Proportional inverse Gaussian distribution: A new tool for analysing continuous proportional data

Abstract

Outcomes in the form of rates, fractions, proportions and percentages often appear in various fields. Existing beta and simplex distributions are frequently unable to exhibit satisfactory performances in fitting such continuous data. This paper aims to develop the normalised inverse Gaussian (N-IG) distribution proposed by Lijoi, Mena & Prünster (2005, Journal of the American Statistical Association, 100, 1278-1291) as a new tool for analysing continuous proportional data in (0, 1) and renames the N-IG as proportional inverse Gaussian (PIG) distribution. Our main contributions include: (i) To overcome the difficulty of an integral in the PIG density function, we propose a novel minorisationmaximisation (MM) algorithm via the continuous version of Jensen's inequality to calculate the maximum likelihood estimates of the parameters in the PIG distribution; (ii) We also develop an MM algorithm aided by the gradient descent algorithm for the PIG regression model, which allows us to explore the relationship between a set of covariates with the mean parameter; (iii) Both the comparative studies and the real data analyses show that the PIG distribution is better when comparing with the beta and simplex distributions in terms of the AIC, the Cramér-von Mises and the Kolmogorov-Smirnov tests. In addition, bootstrap confidence intervals and testing hypothesis on the symmetry of the PIG density are also presented. Simulation studies are conducted and the hospital stay data of Barcelona in 1988 and 1990 are analysed to illustrate the proposed methods.

Bivariate Continuous Negatively Correlated Proportional Models with Applications in Schizophrenia Research

“…The details of constructing the N-EM algorithm are shown in Appendix B.1, and other relevant calculations are given in Supplementary Material A.1 and A.2. Finally, the ML estimates of (θ 0 , θ 1 , θ 2 ) can be obtained by combining (7) and ( 8) when the algorithm stops.…”

“…where H(•) is a concave function, τ(•) is a real-valued function and g(•) is a pdf defined on X ⊆ R [7]. Then, we have log…”

“…By mimicking the construction of beta distributions with gamma variates, Lijoiu et al [6] proposed a so-called normalized inverse Gaussian (IG) distribution by substituting the gamma variates with IG variates, as a new tool for modeling univariate proportional data. Later, Liu et al [7] renamed it as the proportional inverse Gaussian (PIG) distribution and set up regression models. Due to the diversity and dimension enlarger of data, we need to generalize the univariate continuous proportional models to multivariate cases.…”