Newton Algorithms for Analytic Rotation: an Implicit Function Approach

Boik, Robert J.

doi:10.1007/s11336-007-9027-y

Cited by 6 publications

(10 citation statements)

References 38 publications

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“…It was shown in [4] (also see [15,Eq. 28]) that if the less restrictive constraint, T T 1 D = I m , is imposed and minimizes Q( ), then and must satisfy…”

Section: Application To Factor Analysismentioning

confidence: 85%

“…Browne [5] described explicit parameterizations for the rotation matrix T, subject either to T T 1 D = I m or to T T = I m . Implicit parameterizations were described by Boik [4]. Employing either type of parameterization and choosing T to be the minimizer of Q( ) imposes restrictions on and .…”

Section: Application To Factor Analysismentioning

confidence: 99%

“…It can be shown [2,Eq. 17;4] that if T is constrained to be an orthogonal matrix and minimizes Q( ) in (25), then = 0 and must satisfy…”

Section: Application To Factor Analysismentioning

confidence: 99%

“…L; , , − K 3, , and K 4 = − I ⊗ vec I D (3) ; , , ⊗ G * J 21, − D (2) ; , I (k, ) I D (2) ; , ⊗ I J 21, + D (2) ; , I (k, ) K 3, (G * ⊗ G * ) ⊗ I J 21, + n −1 G * E D (4) L; , , , (G * ⊗ G * ⊗ G * )…”

Section: A1 Product and Chain Rulesmentioning

confidence: 99%

See 3 more Smart Citations

An implicit function approach to constrained optimization with applications to asymptotic expansions

Boik

2008

Journal of Multivariate Analysis

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In this article, an unconstrained Taylor series expansion is constructed for scalar-valued functions of vector-valued arguments that are subject to nonlinear equality constraints. The expansion is made possible by first reparameterizing the constrained argument in terms of identified and implicit parameters and then expanding the function solely in terms of the identified parameters. Matrix expressions are given for the derivatives of the function with respect to the identified parameters. The expansion is employed to construct an unconstrained Newton algorithm for optimizing the function subject to constraints.Parameters in statistical models often are estimated by solving statistical estimating equations. It is shown how the unconstrained Newton algorithm can be employed to solve constrained estimating equations. Also, the unconstrained Taylor series is adapted to construct Edgeworth expansions of scalar functions of the constrained estimators. The Edgeworth expansion is illustrated on maximum likelihood estimators in an exploratory factor analysis model in which an oblique rotation is applied after Kaiser row-normalization of the factor loading matrix. A simulation study illustrates the superiority of the two-term Edgeworth approximation compared to the asymptotic normal approximation when sampling from multivariate normal or nonnormal distributions.

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“…It was shown in [4] (also see [15,Eq. 28]) that if the less restrictive constraint, T T 1 D = I m , is imposed and minimizes Q( ), then and must satisfy…”

Section: Application To Factor Analysismentioning

confidence: 85%

Section: Application To Factor Analysismentioning

confidence: 99%

“…It can be shown [2,Eq. 17;4] that if T is constrained to be an orthogonal matrix and minimizes Q( ) in (25), then = 0 and must satisfy…”

Section: Application To Factor Analysismentioning

confidence: 99%

Section: A1 Product and Chain Rulesmentioning

confidence: 99%

See 2 more Smart Citations

An implicit function approach to constrained optimization with applications to asymptotic expansions

Boik

2008

Journal of Multivariate Analysis

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“…The ω-derivative in (3) is of order mn × pq, and represents the arrangement described in [3], also discussed by Boik [2] and others. In the special case where p = q = 1 we obtain the ω-derivative of a vector f with respect to a vector x, and it is an mn × 1 column vector instead of an m × n matrix.…”

Section: Matrix Derivatives: Broad Definitionmentioning

confidence: 99%