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American Journal of Mathematical and Management Sciences Volume 9, 1989 - Issue 1-2: Distributions of Test Statistics: Exact & Asymptotic, Null & Non-Null, Methods/Comparisons/Research Frontiers
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Original Articles

Power Function Studies

E. M. CarterDepartment of Mathematics and StatisticsUniversity of GuelphGuelph, OntarioN1G 2W1
Pages 51-67 | Published online: 14 Aug 2013

  • AndersonG. A. (1965). An asymptotic expansion for the distribution of the latent roots of the estimated covariance matrix. Annals of Mathematical Statistics, 36, 1153–1173.
  • AndersonT. W. and Das GuptaS. (1964a). A montonicity property of the power function of some tests of the equality of two covariance matrices. Annals of Mathematical Statistics, 35, 1059–1063.
  • AndersonT. W. and Das GuptaS. (1964b). Monotonicity of the power functions of some tests of independence between two sets of variates. Annals of Mathematical Statistics, 35, 206–208.
  • BrownG. W. (1939). On the power of the L1-test for equality of several variances. Annals of Mathematical Statistics, 10, 119–128.
  • CarterE. M. and SrivastavaM. S. (1977). Monotonicity of the power functions of the modified likelihood ratio criterion for the homogeneity of variances and of the sphericity test. Journal of Multivariate Analysis, 7, 229–233.
  • CarterE. M. and SrivastavaM. S. (1980). Asymptotic distribution of the latent roots of the noncentral Wishart distribution and the power of the likelihood ratio test for nonadditivity. Canadian Journal of Statistics, 8, 119–134.
  • CarterE. M. and SrivastavaM. S. (1983). Asymptotic non–null distribution for the locally most powerful invariant test for sphericity. Communications in Statistics. Theory and Methods, 12, 1719–1725.
  • Chu, SylviaS. and PillaiK. C. S. (1979). Power comparisons of two sided tests of equality of two covariance matrices based on six criteria. Annals of the Institute of Statistical Mathematics, 31, 185–206.
  • Chu; SylviaS. and PillaiK. C. S. (1980). Some complex variable transformations and exact power comparisons of two-sided tests of equality of two Hermitian matrices. Journal of Statistical Planning and Inference, 4, 267–290.
  • ChouR. and MuirheadR. J. (1979). On some distribution problems in MANOVA and discriminant analysis. Journal of Multivariate Analysis, 9, 410–419.
  • ConsulP. C. (1967a). On the exact distributions of likelihood ratio criteria for testing independence of sets of variates under the null hypothesis, Annals of Mathematical Statistics, 38, 1160–1169.
  • ConsulP. C. (1967b). On the exact distribution of the criteria W for testing sphericity in a p-variate normal distribution, Annals of Mathematical Statistics, 38, 1170–1174.
  • Das GuptaS. (1969). Properties of power functions of some tests concerning dispersion matrices of multivariate normal distributions. Annals of Mathematical Statistics, 40, 697–701.
  • Das GuptaS., AndersonT. W. and MudholkarG. S. (1964). Montonicity of power functions of some tests of the multivariate linear hypothesis. Annals of Mathematical Statistics, 35, 200–205.
  • Das GuptaS. and PerlmanM. (1973). On the power of Wilks' 17-test for MANOVA. Journal of Multivariate Analysis, 8, 220–225.
  • DavisA. W. (1970). Exact distribution of Hotelling's T02 Biometrika, 57, 181–191.
  • DavisA. W. (1971). Percentile approximations for a class of likelihood ratio criteria, Biometrika, 58, 349–356.
  • DavisA.W. (1985). Hotelling's trace under moderate nonnormality, and other applications of the invariant polynomials, Statistical Theory and Data Analysis (MatusitaK. ed.), North-Holland, Amsterdam, 167–180.
  • EatonM. L. and PerlmanM. D. (1974). A monotonicity property of the power function of some invariant tests for MANOVA. Annals of Statistics, 2, 1022–1028.
  • FujikoshiY. (1970). Asymptotic expansions of the distributions of test statistics in multivariate analysis. Journal of the Science of Hiroshima University, Series A-I, 34, 293–299.
  • FujikoshiY. (1973a). Asymptotic formulae for the distributions of three test statistics for multivariate linear hypothesis. Annals of the Institute of Statistical Mathematics, 5, 423–437.
  • FujikoshiY. (1973b). Monotonicity of the power functions of some tests in general MANOVA models. Annals of Statistics, 1, 388–391.
  • FujikoshiY. (1974). On the asymptotic non-null distributions of the LR criterion in a general MANOVA. Canadian Journal of Statistics, 2, 1–12.
  • GiriN. (1968). On the test of equality of two covariance matrices. Annals of Mathematical Statistics, 39, 275–277.
  • GiriN., KieferJ. and SteinC. (1963). Minimax character of Hotelling's T2-test in the simplest case. Annals of Mathematical Statistics, 34, 1524–1535.
  • GnanadesikanR., LauhE., SnyderM. and YaoY. (1965). Efficiency comparisons of certain multivariate analysis of variance test procedures (preliminary report). (abstract). Annals of Mathematical Statistics, 36, 356.
  • HeckD. L. (1960). Charts of some upper percentage points of the distribution of the largest root. Annals of Mathematical Statistics, 31, 625–642.
  • HotellingH. (1931). The generalization of Student's ratio, Annals of Mathematical Statistics, 2, 360–378.
  • HotellingH. (1947). Multivariate quality control, illustrated by air testing of sample bombsights. Techniques of Statistical Analysis, pp 111–184, McGraw–Hill, New York.
  • JamesA. T. (1964). Distributions of matrix variates and latent roots derived from normal samples. Annals of Mathematical Statistics, 35, 475–501.
  • JamesA. T. (1966). Inference on latent roots by calculations of hypergeometric functions of matrix arguments. (Multivariate Analysis I, KrishnaiahP. R., ed.), Academic Press, New York.
  • JamesA. T. (1969). Tests of equality of latent roots of the covariance matrix. (Multivariate Analysis II, KrishnaiahP. R., ed.), Academic Press, New York.
  • JohnS. (1971). Some optimal multivariate tests. Biometrika, 38, 123–127.
  • JohnS. (1972). The distribution of a statistic used for testing sphericity of normal distributions. Biometrika, 39, 169–174.
  • KariyaT. (1978). The general MANOVA problem. Annals of Statistics, 6, 200–214.
  • KariyaT. (1981). A robustness property of Hotelling's T2–test. Annals of Statistics, 9, 210–213.
  • KhatriC. G. and PillaiK. C. S. (1968). On the noncentral distributions of two test criteria in multivariate analysis of variance. Annals of Mathematical Statistics, 39, 215–226.
  • KhatriC. G. and SrivastavaM. S. (1971). On exact non-null distributions of likelihood ratio criteria for sphericity test and equality of two covariance matrices. Sankhyā A, 33, 201–206.
  • KhatriC. G. and SrivastavaM. S. (1974). Asymptotic expansions of the nonnull distr-ibutions of likelihood ratio criteria for covariance matrices. Annals of Statistics, 2, 109–117.
  • KhatriC. G., SrivastavaM. S. and CarterE. M. (1979). The effect of the inequality of variances on the i-test. Sankhyā B, 41, 208–217.
  • KieferJ. and SchwartzR. (1965). Admissible Bayes character of T2, R2, and other fully invariant tests for classical multivariate normal problems, Annals of Mathematical Statistics, 36, 747–770.
  • LawleyD. N. (1938). A generalization of Fisher's z test, Biometrika, 30, 180–187.
  • LawleyD. N. (1956). Test of significance for the latent roots of covariance and correlation matrices. Biometrika, 43, 128–136.
  • LeeY. S. (1971). Asymptotic formulae for the distribution of a multivariate test statistic: power comparisons of certain multivariate tests. Biometrika, 58, 647–651.
  • MathaiA. M. and RathieP. N. (1970). The exact distribution for the sphericity test, Journal of Statistical Research, 4, 140–159.
  • MathaiA. M. and RathieP. N. (1971). The exact distribution of Wilks' criterion, Annals of Mathematical Statistics, 42, 1010–1019.
  • McLarenM. L. (1976). Coefficients of the zonal polynomials. Applied Statistics, 25, 82–87.
  • MikhailN. N. (1965). A comparison of tests of the Wilks-Lawley hypothesis in multivariate analysis. Biometrika, 52, 149–156.
  • MudholkarG. S. (1965). A class of tests with monotone power functions for two problems in multivariate statistical analysis. Annals of Mathematical Statistics, 36, 1794–1801.
  • MuirheadR. J. (1972). The asymptotic noncentral distribution of Hotelling's generalized T02. Annals of Mathematical Statistics, 43, 1491–1497.
  • MuirheadR. J. (1974). Powers of the largest latent root test of σ = I. Communications in Statistics, 3, 513–524.
  • MuirheadR. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York.
  • MuirheadR. J. and ChikuseY. (1975). Asymptotic expansions for the joint and marginal distributions of the latent roots of the covariance matrix. Annals of Mathematical Statistics, 3, 1011–1017.
  • MuirheadR. J. and WaternauxC. M. (1980). Asymptotic distributions in canonical correlation analysis and other multivariate procedures for nonnormal populations. Biometrika, 67, 31–43.
  • NagaoH. (1967). Monotonicity of the modified likelihood ratio test for a co-variance matrix. Journal of the Science of Hiroshima University, Series A–I, 4, 153–247.
  • NagaoH. (1973a). On some test criteria for covariance matrix. Annals of Statistics, 1, 700–709.
  • NagaoH. (1973b). Asymptotic expansions of the distributions of Bartlett's test and sphericity test under the local alternatives. Annals of the Institute of Statistics Mathematics, 25, 407–422.
  • NagaoH. (1974). Asymptotic non-null distributions of two test criteria for equality of covariance matrices under local alternatives. Annals of the Institute of Statistical Mathematics, 26, 396–402.
  • NagarsenkerB. N. and PillaiK. C. S. (1973). The distribution of the sphericity test criterion. Journal of Multivariate Analysis, 3, 226–235.
  • NandaD. N. (1950). Distribution of the sum of roots of a determinantal equation under a certain condition, Annals of Mathematical Statistics, 21, 432–439.
  • PerlmanM. D. (1974). On the monotonicity of the power function of tests based on traces of multivariate beta matrices. Journal of Multivariate Analysis, 4, 22–30.
  • PerlmanM. D. and OlkinI. (1980). Unbiasedness of invariant tests for MANOVA and other multivariate problems. Annals of Statistics, 8, 1326–1341.
  • PillaiK. C. S. (1955). Some new test criteria in multivariate analysis, Annals of Mathematical Statistics, 26, 117–121.
  • PillaiK. C. S. and GuptaA. K. (1969). On the exact distribution of Wilks' criterion. Biometrika, 56, 109–118.
  • PillaiK. C. S. and JayachandranD. (1967). Power comparisons of tests of two multivariate hypotheses based on four criteria, Biometrika, 54, 195–210.
  • PillaiK. C. S. and JayachandranD. (1968). Power comparisons of tests of equality of two covariance matrices based on four criteria. Biometrika, 55, 335–342.
  • RoyS. N. (1945). The individual sampling distributions of the maximum, the minimum and any intermediate of the p-statistics on the null hypotheses. Sankhyā A, 7, 135–158.
  • SarkarS. K. (1984). A note on the power of the likelihood ratio test for MANOVA. Sankhyā A, 46, 303–308.
  • SchatzoffM. (1966). Exact distribution of Wilks' likelihood ratio criterion. Biometrika, 53, 347–358.
  • SrivastavaJ. N. (1964). On the monotonicity property of the three main tests for multivariate analysis of variance. Journal of the Royal Statistical Society B, 26, 77–81.
  • SrivastavaM. S. and AwanH. W. (1982). On the robustness of Hotelling's T2 test and distribution of linear and quadratic forms in sampling from a mixture of two multivariate normal populations. Communications in Statistics. Theory and Methods, 11, 81–107.
  • SrivastavaM. S. and CarterE. M. (1980). Asymptotic expansions for hypergeo-metric functions. Multivariate Analysis V, 337–347, North-Holland, New York.
  • SrivastavaM. S. and KhatriC. G. (1979). An Introduction to Multivariate Statistics. Elsevier-North Holland, New York.
  • SrivastavaM. S., KhatriC. G. and CarterE. M. (1978). On some monotonicity property of the modified likelihood ratio test for the equality of two covariance matrices. Journal of Multivariate Analysis, 8, 262–267.
  • SteinC. (1956). The admissibility of Hotelling's T2-test. Annals of Mathematical Statistics, 27, 616–623.
  • SugiuraN. (1969). Asymptotic expansions of the distributions of the likelihood ratio criteria for covariance matrix. Annals of Mathematical Statistics, 40, 2051–2063.
  • SugiuraN. (1972). Locally best invariant test for sphericity and the limiting distributions. Annals of Mathematical Statistics, 43, 1312–1316.
  • SugiuraN. (1973a). Further asymptotic formulas for the non-null distributions of three statistics for multivariate linear hypothesis. Annals of the Institute of Statistical Mathematics, 25, 153–163.
  • SugiuraN. (1973b). Asymptotic non-null distributions of the likelihood ratio criteria for covariance matrix under local alternatives. Annals of Statistics, 1, 718–728.
  • SugiuraN. and NagaoH. (1968). Unbiasedness of some test criteria for the equality of one or two covariance matrices. Annals of Mathematical Statistics, 39, 1686–1692.
  • SugiyamaT. (1972). Percentile points of the largest latent root of a matrix and power calculations for testing σ = I. Journal of the Japan Statistical Society, 3, 1–8.
  • TangP. C. (1938). The power function of the analysis of variance tests with tables and illustrations of their use. Statistical Research Memoir, 2, 126–157.
  • TikuM. L. and SinghM. (1982). Robust statistics for testing mean vectors of multivariate distributions. Communications in Statistics. Theory and Methods, 11, 985–1001.
  • WaternauxC. M. (1976). Asymptotic distribution of the sample roots for a nonnormal population. Biometrika, 63, 639–645.
  • WilksS. S. (1946). Sample criteria for testing equality of means, equality of variances and equality of covariances in a normal multivariate distribution. Annals of Mathematical Statistics, 17, 257–281.

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