Under certain regularity conditions, the Fisher Information Matrix may also be written in the following form, which is often more convenient for computation:
12.
Under certain regularity conditions, it can be shown that the first moment of the score ( that is, its expected value ) is 0:
13.
If is twice differentiable with respect to " ? ", and under certain regularity conditions, then the Fisher information may also be written as
14.
The left Haar measure satisfies the inner regularity condition for all ?-finite Borel sets, but may not be inner regular for " all"
15.
Under some weak regularity conditions on the mapping \ Phi for any function f : S \ to \ mathbf { R }, we have the almost sure convergence
16.
In order for a minimum point x ^ * to satisfy the above KKT conditions, the problem should satisfy some regularity conditions; the most used ones are listed below:
17.
However, as the volume is assumed to contain contiguous tissue fibers, it may be preferable to estimate the volume of diffusion tensors in its entirety by imposing regularity conditions on the underlying field of tensors.
18.
In 1976 Verdier developed a useful regularity condition on stratified sets that the Chinese-Australian mathematician Tzee-Char Kuo had previously shown implied the Whitney conditions for subanalytic sets ( such as real or complex analytic varieties ).
19.
The regularity condition is a kind of Newton polygon condition, in the sense that the allowed poles are in a region, when plotted against " i ", bounded by a line at 45?to the axes.
20.
These statements are true for a single f, by standard CLT arguments under regularity conditions, and the difficulty in the Empirical Processes comes in because joint statements are being made for all f \ in \ mathcal { F }.
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