754 0.816 0.879 0.991

6 .618 .690 .753 .825 .940

7 .572 .642 .702 .771 .891

8 .535 .602 .660 .725 .847

9 .504 .569 .624 .687 .808

 

10 .478 .540 .594 .655 .775

11 .456 .516 .567 .627 .743

12 .437 .494 .544 .602 .716

13 .420 .475 .524 .580 .692

14 .405 .458 .505 .560 .669

 

15 .391 .443 .489 .542 .649

16 .379 .429 .474 .525 .630

17 .367 .417 .460 .510 .613

18 .357 .405 .447 .496 .597

19 .348 .394 .436 .484 .583

 

20 .339 .385 .425 .472 .569

21 .331 .375 .415 .461 .556

22 .323 .367 .405 .451 .544

23 .316 .359 .397 .441 .533

24 .309 .351 .389 .432 .522

 

25 .303 .344 .381 .423 .512

30 .277 .315 .348 .387 .470

35 .256 .292 .323 .359 .436

40 .240 .273 .302 .336 .409

45 .226 .257 .285 .318 .386

 

50 .214 .244 .270 .301 .367

100 .15 .17 .19 .21 .26

 

4.605 5.991 7.378 9.210 13.816

 

*Based on Table 2 of Stephens (1969d) with the kind permission of the author and the editor of J. Amer. Statist. Ass. and on Batschelet (1971) with the kind permission of the author and the publisher, Amer. Inst. Biol. Sciences and Dr. W.T. Keeton, New York University.