By Shun-ichi Amari

**From the reviews:** "In this Lecture be aware quantity the writer describes his differential-geometric method of parametrical statistical difficulties summarizing the consequences he had released in a sequence of papers within the final 5 years. the writer presents a geometrical framework for a *special* category of try and estimation methods for curved exponential households. ... ... the cloth and concepts offered during this quantity are vital and it's endorsed to every body attracted to the relationship among data and geometry ..." #*Metrika*#1 "More than hundred references are given exhibiting the becoming curiosity in differential geometry with appreciate to stats. The publication can purely strongly be advised to a geodesist because it bargains many new insights into facts on a well-recognized ground." #*Manuscripta Geodaetica*#2

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**Extra info for Differential-Geometrical Methods in Statistics**

**Example text**

In other words, the the basis vector di(a') at Ta , correponds to di(a) at Ta in 6-coordinate distributions form This system. a mixture is because family. 21). 6. Curvature and torsion In a manifold with an affine connection, the tangent space Ta at a can be mapped by an affine transformation to the tangent space T a , at a' along a curve a(t) connecting a and a'. This affine transformation depends in general on the curve connecting the two points. closed, A curve a(t), i. , a (to) = t € [to' is called a a (t l ) holds (to f given, tangent space Ta at a along the loop a(t).

Let a be a scalar parameter. dj R. (x, e)}dkR,(x, e)]. 26) when a = - 1. The covariant derivative with respect to the a-connection is denoted by v(a) . Let us define a third-order tensor by T ij k (e) = E [ aiR, (x, e) a j R, (x, e) ok R, (x, e) 1 . 28) Notice that its components change as _ -i-j-k Taay - BaBaByTijk under coordinate tensor. Such a quantity is called a The a-connection can be written as = r(l) + ijk r~<;tk) ~J which transformations. 29) calculating the coefficients of the a-connections.

K = m+l. m+2. n. to each A(u). a pair (u. v) specifies uniquely a point in S. This point is in the submanifold A(u) rigging u E M and has the coordinates v in A(u) one-dimensional. m = 1. ll. where M is is two-dimensional. ). Conversely. every point in S (or at least in a neighborhood of M) can uniquely be specified by a pair (u. v). a family A {A(u) I u E M} is More precisely. we say that smooth. when there exists a coordinate system v in each A(u) such that the pair (u. v) forms an allowable coordinate system of S.