Abstract
This paper presents properties of the inverse gamma distribution and how it can be used as a survival distribution. A result is included that shows that the inverse gamma distribution always has an upside-down bathtub (UBT) shaped hazard function, thus, adding to the limited number of available distributions with this property. A review of the utility of UBT distributions is provided as well. Probabilistic properties are presented first, followed by statistical properties to demonstrate its usefulness as a survival distribution. As the inverse gamma distribution is discussed in a limited and sporadic fashion in the literature, a summary of its properties is provided in an appendix.
This paper, originally published in The Journal of Quality Technology, Volume 43 in 2011, is another paper that relied very heavily on APPL procedures. The procedure , which derives the PDF of transformations of random variables was key to exploring various uses for the inverse of the gamma distributions. The procedure also helped derive the newly found distribution, which the author calls the Gamma Ratio distribution, a new one-parameter lifetime distribution. Creating distributions and determining their many probabilistic properties is exactly what APPL excels at.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Aalen, O. O., & Gjessing, H. K. (2001). Understanding the shape of the hazard rate: A process point of view. Statistical Sciences, 16(1), 1–22.
Bae, S. J., Kuo, W., & Kvam, P. H. (2007). Degradation models and implied lifetime distributions. Reliability Engineering and System Safety, 92, 601–608.
Cox, D. R., & Oakes, D. (1984). Analysis of survival data. London: Chapman & Hall/CRC.
Crowder, M. J., Kimber, A. C., Smith, R. L., & Sweeting, T. J. (1991). Statistical analysis of reliability data. London: Chapman & Hall/CRC.
Evans, G., Hastings, N., & Peacock, B. (2000). Statistical distributions (3rd ed.) New York: Wiley.
Gehan, E. A. (1965). A generalized Wilcoxon test for comparing arbitrarily singly-censored samples. Biometrika, 52, Parts 1 and 2, 203–223.
Gelman, A., Carling, J., Stern, H., & Rubin, D. (2004). Bayesian data analysis (2nd ed.). New York: Chapman & Hall/CRC.
Glen, A., & Leemis, L. (1997). The arctangent survival distribution. The Journal of Quality and Technology, 29(2), 205–210.
Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions (2nd ed., Vol. 2). New York: Wiley.
Kleiber, C., & Kotz, S. (2003). Statistical size distributions in economics and actuarial sciences. New York: Wiley.
Koch, K. (2007). Introduction to Bayesian statistics. Heidelberg: Springer.
Lai, C., & Xie, M. (2006). Stochastic aging and dependence for reliability. New York: Springer.
Lan, Y., & Leemis, L. (2008). The Logistic–Exponential survival distribution. Naval Research Logistics, 55, 254–264.
Lawless, J. F. (1980). Inference in the generalized gamma and log-gamma distributions. Technometrics, 22(3), 409–419.
Leemis, L., & Park, S. (2006). Discrete event simulation: A first course. Upper Saddle River: Pearson Prentice–Hall.
Lieblein, J., & Zelen, M. (1956). Statistical investigation of the fatigue life of deep-groove ball bearings. Journal of Research of the National Bureau of Standards, 57, 273–316.
Marshall, A. W., & Olkin, I. (2007). Life distributions: Structure of nonparametric, semiparametric, and parametric families. New York: Springer.
Meeker, W., & Escobar, L. (1998). Statistical methods for reliability data. New York: Wiley.
Milevsky, M. A., & Posner, S. E. (1998). Asian options, the sum of log-normals, and the reciprocal gamma distribution. The Journal of Financial and Quantitative Analysis, 33(3), 203–218.
Phillips, M. J., & Sweeting, T. J. (1996). Estimation for censored exponential data when the censoring times are subject to error. Journal of the Royal Statistical Society, Series B (Methodological), 58(4), 775–783.
Poirier, D. J. (1995). Intermediate statistics and econometrics. Cambridge: MIT Press.
Robert, C. (2007). The Bayesian choice (2nd ed.). New York: Springer.
Vargo, E., Pasupathy, R., & Leemis, L. (2010). Moment-ratio diagrams for univariate distributions. Journal of Quality Technology, 42(3), 276–286.
Witkovsky, V. (2001). Computing the distribution of a linear combination of inverted gamma variables. Kybernetika, 37(1), 79–90.
Witkovsky, V. (2002). Exact distribution of positive linear combinations of inverted Chi-square random variables with odd degrees of freedom. Statistics & Probability Letters, 56, 45–50.
Wolfram Software (2009). Mathematica Version 7.
Zellner, A. (1971). An introduction to bayesian inference in econometrics. New York: Wiley.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Glen, A.G. (2017). On the Inverse Gamma as a Survival Distribution. In: Glen, A., Leemis, L. (eds) Computational Probability Applications. International Series in Operations Research & Management Science, vol 247. Springer, Cham. https://doi.org/10.1007/978-3-319-43317-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-43317-2_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-43315-8
Online ISBN: 978-3-319-43317-2
eBook Packages: Business and ManagementBusiness and Management (R0)