On optimal tests for isotropy against the symmetric wrapped stable-circular uniform mixture family

The family of Symmetric Wrapped Stable (SWS) distributions can be widely used for modelling circular data. Mixtures of Circular Uniform (CU) with the former also have applications as a larger family of circular distributions to incorporate possible outliers. Restricting ourselves to such a mixture, we derive the locally most powerful invariant (LMPI) test for the hypothesis of isotropy or randomness of directions-expressed in terms of the null value of the mixing proportion, p, in the model. Global monotonicity of the power function of the test is established. The test is also consistent. Power values of the test for some selected parameter combinations, obtained through simulation reveal quite encouraging performances even for moderate sample sizes. The P 3 approach (SenGupta, 1991; Pal & SenGupta, 2000) for unknown p and rho and the nonregular case of unknown a, the index parameter, are also discussed. A real-life example is presented to illustrate the inadequacy of the circular normal distribution as a circular model. This example is also used to demonstrate the applications of the LMPI test, optimal P 3 test and a Daviesmotivated test (Davies, 1977, 1987). Finally, a goodnessof-fit test performed on the data establishes the plausibility of the above SWS-CU mixture model for real-life problems encountered in practical situations.