If the log likelihood function is twice differentiable with respect to the parameter ?, and under certain regularity conditions, then the Fisher information may also be written as follows ( which is often a more convenient form for calculation purposes ):
22.
In a more general set-up, for a multi-input-multi-output production processes, one may assume technology can be represented via some technology set, call it \ T, which must satisfy some regularity conditions of production theory.
23.
Figueiredo and Jain note that convergence to'meaningless'parameter values obtained at the boundary ( where regularity conditions breakdown, e . g ., Ghosh and Sen ( 1985 ) ) is frequently observed when the number of model components exceeds the optimal / true one.
24.
The theorem also holds if balls are replaced, in the definition of the derivative, by families of sets with diameter tending to zero satisfying the " Lebesgue's regularity condition ", defined above as " family of sets with bounded eccentricity ".
25.
Parametric approaches have the advantage that likelihood theory provides an'off the shelf'approach to inference ( although for mixture models such as the \ varepsilon-contamination model, the usual regularity conditions might not apply ), and it is possible to build simulation models from the fit.
26.
It can also be proved that there exists a unique ( up to multiplication by a positive constant ) right-translation-invariant Borel measure \ nu satisfying the above regularity conditions and being finite on compact sets, but it need not coincide with the left-translation-invariant measure \ mu.
