Abstract
Recursive formulas are provided for computing probabilities of a multinomial distribution. Firstly, a recursive formula is provided for computing rectangular probabilities which include the cumulative distribution function as a special case. These rectangular probabilities can be used to provide goodness-of-fit tests for the cell probabilities. The probability that a certain cell count is the maximum of all cell counts is also considered, which can be used to assess the probability that the maximum cell count corresponds to the cell with the maximum probability. Finally, a recursive formula is provided for computing the probability that the cell counts satisfy a certain ordering, which can be used to assess the probability that the ordering of the cell counts corresponds to the ordering of the cell probabilities. The computational intensity of these recursive formulas is linear in the number of cells, and they provide opportunities for calculating probabilities that would otherwise be computationally challenging. Some examples of the applications of these formulas are provided.
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References
Ahmad IA (1985) On the Poisson approximation of multinomial probabilities. Stat Probab Lett 3:55–56
Bromaghin JF (1993) Sample size determination for interval estimation of multinomial probabilities. Am Stat 47(3):203–206
Cai Y, Krishnamoorthy K (2006) Exact size and power properties of five tests for multinomial proportions. Commun Stat Simul Comput 35:149–160
Cressie N, Read TRC (1984) Multinomial goodness-of-fit tests. J R Stat Soc Ser B 46:440–464
Fitzpatrick S, Scott A (1987) Quick simultaneous confidence intervals for multinomial proportions. J Am Stat Assoc 82(399):875–878
Frey J (2009) An algorithm for computing rectangular multinomial probabilities. J Stat Comput Simul 79:1483–1489
Frey J (2012) Testing for positive evidence of equally likely outcomes. J Multivar Anal 103:48–57
Good IJ (1957) Saddle-point methods for the multinomial distribution. Ann Math Stat 4:861–881
Goodman LA (1965) On simultaneous confidence intervals for multinomial proportions. Technometrics 7:247–254
Hayter AJ (2006) Recursive integration methodologies with statistical applications. J Stat Plan Inference 136:2284–2296
Katti SK, Sastry AN (1965) Biological examples of small expected frequencies and the chi-square test. Biometrics 21:49–54
Lebrun R (2013) Efficient time/space algorithm to compute rectangular probabilities of multinomial, multivariate hypergeometric and multivariate Polya distributions. Stat Comput 23:615–623
Levin B (1981) A representation for multinomial cumulative distribution functions. Ann Math Stat 9(5):1123–1126
Mallows CL (1968) An inequality involving multinomial probabilities. Biometrika 55(2):422–424
Pearson K (1900) On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can reasonably be supposed to have arisen from random sampling. Philos Mag 5th Ser 50:157–175
Queensberry CP, Hurst DC (1964) Large sample simultaneous confidence intervals for multinational proportions. Technometrics 6:191–195
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I would like to sincerely thank each of the reviewers for their comments and suggestions that have resulted in a much improved version of this manuscript.
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Hayter, A.J. Recursive formulas for multinomial probabilities with applications. Comput Stat 29, 1207–1219 (2014). https://doi.org/10.1007/s00180-014-0487-0
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DOI: https://doi.org/10.1007/s00180-014-0487-0