Matrix Differentiation

David A. Harville ()
Additional contact information
David A. Harville: IBM T.J. Watson Research Center, Mathematical Sciences Department

Chapter 15 in Matrix Algebra From a Statistician’s Perspective, 1997, pp 289-335 from Springer

Abstract: Abstract It is natural and convenient to use matrix notation and terminology in defining and discussing certain functions of one or more variables. Earlier (in Chapter 14), matrix notation and terminology were used in defining and discussing linear, bilinear, and quadratic forms. In some cases, the use of matrix notation and terminology is essentially unavoidable—consider, for example, a case where the determinant of an m × m matrix is regarded as a function of its m2 elements. Matrix differentiation is the derivation of the first-, second-, or higher-order partial derivatives of a function or functions that have been expressed in terms of matrices. In deriving, presenting, and discussing the partial derivatives of such functions, it is natural, convenient, and in some cases necessary to employ matrix notation and terminology. Not only may the functions be expressed in terms of matrices, but the functions may comprise the elements of a vector or matrix, as would be the case if the elements of the inverse of an m×m (nonsingular) matrix A were regarded as functions of the elements of A. Matrix differentiation is of considerable importance in statistics. It is especially useful in connection with the maximum likelihood estimation of the parameters in a statistical model. The maximum likelihood estimates of the model’s parameters satisfy the equations (known as the likelihood equations) obtained by equating to zero the first-order partial derivatives (with respect to the model’s parameters) of the logarithm of the so-called likelihood function—in many important cases, the likelihood function involves the determinant and/or inverse of a matrix. Further, an approximation (suitable for large samples) to the variance-covariance matrix of the maximum likelihood estimators can be obtained by inverting the matrix (known as the information matrix) whose ij th element is–1 times the second-order partial derivative (with respect to the i th and j th parameters) of the logarithm of the likelihood function.

Date: 1997
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-0-387-22677-4_15

Ordering information: This item can be ordered from
http://www.springer.com/9780387226774

DOI: 10.1007/0-387-22677-X_15

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().