Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Computation of a multivariate $ F$ distribution


Authors: D. E. Amos and W. G. Bulgren
Journal: Math. Comp. 26 (1972), 255-264
MSC: Primary 65D20
DOI: https://doi.org/10.1090/S0025-5718-1972-0298881-9
MathSciNet review: 0298881
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Methods for evaluating the joint cumulative probability integral associated with random variables $ {F_k} = ({X_k}/{r_k})/(Y/s),k = 1,2, \cdots ,n$, are considered where the $ {X_k}$ and $ Y$ are independently $ {\chi ^2}({r_k})$ and $ {\chi ^2}(s)$, respectively. For $ n = 2$, series representations in terms of incomplete beta distributions are given, while a quadrature with efficient procedures for the integrand is presented for $ n \geqq 2$. The results for $ n = 2$ are applied to the evaluation of the correlated bivariate $ F$ distribution.


References [Enhancements On Off] (What's this?)

  • [1] Donald E. Amos, Significant digit computation of certain distribution functions, Proc. Sympos. on Empirical Bayes Estimation and Computing in Statistics (Texas Tech. Univ., Lubbock, Texas, 1969) Texas Tech. Press, Lubbock, Tex., 1970, pp. 165–180. MR 0258241
  • [2] D. E. Amos & S. L. Daniel, Evaluation of Probabilities Associated with a Fixed Sample Selection Procedure, Sandia Corporation Research Report SC-RR-69-752 (available through Division 3151, Sandia Corporation, Albuquerque, N. M. 87115).
  • [3] D. E. Amos & S. L. Daniel, Significant Digit Incomplete Beta Ratios, Sandia Corporation Development Report SC-DR-69-591 (available through Division 3151, Sandia Corporation, Albuquerque, N. M. 87115).
  • [4] P. Appell & J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques Polynomes d'Hermite, Gauthier-Villars, Paris, 1926.
  • [5] L. D. Broemeling, ``Confidence regions for variance ratios of random models,'' J. Amer. Statist. Assoc., v. 64, 1969, pp. 660-664.
  • [6] V. Chew, ``Simultaneous prediction intervals,'' Technometrics, v. 10, 1968, pp. 323330.
  • [7] Benjamin Epstein, Estimation from life test data, Technometrics 2 (1960), 447–454. MR 0116447, https://doi.org/10.2307/1266453
  • [8] A. Erdélyi, et al., Higher Transcendental Functions. Vol. 1, McGraw-Hill, New York, 1953. MR 15, 419.
  • [9] A. Erdélyi, et al., Higher Transcendental Functions. Vol. 2, McGraw-Hill, New York, 1953. MR 15, 419.
  • [10] John Leroy Folks and Charles E. Antle, Straight line confidence regions for linear models, J. Amer. Statist. Assoc. 62 (1967), 1365–1374. MR 0221670
  • [11] W. Gautschi, ``Algorithm 222--incomplete beta function ratios,'' Comm. Assoc. Comput. Mach., v. 7, 1964, pp. 143-144; ``Certification of Algorithm 22,'' ibid., 1964, p. 244.
  • [12] Walter Gautschi, Computational aspects of three-term recurrence relations, SIAM Rev. 9 (1967), 24–82. MR 0213062, https://doi.org/10.1137/1009002
  • [13] Shanti S. Gupta, On a selection and ranking procedure for gamma populations, Ann. Inst. Statist. Math. 14 (1962/1963), 199–216. MR 0157457, https://doi.org/10.1007/BF02868642
  • [14] Shanti S. Gupta and Milton Sobel, On the smallest of several correlated 𝐹 statictics, Biometrika 49 (1962), 509–523. MR 0160300, https://doi.org/10.1093/biomet/49.3-4.509
  • [15] John E. Hewett, ``A Note on prediction intervals based on partial observations in certain life test experiments,'' Technometrics, v. 10, 1968, pp. 850-853.
  • [16] J. E. Hewett & W. G. Bulgren, ``Inequalities for some multivariate $ f$-distributions with applications,'' Technometrics, v. 13, 1971, pp. 397-402.
  • [17] D. R. Jensen, An inequality for a class of bivariate chi-square distributions, J. Amer. Statist. Assoc. 64 (1969), 333–336. MR 0240896
  • [18] C. G. Khatri, On certain inequalities for normal distributions and their applications to simultaneous confidence bounds, Ann. Math. Statist. 38 (1967), 1853–1867. MR 0220392, https://doi.org/10.1214/aoms/1177698618
  • [19] A. W. Kimball, On dependent tests of significance in the analysis of variance, Ann. Math. Statistics 22 (1951), 600–602. MR 0044091
  • [20] P. R. Krishnaiah, On the simultaneous 𝐴𝑁𝑂𝑉𝐴 and 𝑀𝐴𝑁𝑂𝑉𝐴 tests, Ann. Inst. Statist. Math. 17 (1965), 35–53. MR 0192592, https://doi.org/10.1007/BF02868151
  • [21] P. R. Krishnaiah & J. V. Armitage, Probability Integrals of the Multivariate $ F$ Distribution, with Tables and Applications, ARL 65-236 Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio, November 1965.
  • [22] P. R. Krishnaiah, Peter Hagis Jr., and Leon Steinberg, A note on the bivariate chi distribution, SIAM Rev. 5 (1963), 140–144. MR 0153069, https://doi.org/10.1137/1005034
  • [23] Jack Nadler, Inverse binomial sampling plans when an expontential distribution is sampled with censoring, Ann. Math. Statist. 31 (1960), 1201–1204. MR 0120729, https://doi.org/10.1214/aoms/1177705691
  • [24] K. R. Nair, The Studentized form of the extreme mean square test in the analysis of variance, Biometrika 35 (1948), 16–31. MR 0025124, https://doi.org/10.1093/biomet/35.1-2.16
  • [25] D. B. Owen, Some current problems in statistics requiring numerical results, Proc. Sympos. on Empirical Bayes Estimation and Computing in Statistics (Texas Tech. Univ., Lubbock, Texas, 1969) Texas Tech. Press, Lubbock, Tex., 1970, pp. 155–164. MR 0258226
  • [26] Paul Terrel Pope, On the Stepwise Construction of a Prediction Equation, Technical Report No. 37, Dept. of Statistics, SMU, 1969.
  • [27] George C. Tiao and Irwin Cuttman, The inverted Dirichlet distribution with applications, J. Amer. Statist. Assoc. 60 (1965), 793–805. MR 0185715

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65D20

Retrieve articles in all journals with MSC: 65D20


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1972-0298881-9
Keywords: Dirichlet distribution, maximum $ F$ distribution, multivariate $ {t^2}$ distribution, $ F$ with correlation, ranking and selection
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society