The Fisher information matrix I([GAMMA]) is given by the K x K symmetric matrix whose (u, v)th element is the covariance between uth and vth first

partial derivatives of the log-likelihood:

The

partial derivatives of the magnetic components (2) in the laboratory system are

The first

partial derivative of C3S with respect to f becomes:

After making use of relation (10), it is very easy to show that the

partial derivatives of the equilibrium quantities of loans ([L.sub.i]) with respect to parameters [alpha] and r and variables [D.sub.i] and [D.sub.j] have as follows:

The Jacobian determinant is a combination of

partial derivatives. Since these derivatives are evaluated by the BIEM, singularities occur when the unknown derivatives are located on the boundaries.

The way each of the three functionally relevant parameters E, D, and K depends upon each of the three governing growth-related parameters [alpha], [rho], and [delta] may be quantified by considering the corresponding

partial derivatives [partial derivative]E/[partial derivative][alpha], [partial derivative]E/[partial derivative][rho], [partial derivative]E/[partial derivative][delta], [partial derivative]D/[partial derivative][alpha], [partial derivative]D/[partial derivative][rho], [partial derivative]D/[partial derivative][delta], [partial derivative]K/[partial derivative][alpha], [partial derivative]K/[partial derivative][rho], [partial derivative]K/[partial derivative][delta].

The assumption that u [member of] [C.sup.2] enables the examination of

partial derivatives signs.

The following

partial derivatives obtained from differentiating the functions [F.sub.1] and [F.sub.2] are needed to determine the sign of [absolute value of [J.sub.1]]:

The notation [U.sub.X]k = [[[partial derivative].sup.k]U(X,t)]/[[partial derivative][X.sup.k]] takes care of all possible

partial derivatives where k is the highest

partial derivatives of U w.

where the

partial derivatives within the parentheses are normalized by the Compton radius [r.sub.c].

More formally, the expansion of the preceding determinant in (45) verifies that a symmetric matrix is obtained for [partial derivative][h.sub.i](P,k)/[partial derivative][P.sub.j] due to the commutation of second-order

partial derivatives of [P.sub.j]

where

partial derivatives of potential are the following: