International Statistical Review

Percentage Points of the Multivariate t Distribution

Saralees Nadarajah and Samuel Kotz

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Abstract

The known methods for computing percentage points of multivariate t distributions are reviewed. We believe that this review will serve as an important reference and encourage further research activities in the area.

Article information

Source
Internat. Statist. Rev., Volume 74, Number 1 (2006), 15-30.

Dates
First available in Project Euclid: 29 March 2006

Permanent link to this document
https://projecteuclid.org/euclid.isr/1143654384

Zentralblatt MATH identifier
1142.62356

Keywords
Multivariate normal distribution Multivariate t distribution Percentage points

Citation

Nadarajah, Saralees; Kotz, Samuel. Percentage Points of the Multivariate t Distribution. Internat. Statist. Rev. 74 (2006), no. 1, 15--30. https://projecteuclid.org/euclid.isr/1143654384


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References

  • [1] Ahner, C. & Passing, H. (1983). Berechnung der Multivariaten t-Verteilung und simultane Vergleiche gegen eine Kontrolle bei ungleichen Gruppenbesetzungen. EDV in Medizin und Biologie, 14, 113-120.
  • [2] Amos, D.E. (1978). Evaluation of some cumulative distribution functions by numerical evaluation. SIAM Review, 20, 778-800.
  • [3] Ando, A. & Kaufman, G.W. (1965). Bayesian analysis of the independent multi-normal process-neither mean nor precision known. Journal of the American Statistical Association, 60, 347-358.
  • [4] Armitage, J.V. & Krishnaiah, R.R. (1965). Tables of percentage points of multivariate t distribution (abstract). Annals of Mathematical Statistics, 36, 726.
  • [5] Bechhofer, R.E. & Dunnett, C.W. (1988). Tables of percentage points of multivariate t distributions. In Selected Tables in Mathematical Statistics, 11, Eds. R.E. Odeh and J.M. Davenport. American Mathematical Society, Providence, Rhode Island.
  • [6] Bechhofer, R.E., Dunnett, C.W. & Sobel, M. (1954). A two-sample multiple-decision procedure for ranking means of normal populations with a common unknown variance. Biometrika, 41, 170-176.
  • [7] Blattberg, R.C. & Gonedes, N.J. (1974). A comparison of the stable and Student distributions as statistical models for stock prices. Journal of Business, 47, 224-280.
  • [8] Bowden, D.C. & Graybill, F.A. (1966). Confidence bands of uniform and proportional width for linear models. Journal of the American Statistical Association, 61, 182-198.
  • [9] Chen, H.J. (1979). Percentage points of multivariate t distribution with zero correlations and their application. Biometrical Journal, 21, 347-360.
  • [10] Chien, J.-T. (2002). A Bayesian prediction approach to robust speech recognition and online environmental testing. Speech Communication, 37, 321-334.
  • [11] Cornish, E.A. & Fisher, R.A. (1950). Moments and cumulants in the specification of distributions. In Contributions to Mathematical Statistics. New York: John Wiley and Sons.
  • [12] DasGupta, A., Ghosh, J.K. & Zen, M.M. (1995). A new general method for constructing confidence sets in arbitrary dimensions: with applications. Annals of Statistics, 23, 1408-1432.
  • [13] Dickey, J.M. (1967). Matric-variate generalizations of the multivariate t distribution and the inverted multivariate t distribution. Annals of Mathematical Statistics, 38, 511-518.
  • [14] Dunn, O.J. (1958). Estimation of the means of dependent variables. Annals of Mathematical Statistics, 29, 1095-1111.
  • [15] Dunn, O.J. (1961). Multiple comparison among means. Journal of the American Statistical Association, 56, 52-64.
  • [16] Dunn, O.J. & Massey, F.J. (1965). Estimation of multiple contrasts using t-distributions. Journal of the American Statistical Association, 60, 573-583.
  • [17] Dunnett, C.W. (1955). A multiple comparison procedure for comparing several treatments with a control. Journal of the American Statistical Association, 50, 1096-1121.
  • [18] Dunnett, C.W. (1964). New tables for multiple comparisons with a control. Biometrics, 20, 482-491.
  • [19] Dunnett, C.W. (1985). Multiple comparisons between several treatments and a specified treatment. In Linear Statistical Inference, Lecture Notes in Statistics No. 35, Eds. T. Cali\'{n}ski and W. Klonecki, pp. 39-46. New York: Springer-Verlag.
  • [20] Dunnett, C.W. & Sobel, M. (1954). A bivariate generalization of Student's t-distribution with tables for certain special cases. Biometrika, 41, 153-169.
  • [21] Dunnett, C.W. & Sobel, M. (1955). Approximations to the probability integral and certain percentage points of a multivariate analogue of Student's t-distribution. Biometrika, 42, 258-260.
  • [22] Dunnett, C.W. & Tamhane, A.C. (1990). A step-up multiple test procedure. Technical Report 90-1, Department of Statistics, Northwestern University.
  • [23] Dunnett, C.W. & Tamhane, A.C. (1991). Step-down multiple tests for comparing treatments with a control in unbalanced one-way layouts. Statistics in Medicine, 10, 939-947.
  • [24] Dunnett, C.W. & Tamhane, A.C. (1992). A step-up multiple test procedure. Journal of the American Statistical Association, 87, 162-170.
  • [25] Dunnett, C.W. & Tamhane, A.C. (1995). Step-up multiple testing of parameters with unequally correlated estimates. Biometrics, 51, 217-227.
  • [26] Dutt, J.E., Mattes, K.D., Soms, A.P. & Tao, L.C. (1976). An approximation to the maximum modulus of the trivariate T with a comparison to exact values. Biometrics, 32, 465-469.
  • [27] Dutt, J.E., Mattes, K.D. & Tao, L.C. (1975). Tables of the trivariate t for comparing three treatments to a control with unequal sample sizes. G.D. Searle and Company, Math. and Statist. Services, TR-3.
  • [28] Edwards, D.E. & Berry, J.J. (1987). The efficiency of simulation based multiple comparisons. Biometrics, 43, 913-928.
  • [29] Fernandez, C. & Steel, M.F.J. (1999). Multivariate Student-t regression models: pitfalls and inference. Biometrika, 86, 153-167.
  • [30] Fisher, R.A. (1941). The asymptotic approach to Behren's integral with further tables for the d-test of significance. Ann. Eugen., Lond., 11, 141. \pagebreak
  • [31] Fraser, D.A.S. & Haq, M.S. (1969). Structural probability and prediction for the multivariate model. Journal of the Royal Statistal Society, 31, 317-331.
  • [32] Freeman, H. & Kuzmack, A. (1972). Tables of multivariate t in six or more dimensions. Biometrika, 59, 217-219.
  • [33] Freeman, H., Kuzmack, A. & Maurice, R. (1967). Multivariate t and the ranking problem. Biometrika, 54, 305-308.
  • [34] Geisser, S. & Cornfield, J. (1963). Posterior distributions for multivariate normal parameters. Journal of the Royal Statistical Society B, 25, 368-376.
  • [35] Genz, A. & Bretz, F. (1999). Numerical computation of multivariate t probabilities with application to power calculation of multiple contrasts. Journal of Statistical Computation and Simulation, 63, 361-378.
  • [36] Goldberg, H. & Levine, H. (1946). Approximate formulas for the percentage points and normalization of t and χ^2. Annals of Mathematical Statistics, 17, 216.
  • [37] Graybill, F.A. & Bowden, D.C. (1967). Linear segment confidence bands for simple linear models. Journal of the American Statistical Association, 62, 403-408.
  • [38] Gupta, S.S. (1963). Probability integrals of multivariate normal and multivariate t. Annals of Mathematical Statistics, 34, 792-828.
  • [39] Gupta, S.S., Nagel, K. & Panchapakesan, S. (1973). On the order statistics from equally correlated normal random variables. Biometrika, 60, 403-413.
  • [40] Gupta, S.S., Panchapakesan, S. & Sohn, J.K. (1985). On the distribution of the studentized maximum of equally correlated normal random variables. Communications in Statistics-Simulation and Computation, 14, 103-135.
  • [41] Gupta, S.S. & Sobel, M. (1957). On a statistic which arises in selection and ranking problems. Annals of Mathematical Statistics, 28, 957-967.
  • [42] Hahn, G.J. & Hendrickson, R.W. (1971). A table of percentage points of the distribution of the largest absolute value of k Student t variates and its application. Biometrika, 58, 323-332.
  • [43] Halperin, M., Greenhouse, S.W., Cornfield, J. & Zalokar, J. (1955). Tables of percentage points for the studentized maximum absolute deviate in normal samples. Journal of the American Statistical Association, 50, 185-195.
  • [44] Hochberg, Y. & Tamhane, A.C. (1987). Multiple Comparison Procedures\/. New York: John Wiley and Sons.
  • [45] Hsu, J.C. (1992). The factor analytic approach to simultaneous inference in the general linear model. Journal of Computational and Graphical Statistics, 1, 151-168.
  • [46] Hsu, J.C. & Nelson, B.L. (1998). Multiple comparisons in the general linear model. Journal of Computational and Graphical Statistics, 7, 23-41.
  • [47] Iyengar, S. (1988). Evaluation of normal probabilities of symmetric regions. SIAM Journal on Scientific and Statistical Computing, 9, 418-423.
  • [48] Jensen, D.R. (1994). Closure of multivariate t and related distributions. Statistics and Probability Letters, 20, 307-312.
  • [49] Krishnaiah, P.R. & Armitage, J.V. (1966). Tables for multivariate t distribution. Sankhy\=a B, 28, 31-56.
  • [50] Kwong, K.-S. (2001a). A modified Dunnett and Tamhane step-up approach for establishing superiority/equivalence of a new treatment compared with k standard treatments. Journal of Statistical Planning and Inference, 97, 359-366.
  • [51] Kwong, K.-S. (2001b). An algorithm for construction of multiple hypothesis testing. Computational Statistics, 16, 165-171.
  • [52] Kwong, K.-S. & Iglewicz, B. (1996). On singular multivariate normal distribution and its applications. Computational Statistics and Data Analysis, 22, 271-285.
  • [53] Kwong, K.-S. & Liu, W. (2000). Calculation of critical values for Dunnett and Tamhane's step-up multiple test procedure. Statistical Probability Letters, 49, 411-416.
  • [54] Lauprete, G.J., Samarov, A.M. & Welsch, R.E. (2002). Robust portfolio optimization. Metrika, 55, 139-149.
  • [55] Lee, R.E. & Spurrier, J.D. (1995). Successive comparisons between ordered treatments. Journal of Statistical Planning and Inference, 43, 323-330.
  • [56] Liu, C. (1995). Missing data imputation using the multivariate t distribution. Journal of Multivariate Analysis, 53, 139-158.
  • [57] Liu, C. (1996). Bayesian robust multivariate linear regression with incomplete data. Journal of the American Statistical Association, 91, 1219-1227.
  • [58] Liu, C. & Rubin, D.B. (1995). ML estimation of the multivariate t distribution with unknown degrees of freedom. Statistica Sinica, 5, 19-39.
  • [59] Liu, W., Miwa, T. & Hayter, A.J. (2000). Simultaneous confidence interval estimation for successive comparisons of ordered treatment effects. Journal of Statistical Planning and Inference, 88, 75-86.
  • [60] McCann, M. & Edwards, D. (1996). A path inequality for the multivariate t distribution, with applications to multiple comparisons. Journal of the American Statistical Association, 91, 211-216.
  • [61] McLachlan, G.J. & Peel, D. (1998). Robust cluster analysis via mixtures of multivariate t-distributions. In Lecture Notes in Computer Science, 1451, Eds. A. Amin, D. Dori, P. Pudil and H. Freeman, pp. 658-666. Berlin: Springer-Verlag.
  • [62] McLachlan, G.J., Peel, D., Basford, K.E. & Adams, P. (1999). Fitting of mixtures of normal and t components. Journal of Statistical Software, 4.
  • [63] Nason, G.P. (2000). Analytic formulae for projection indices in a robustness experiment. Technical Report 00:06, Department of Mathematics, University of Bristol.
  • [64] Nason, G.P. (2001). Robust projection indices. Journal of the Royal Statistical Society B, 63, 551-567.
  • [65] Paulson, E. (1952). On the comparison of several experimental categories with a control. Annals of Mathematical Statistics, 23, 239-246.
  • [66] Pearson, K. (1923). On non-skew frequency surfaces. Biometrika, 15, 231.
  • [67] Pearson, K. (1931). Tables for Statisticians and Biometricians, Part II. London: Cambridge University Press for the Biometrika Trust.
  • [68] Peel, D. & McLachlan, G.J. (2000). Robust mixture modelling using the t distribution. Statistics and Computing, 10, 339-348. \pagebreak
  • [69] Pillai, K.C.S. & Ramachandran, K.V. (1954). Distribution of a Studentized order statistic. Annals of Mathematical Statistics, 25, 565-571.
  • [70] Press, S.J. (1972). Applied Multivariate Analysis\/. New York: Holt, Rinehart and Winston, Inc.
  • [71] Rausch, W. & Horn, M. (1988). Applications and tabulations of the multivariate t distribution with ρ= 0. Biometrical Journal, 30, 595-605.
  • [72] Rubin, D.B. (1983). Iteratively reweighted least squares. In Encyclopedia of Statistical Sciences 4, Eds. S. Kotz and N. L. Johnson, pp. 272-275. New York: John Wiley and Sons.
  • [73] Siotani, M. (1964). Interval estimation for linear combinations of means. Journal of the American Statistical Association, 59, 1141-1164.
  • [74] Somerville, P.N. (1997). Multiple testing and simultaneous confidence intervals: Calculation of constants. Computational Statistics and Data Analysis, 25, 217-233.
  • [75] Somerville, P.N. (1998). Numerical computation of multivariate normal and multivariate t probabilities over convex regions. Journal of Computational and Graphical Statistics, 7, 529-544.
  • [76] Somerville, P.N., Miwa, T., Liu, W. & Hayter, A. (2001). Combining one-sided and two-sided confidence interval procedures for successive comparisons of ordered treatment effects. Biometrical Journal, 43, 533-542.
  • [77] Spurrier, J.D. & Isham, S.P. (1985). Exact simultaneous confidence intervals for pairwise comparisons of three normal means. Journal of the American Statistical Association, 80, 438-442.
  • [78] Steffens, F.E. (1969). Critical values for bivariate Student t-tests. Journal of the American Statistical Association, 64, 637-646.
  • [79] Sutradhar, B.C. (1990). Discrimination of observations into one of two t populations. Biometrics, 46, 827-835.
  • [80] Sutradhar, B.C. & Ali, M.M. (1986). Estimation of the parameters of a regression model with a multivariate t error variable. Communications in Statistics-Theory and Methods, 15, 429-450.
  • [81] Sweeting, T.J. (1984). Approximate inference in location-scale regression models. Journal of the American Statistical Association, 79, 847-852.
  • [82] Sweeting, T.J. (1987). Approximate Bayesian analysis of censored survival data. Biometrika, 74, 809-816.
  • [83] Tiao, G.C. & Zellner, A. (1964). On the Bayesian estimation of multivariate regression. Journal of the Royal Statistical Society B, 26, 277-285.
  • [84] Tong, Y.L. (1970). Some probability inequalities of multivariate normal and multivariate t. Journal of the American Statistical Association, 65, 1243-1247.
  • [85] Trout, J.R. & Chow, B. (1972). Table of the percentage points of the trivariate t distribution with an application to uniform confidence bands. Technometrics, 14, 855-879.
  • [86] Yang, Z.Q. & Zhang, C.M. (1997). Dimension reduction and L_1-approximation for evaluations of multivariate normal integrals. Chinese Journal of Numerical Mathematics and Applications, 19, 82-95.
  • [87] Zellner, A. (1971). An Introduction to Bayesian Inference in Econometrics\/. New York: John Wiley and Sons.
  • [88] Zellner, A. (1976). Bayesian and non-Bayesian analysis of the regression model with multivariate Student-t error terms. Journal of the American Statistical Association, 71, 400-405.