Resumen de On the asymptotic distribution of proportions of multinomial count data
Jan Graffelman, Juan José Egozcue Rubí, María Isabel Ortego Martínez
Compositional data is often transformed by taking log-ratios prior to the statistical analysis. The transformed variables are usually assumed to be approximately normally distributed. In this situation, random compositions follow (approximately) a normal distribution on the simplex. This justifies the posterior use of standard statistical techniques that rely on the normality assumption, such as Student t-tests, regression modeling, analysis of variance and other techniques and models. As a consequence, results supporting the asymptotic normality of log-ratio transformations are worth to be studied. In this contribution, a multinomial sampling scheme is considered. The distribution of the proportions of the counts in each category are transformed by an isometric log-ratio transformation (ilr). The asymptotic distribution of these coordinates is shown to be normal, and its mean and covariance parameters are expressed as functions of the multinomial probabilities. The theory is based on the known mean and variance of a multinomial variable and its asymptotic normal distribution. Following a previous study for trinomial variables (Graffelman, 2011), the delta-method, applied to the maximum likelihood estimators of frequencies, provides the asymptotic normal distribution of the isometric log-ratio coordinates. As a consequence, proportions from multinomial counts are asymptotically normal on the simplex and its parameters (center and variability) depend only on the multinomial parameters. Theoretical and simulated examples are presented.
