7.1 Point estimation of center and variability
In Chapters 5 and 6, center and variability of a sample and a random composition were defined. In particular, the normal distribution on the simplex was characterized by the mean vector and covariance matrix of coordinates with respect to a given basis of the simplex (Section 6.3.1). It is thus pertinent to consider which is the relationship between these sample statistics and distributional parameters and, in particular, how the latter can be estimated. To cut a long story short, the center parameter is estimated with the sample center, the coordinate mean parameter with the sample mean, and the covariance matrix parameter with the sample covariance. The theory of parameter estimation is mainly based on the concept of likelihood function both in frequentist and Bayesian statistics (see, e.g., Jeffreys, 1961; Box and Tiao, 1973; Robert, 1994; Shao, 1998). The estimation of center and variability of a random composition can also be developed from this point of view.
In general, if a random variable follows a known probability model that depends on a vector of unknown parameters and a random sample