In this chapter we discuss how to estimate the distribution of the market invariants from empirical observations. In Section 4.1 we define the concept of estimator and optimality criteria to evaluate an estimator. In Section 4.2 we introduce nonparametric estimators. In Section 4.3 we discuss maximum likelihood estimators and we compute the maximum likelihood estimators of location, dispersion, and factor loadings under the assumption that the market invariants are elliptically distributed. In Section 4.4 we discuss the shrinkage estimators for the location parameters, the dispersion parameters and the factor loadings of a linear model. In Section 4.5 we discuss robust estimation. In Section 4.6 we conclude with a series of practical tips to improve the estimation of the distribution of the market invariants in specific situations.
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