Abstract
Matrix inequalities play a large role in econometrics. Such inequalities essentially come in two kinds. One is concerned with inequalities of the scalar type involving scalar functions of matrices. A fine overview is offered by chapter 11 of the often-cited reference book of Magnus and Neudecker (1988). The title of this chapter refers to “Matrix inequalities” but the results given, though involving matrices, are in scalar form. Without distracting from the usefulness and the comprehensiveness of these results, an equally challenging type of inequalities are of the matrix type. Again, we may distinguish two types here. One involves element-by-element inequalities. These find many applications in mathematical economics. See e.g. Takayama (1974) for an overview. In econometrics (and in multivariate statistics in general) another type of matrix inequality occurs frequently. One is then concerned with so-called Löwner orderings, i.e., A > B means that A - B is a positive definite matrix, and A ≥ B means that A - B is a positive semidefinite (or nonnegative definite) matrix.
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Bekker, P., Wansbeek, T. (2000). Matrix Inequality Applications in Econometrics. In: Heijmans, R.D.H., Pollock, D.S.G., Satorra, A. (eds) Innovations in Multivariate Statistical Analysis. Advanced Studies in Theoretical and Applied Econometrics, vol 36. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-4603-0_3
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DOI: https://doi.org/10.1007/978-1-4615-4603-0_3
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