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Keywords:
noncentral $t$-distribution; cumulative distribution function (CDF); noncentrality parameter; extreme tail probability; MATLAB algorithm
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, File Exchange: www.mathworks.com/matlabcentral/fileexchange/41790-nctcdfvw.
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
[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. DOI 10.1016/S0167-9473(02)00283-9 | MR 1985338
[4] Bristow, P. A., Maddock, J.: DistExplorer: Statistical Distribution Explorer. Boost Software License, Edition: Version 1.0., 2012 http://sourceforge.net/projects/distexplorer/
[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 http://www.advanpix.com
[8] Inglot, T.: Inequalities for quantiles of the chi-square distribution. Probability and Mathematical Statistics 30 (2010), 339–351. MR 2792589 | Zbl 1231.62092
[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. DOI 10.1016/j.anihpb.2005.05.002 | MR 2259976
[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. MR 1326603
[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. MR 2710174
[13] Krishnamoorthy, K., Mathew, T.: Statistical Tolerance Regions: Theory, Applications, and Computation. John Wiley & Sons, New York, 2009. MR 2500599
[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 http://www.boost.org
[16] The MathWorks Inc.: MATLAB Edition: Version 8.0.0.783 (R2012b). Natick, Massachusetts, 2012 http://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 http://www.R-project.org
[18] SAS Institute Inc.: PROBT Function. SAS(R) 9.3 Functions and CALL Routines: Reference. 2013 http://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 http://www.wolfram.com/mathematica/

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