A formula for the curvature of the likelihood surface of a sample drawn from a distribution admitting sufficient statistics
Abstract
1. SUMMARY In this paper is given a formula for the curvature of the ‘likelihood surface’ (see §3) of a sample drawn from a distribution admitting sufficient statistics for the parameters, at the point represented by the maximum likelihood estimates (m.I.e.) of the parameters. More precisely it is shown in the case of two parameters that the negative of the trace and the determinant of Fisher's information matrix, respectively, measure the first and second (Gaussian) curvatures of the ’likelihood surface‘ at the point represented by the m.l.e.'s of the parameters. In the case of m parameters an expression for the Riemann curvature of the ‘likelihood hypersurface’ (see § 4) of a sample at the point represented by the m.l.e.'s is obtained in terms of the elements of Fisher's information matrix. An expression for Riemann's curvature invariant is also obtained. A large sample interpretation is given.
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