The exact and near-exact distributions of the main likelihood ratio test statistics used in the complex multivariate normal setting

Test

Volumn 24, Issue 2, 2015, Pages 386-416

  Coelho, Carlos A ^a   Arnold, Barry C ^b   Marques, Filipe J ^a  

^a UNIVERSIDADE NOVA DE LISBOA   (Portugal)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Covariance matrix; Equality of covariance matrices; Equality of mean vectors; Expected value matrix; Fourier transforms; Generalized integer gamma (GIG) distribution; Generalized near integer gamma (GNIG) distribution; Independence; Mixtures; Sphericity; Statistical distributions (distribution functions)

Indexed keywords

EID: 84929841661     PISSN: 11330686     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11749-014-0418-y     Document Type: Article

References (32)

1. Anderson TW (2003) An introduction to multivariate statistical analysis, 3rd edn. Wiley, New York
2. Box GEP (1949) A general distribution theory for a class of likelihood criteria. Biometrika 36:317–346
3. Brillinger DR (2001) Time series: data analysis and theory. SIAM, Philadelphia
4. Brillinger DR, Krishnaiah PR (eds) (1982) Handbook of statistics. Time series in frequency domain, vol 2. North Holland, Amsterdam
5. Coelho CA (1998) The generalized integer gamma distribution—a basis for distributions in multivariate statistics. J Multivar Anal 64:86–102
6. Coelho CA, Marques FJ (2009) The advantage of decomposing elaborate hypotheses on covariance matrices into conditionally independent hypotheses in building near-exact distributions for the test statistics. Linear Algebra Appl 430:2592–2606
7. Coelho CA, Marques FJ (2012) Near-exact distributions for the likelihood ratio test statistic to test equality of several variance–covariance matrices in elliptically contoured distributions. Comput Stat 27:627–659
8. Conradsen K (2012) Multivariate analysis of polarimetric SAR images. In: LINSTAT’12-international conference trends and perspectives in linear statistical inference and IWMS’12–21st international workshop on matrices and statistics. Abstract Book, pp 87–88
9. Conradsen K, Nielsen AA, Schou J, Skiver H (2003) A test statistic in the complex Wishart distribution and its application to change detection in polarimetric SAR data. IEEE Trans Geosci Remote Sens 41:4–19
10. Fang C, Krishnaiah PR, Nagarsenker BN (1982) Asymptotic distributions of the likelihood ratio test statistics for covariance structures of the complex multivariate normal distributions. J Multivar Anal 12:597–611
11. Goodman NR (1963a) Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction). Ann Math Stat 34:152–177
12. Goodman NR (1963b) The distribution of the determinant of a complex Wishart distributed matrix. Ann Math Stat 34:178–180
13. Gupta AK (1971) Distribution of Wilks’ likelihood-ratio criterion in the complex case. Ann Inst Stat Math 23:77–87
14. Gupta AK (1976) Nonnull distribution of Wilks’ statistic for MANOVA in the complex case. Commun Stat Simul Comput 5:177–188
15. Gupta AK, Rathie PN (1983) Nonnull distributions of Wilks’ Lambda in the complex case. Statistica 43:445–450
16. Jensen ST (1988) Covariance hypotheses which are linear in both the covariance and the inverse covariance. Ann Stat 16:302–322
17. Khatri CG (1965) Classical statistical analysis based on a certain multivariate complex Gaussian distribution. Ann Math Stat 36:98–114
18. Krishnaiah PR, Lee JC, Chang TC (1976) The distributions of the likelihood ratio statistics for tests of certain covariance structures of complex multivariate normal populations. Biometrika 63:543–549
19. Lehmann NH, Fishler E, Haimovich AM, Blum RS, Chizhik D, Cimini LJ, Valenzuela RA (2007) Evaluation of transmit diversity in MIMO-radar direction finding. IEEE Trans Signal Process 55:2215–2225
20. Marques FJ, Coelho CA, Arnold BC (2011) A general near-exact distribution theory for the most common likelihood ratio test statistics used in multivariate analysis. Test 20:180–203
21. Mathai AM (1973) A few remarks about some recent articles on the exact distributions of multivariate test criteria: I. Ann Inst Stat Math 25:557–566
22. Nagar DK, Jain SK, Gupta AK (1985) Distribution of LRC for testing sphericity of a complex multivariate Gaussian model. Int J Math Math Sci 8:555–562
23. Nagarsenker BN, Das MM (1975) Exact distribution of sphericity criterion in the complex case and its percentage points. Commun Stat 4:363–374
24. Nagarsenker BN, Nagarsenker PB (1981) Distribution of the likelihood ratio statistic for testing sphericity structure for a complex normal covariance matrix. Sankhya Ser B 43:352–359
25. Pannu NS, Coy AJ, Read J (2003) Application of the complex multivariate normal distribution to crystallographic methods with insights into multiple isomorphous replacement phasing. Acta Crystallogr D Biol Crystallogr 59:1801–1808
26. Pillai KCS, Jouris GM (1971) Some distribution problems in the multivariate complex Gaussian case. Ann Math Stat 42:517–525
27. Pillai KCS, Nagarsenker BN (1971) On the distribution of the sphericity test criterion in classical and complex normal populations having unknown covariance matrices. Ann Math Stat 42:764–767
28. Shumway RH, Stoffer DS (2006) Time series analysis and its applications: with R examples, 2nd edn. Springer, New York
29. Tang J, Gupta AK (1986) Exact distribution of certain general test statistics in multivariate analysis. Aust J Stat 28:107–114
30. Tricomi FG, Erdélyi A (1951) The asymptotic expansion of a ratio of gamma functions. Pac J Math 1:133–142
31. Turin GL (1960) The characteristic function of Hermitian quadratic forms in complex normal variables. Biometrika 47:199–201
32. Wooding RA (1956) The multivariate distribution of complex normal variables. Biometrika 43:212–215