Logarithmic concavity of the inverse incomplete beta function with respect to the first parameter

  • Dimitris Askitis

Abstract

The beta distribution is a two-parameter family of probability distributions whose distribution function is the (regularised) incomplete beta function. In this paper, the inverse incomplete beta function is studied analytically as a univariate function of the first parameter. Monotonicity, limit results and convexity properties are provided. In particular, logarithmic concavity of the inverse incomplete beta function is established. In addition, we provide monotonicity results on inverses of a larger class of parametrised distributions that may be of independent interest.

References

Adell, J. A. and Jodrá, P., On a Ramanujan equation connected with the median of the gamma distribution, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3631–3644. https://doi.org/10.1090/S0002-9947-07-04411-X

Alm, S. E., Monotonicity of the difference between median and mean of gamma distributions and of a related Ramanujan sequence, Bernoulli 9 (2003), no. 2, 351–371. https://doi.org/10.3150/bj/1068128981

Andrews, G. E., Askey, R., and Roy, R., Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. https://doi.org/10.1017/CBO9781107325937

Berg, C. and Pedersen, H. L., The Chen-Rubin conjecture in a continuous setting, Methods Appl. Anal. 13 (2006), no. 1, 63–88. https://doi.org/10.4310/MAA.2006.v13.n1.a4

Berg, C. and Pedersen, H. L., Convexity of the median in the gamma distribution, Ark. Mat. 46 (2008), no. 1, 1–6. https://doi.org/10.1007/s11512-006-0037-2

Çınlar, E., Probability and stochastics, Graduate Texts in Mathematics, vol. 261, Springer, New York, 2011. https://doi.org/10.1007/978-0-387-87859-1

Chen, J. and Rubin, H., Bounds for the difference between median and mean of gamma and Poisson distributions, Statist. Probab. Lett. 4 (1986), no. 6, 281–283. https://doi.org/10.1016/0167-7152(86)90044-1

Choi, K. P., On the medians of gamma distributions and an equation of Ramanujan, Proc. Amer. Math. Soc. 121 (1994), no. 1, 245–251. https://doi.org/10.2307/2160389

Gupta, R. D. and Kundu, D., Generalized exponential distributions, Aust. N. Z. J. Stat. 41 (1999), no. 2, 173–188. https://doi.org/10.1111/1467-842X.00072

Johnson, N. L., Kotz, S., and Balakrishnan, N., Continuous univariate distributions. Vol. 2, second ed., Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1995.

Karp, D. B., Normalized incomplete beta function: Log-concavity in parameters and other properties, J. Math. Sci. (N.Y.) 217 (2016), no. 1, 91–107. https://doi.org/10.1007/s10958-016-2958-z

Klenke, A., Probability theory: A comprehensive course, second ed., Universitext, Springer, London, 2014. https://doi.org/10.1007/978-1-4471-5361-0

Koumandos, S. and Pedersen, H. L., On the asymptotic expansion of the logarithm of Barnes triple gamma function, Math. Scand. 105 (2009), no. 2, 287–306. https://doi.org/10.7146/math.scand.a-15119

Krantz, S. G. and Parks, H. R., The implicit function theorem: History, theory, and applications, Birkhäuser Boston, 2002. https://doi.org/10.1007/978-1-4612-0059-8

Nadarajah, S., Exponentiated beta distributions, Comput. Math. Appl. 49 (2005), no. 7-8, 1029–1035. https://doi.org/10.1016/j.camwa.2004.11.008

NIST digital library of mathematical functions, http://dlmf.nist.gov/, Release 1.0.24 of 2019-09-15, F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.

Payton, M. E., Young, L. J., and Young, J. H., Bounds for the difference between median and mean of beta and negative binomial distributions, Metrika 36 (1989), no. 6, 347–354. https://doi.org/10.1007/BF02614111

Temme, N. M., Asymptotic inversion of the incomplete beta function, J. Comput. Appl. Math. 41 (1992), no. 1–2, 145–157. https://doi.org/10.1016/0377-0427(92)90244-R
Published
2021-02-17
How to Cite
Askitis, D. (2021). Logarithmic concavity of the inverse incomplete beta function with respect to the first parameter. MATHEMATICA SCANDINAVICA, 127(1), 111-130. https://doi.org/10.7146/math.scand.a-121924
Section
Articles