By Paul Marriott, Mark Salmon
Differential geometry has turn into a regular device within the research of statistical versions, delivering a deeper appreciation of latest methodologies and highlighting the problems that may be hidden in an algebraic improvement of an issue. This quantity is the 1st to use those thoughts to econometrics. An introductory bankruptcy offers a short instructional for these strange with the instruments of differential geometry. the subsequent chapters supply functions of geometric how to sensible recommendations and supply perception into difficulties of econometric inference.
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Additional info for Applications of Differential Geometry to Econometrics
These are not formally distance functions as they do not obey the standard axioms for a distance, such as symmetry or the triangle inequality. For a clari®cation of their geometric role, see Critchley, Marriott and Salmon (1994). Amari's projection theorem connects these divergence functions with the geodesic structure. We give a simpli®ed version of his theorem here, for exponential families. For more details see Amari (1990, p. 89). Theorem Let p x j be a full exponential family, and p x j a curved subfamily.
The likethe conditional resolution form x ; lihood can then be written as a: ` Y x ` Y ; We use the following notation `i @ a; ` Y ; @i and `;i @ a: ` Y ; @i 22 In this notation it can be shown that gij À`ij `i; j is the observed Fisher information metric, and Tijk `ijk is the skewness tensor. 6 Three key results In this section we look at the key mathematical theorems of statistical manifolds; the applications to statistics are shown in sections 7 and 8.
These are called the Christoffel symbols for the connection. Note that Condition 5 implies that, for all i; j and k, Àkij Àkji ; hence the connection is called symmetric. It is important to note that the Christoffel symbols do not transform as tensors. If Àcab are the Christoffel symbols relative to the -parameters and À~ kij relative to the -parameters, then the two sets of components are connected according to the rule Àcab @k c @2ab k @a i @b j À~ kij : 16 One consequence of this not being a tensor rule is that Christoffel symbols can be identically zero with respect to one parameterisation, but non-zero in another.