# The distribution of the characteristic roots of*S* _{1} *S* _{2} ^{−1} under violations in the complex case and power comparisons of four tests

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## Summary

The joint density function of the latent roots of**S**_{1} **S**_{2} ^{−1} under violations is obtained where**S**_{1} has a complex non-central Wishart distribution**W**_{ c }(*p*,*n* _{1},**Σ**_{1},* Ω*) and

**S**_{2}, an independent complex central Wishart,

**W**_{ c }(

*p*,

*n*

_{2},

**Σ**_{2}, 0). The density and moments of Hotelling's trace are also derived under violations. Further, the non-null distributions of the following four criteria in the two-roots case are studied for tests of three hypotheses: Hotelling's trace, Pillai's trace, Wilks' criterion and Roy's largest root. In addition, tabulations of powers are carried out and power comparisons for tests of each of three hypotheses based on the four criteria are made in the complex case extending such work of Pillai and Jayachandran in the classical Gaussian case. The findings in the complex Gaussian are generally similar to those in the classical.

### AMS classifications

62H10 62H15### Key words and phrases

Complex Gaussian Characteristic roots distribution under violations non-normality unequal covariance matrices Hotelling's trace Pillai's trace Wilks' criterion Roy's largest root power comparisons tests of three hypotheses tabulations## Preview

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### References

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