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A note on computing extreme tail probabilities of the noncentral $$t$$-distribution with large noncentrality parameter. (English) Zbl 1302.62034
Summary: The noncentral $$t$$-distribution is a generalization of the Student’s $$t$$-distribution. In this paper we suggest an alternative approach for computing the cumulative distribution function (CDF) of the noncentral $$t$$-distribution which is based on a direct numerical integration of a well behaved function. With a double-precision arithmetic, the algorithm provides highly precise and fast evaluation of the extreme tail probabilities of the noncentral $$t$$-distribution, even for large values of the noncentrality parameter $$\delta$$ and the degrees of freedom $$\nu$$. The implementation of the algorithm is available at the MATLAB Central [http://www.mathworks.com/matlabcentral/fileexchange/41790-nctcdfvw].
##### MSC:
 62E15 Exact distribution theory in statistics 62-04 Software, source code, etc. for problems pertaining to statistics
##### Software:
advanpix; Boost; Mathematica; Matlab; NCTCDFVW; PROBT; R; StInt
Full Text:
##### References:
 [1] Abramowitz M., Stegun I. A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Tenth Edition, National Bureau of Standards, 1972. · Zbl 0543.33001 · www.cs.bham.ac.uk [2] Airey, J. R., Irwin, J. O., Fisher, R. A.: Introduction to Tables of $$Hh$$ Functions. British Association for the Advancement of Science, Mathematical Tables 1, XXIV-XXXV, 1931. [3] Benton D., Krishnamoorthy K.: Computing discrete mixtures of continuous distributions: noncentral chisquare, noncentral $$t$$ and the distribution of the square of the sample multiple correlation coefficient. Computational Statistics & Data Analysis 43 (2003), 249-267. · Zbl 1429.62058 · doi:10.1016/S0167-9473(02)00283-9 [4] Bristow, P. A., Maddock, J.: DistExplorer: Statistical Distribution Explorer. Boost Software License, Edition: Version 1.0., 2012 · sourceforge.net [5] Guenther, W. C.: Evaluation of probabilities for the noncentral distributions and the difference of two $$t$$ variables with a desk calculator. Journal of Statistical Computation and Simulation 6 (1978), 199-206. · doi:10.1080/00949657808810188 [6] Hahn, G. J., Meeker, G. J.: Statistical Intervals: A Guide for Practitioners. John Wiley & Sons, New York, 1991. · Zbl 0850.62763 [7] Holoborodko P.: Multiprecision Computing Toolbox for MATLAB. Advanpix, Yokohama. Edition: Version 3.4.3, 2013 · www.advanpix.com" [8] Inglot, T.: Inequalities for quantiles of the chi-square distribution. Probability and Mathematical Statistics 30 (2010), 339-351. · Zbl 1231.62092 · www.math.uni.wroc.pl [9] Inglot, T., Ledwina, T.: Asymptotic optimality of a new adaptive test in regression model. Annales de l’Institut Henri Poincaré 42 (2006), 579-590. · Zbl 1098.62053 · doi:10.1016/j.anihpb.2005.05.002 · numdam:AIHPB_2006__42_5_579_0 · eudml:77909 [10] Janiga, I., Garaj, I.: One-sided tolerance factors of normal distributions with unknown mean and variability. Measurement Science Review 8 (2006), 12-16. [11] Johnson, N. L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions, Volume 2. Second Edition, John Wiley & Sons, New York, 1995. · Zbl 0821.62001 [12] Kim, J.: Efficient Confidence Inteval Methodologies for the Noncentrality Parameters of the Noncentral $$T$$-Distributions. PhD Thesis, H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 2007. [13] Krishnamoorthy, K., Mathew, T.: Statistical Tolerance Regions: Theory, Applications, and Computation. John Wiley & Sons, New York, 2009. · Zbl 1291.60001 · doi:10.1002/9780470473900 [14] Lenth, R. V.: Algorithm AS 243 - Cumulative distribution function of the non-central $$t$$ distribution. Applied Statistics 38 (1989), 185-189. · doi:10.2307/2347693 [15] Maddock, J., Bristow, P. A., Holin, H., Zhang, X., Lalande, B., Rade, J., Sewani, G., van den Berg, T., Sobotta, B.: Noncentral $$T$$ Distribution. Boost C++ Libraries, Edition: Version 1.53.0, 2012 · www.boost.org" [16] The MathWorks Inc.: MATLAB Edition: Version 8.0.0.783 (R2012b). Natick, Massachusetts, 2012 · www.mathworks.com" [17] R Development Core Team.: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Edition: Version 3.0.0, Vienna, Austria, 2013 · www.R-project.org" [18] SAS Institute Inc.: PROBT Function. SAS(R) 9.3 Functions and CALL Routines: Reference. 2013 · support.sas.com [19] Student: The probable error of a mean. Biometrika 6 (1908), 1-25. · doi:10.1093/biomet/6.1.1 [20] Wolfram Research, Inc.: Mathematica Edition: Version 9.0. Wolfram Research, Inc., Champaign, Illinois, 2013 · www.wolfram.com
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