Mean and variance of ratios of proportions from categories of a multinomial distribution.

*(English)*Zbl 1394.62012Summary: Ratio distribution is a probability distribution representing the ratio of two random variables, each usually having a known distribution. Currently, there are results when the random variables in the ratio follow (not necessarily the same) Gaussian, Cauchy, binomial or uniform distributions. In this paper we consider a case, where the random variables in the ratio are joint binomial components of a multinomial distribution. We derived formulae for mean and variance of this ratio distribution using a simple Taylor-series approach and also a more complex approach which uses a slight modification of the original ratio. We showed that the more complex approach yields better results with simulated data. The presented results can be directly applied in the computation of confidence intervals for ratios of multinomial proportions.

##### MSC:

62E15 | Exact distribution theory in statistics |

62H05 | Characterization and structure theory for multivariate probability distributions; copulas |

60E05 | Probability distributions: general theory |

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\textit{F. Duris} et al., J. Stat. Distrib. Appl. 5, Paper No. 2, 20 p. (2018; Zbl 1394.62012)

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##### References:

[1] | Aho, K; Bowyer, RT, Confidence intervals for ratios of proportions: implications for selection ratios, Methods Ecol. Evol, 6, 121-132, (2015) |

[2] | Alghamdi, N: Confidence intervals for ratios of multinomial proportions (2015). Masterâ€™s thesis, University if Nebraska at Omaha. |

[3] | Basu, A; Lochner, RH, On the distribution of the ratio of two random variables having generalized life distributions, Technometrics, 13, 281-287, (1971) · Zbl 0217.51002 |

[4] | Bonett, DG; Price, RM, Confidence intervals for a ratio of binomial proportions based on paired data, Stat. Med, 25, 3039-3047, (2006) |

[5] | Chiu, RW; Chan, KA; Gao, Y; Lau, VY; Zheng, W; Leung, TY; Foo, CH; Xie, B; Tsui, NB; Lun, FM; etal., Noninvasive prenatal diagnosis of fetal chromosomal aneuploidy by massively parallel genomic sequencing of DNA in maternal plasma, Proc. Natl. Acad. Sci., 105, 20458-20463, (2008) |

[6] | Culley, TM; Wallace, LE; Gengler-Nowak, KM; Crawford, DJ, A comparison of two methods of calculating gst, a genetic measure of population differentiation, Am. J. Bot., 89, 460-465, (2002) |

[7] | Fieller, E, The distribution of the index in a normal bivariate population, Biometrika, 24, 428-440, (1932) · Zbl 0006.02103 |

[8] | Frishman, F: On the arithmetic means and variances of products and ratios of random variables (1971). Technical report, DTIC Document. |

[9] | Geary, R, The frequency distribution of the quotient of two normal variates, J. R. Stat. Soc., 93, 442-446, (1930) · JFM 56.1094.01 |

[10] | Goodman, LA, On simultaneous confidence intervals for multinomial proportions, Technometrics, 7, 247-254, (1965) · Zbl 0131.17701 |

[11] | Graham, RL, Knuth, DE, Patashnik, O: Concrete Mathematics: A Foundation for Computer Science, 2nd edn, p. 492. Addison-Wesley Longman Publishing Co., Inc., Boston (1994). exercise 42. · Zbl 0836.00001 |

[12] | Hinkley, DV, On the ratio of two correlated normal random variables, Biometrika, 56, 635-639, (1969) · Zbl 0183.48101 |

[13] | Koopman, P, Confidence intervals for the ratio of two binomial proportions, Biometrics, 40, 513-517, (1984) |

[14] | Korhonen, PJ; Narula, SC, The probability distribution of the ratio of the absolute values of two normal variables, J. Stat. Comput. Simul., 33, 173-182, (1989) · Zbl 0726.62024 |

[15] | Lau, TK; Chen, F; Pan, X; Pooh, RK; Jiang, F; Li, Y; Jiang, H; Li, X; Chen, S; Zhang, X, Noninvasive prenatal diagnosis of common fetal chromosomal aneuploidies by maternal plasma DNA sequencing, J. Matern. Fetal Neonatal Med., 25, 1370-1374, (2012) |

[16] | Marsaglia, G, Ratios of normal variables, J. Stat. Softw., 16, 1-10, (2006) |

[17] | Mekic, E; Sekulovic, N; Bandjur, M; Stefanovic, M; Spalevic, P, The distribution of ratio of random variable and product of two random variables and its application in performance analysis of multi-hop relaying communications over fading channels, Przegl. Elektrotechniczny, 88, 133-137, (2012) |

[18] | Minarik, G; Repiska, G; Hyblova, M; Nagyova, E; Soltys, K; Budis, J; Duris, F; Sysak, R; Bujalkova, MG; Vlkova-Izrael, B; etal., Utilization of benchtop next generation sequencing platforms ion torrent pgm and miseq in noninvasive prenatal testing for chromosome 21 trisomy and testing of impact of in silico and physical size selection on its analytical performance, PloS ONE, 10, 0144811, (2015) |

[19] | Nadarajah, S, On the product and ratio of Laplace and Bessel random variables, J. Appl. Math., 2005, 393-402, (2005) · Zbl 1092.60005 |

[20] | Nadarajah, S; Kotz, S, On the ratio of Pearson type vii and Bessel random variables, Adv. Decis. Sci., 2005, 191-199, (2005) · Zbl 1196.62013 |

[21] | Nadarajah, S; Kotz, S, On the product and ratio of gamma and Weibull random variables, Econ. Theory, 22, 338-344, (2006) · Zbl 1139.62302 |

[22] | Nelson, W, Statistical methods for the ratio of two multinomial proportions, Am. Stat., 26, 22-27, (1972) |

[23] | Pham-Gia, T, Distributions of the ratios of independent beta variables and applications, Commun. Stat. Theory Methods, 29, 2693-2715, (2000) · Zbl 1107.62309 |

[24] | Piegorsch, WW; Richwine, KA, Large-sample pairwise comparisons among multinomial proportions with an application to analysis of mutant spectra, J. Agric. Biol. Environ. Stat., 6, 305-325, (2001) |

[25] | Piper, J; Rutovitz, D; Sudar, D; Kallioniemi, A; Kallioniemi, O-P; Waldman, FM; Gray, JW; Pinkel, D, Computer image analysis of comparative genomic hybridization, Cytometry, 19, 10-26, (1995) |

[26] | Poorter, H; Garnier, E, Plant growth analysis: an evaluation of experimental design and computational methods, J. Exp. Bot., 47, 1343-1351, (1996) |

[27] | Press, SJ, The t-ratio distribution, J. Am. Stat. Assoc., 64, 242-252, (1969) |

[28] | Price, RM; Bonett, DG, Confidence intervals for a ratio of two independent binomial proportions, Stat. Med., 27, 5497-5508, (2008) |

[29] | Provost, S, On the distribution of the ratio of powers of sums of gamma random variables, Pak. J. Stat., 5, 157-174, (1989) · Zbl 0707.62029 |

[30] | Quesenberry, CP; Hurst, D, Large sample simultaneous confidence intervals for multinomial proportions, Technometrics, 6, 191-195, (1964) · Zbl 0129.32605 |

[31] | Sakamoto, H, On the distributions of the product and the quotient of the independent and uniformly distributed random variables, Tohoku Math. J. First Ser., 49, 243-260, (1943) · Zbl 0063.06652 |

[32] | Sehnert, AJ; Rhees, B; Comstock, D; de Feo, E; Heilek, G; Burke, J; Rava, RP, Optimal detection of fetal chromosomal abnormalities by massively parallel DNA sequencing of cell-free fetal DNA from maternal blood, Clin. Chem., 57, 1042-1049, (2011) |

[33] | Van Kempen, G; Van Vliet, L, Mean and variance of ratio estimators used in fluorescence ratio imaging, Cytometry, 39, 300-305, (2000) |

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