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Binary Quadratic Forms: A Historical View (EJ769604)
Author(s):
Khosravani, Azar N.; Beintema, Mark B.
Source:
Mathematics and Computer Education, v40 n3 p226-236 Fall 2006
Pub Date:
2006-00-00
Pub Type(s):
Journal Articles; Reports - Descriptive
Peer-Reviewed:
Yes
Descriptors:
Expository Writing; Equations (Mathematics); Mathematical Logic; Predictive Validity; Number Concepts; Intellectual History; Demonstrations (Educational); Mathematical Concepts
Abstract:
We present an expository account of the development of the theory of binary quadratic forms. Beginning with the formulation and proof of the Two-Square Theorem, we show how the study of forms of the type x[squared] + ny[squared] led to the discovery of the Quadratic Reciprocity Law, and how this theorem, along with the concept of reduction relates to the general representation problem
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2. Limited Information Goodness-of-Fit Testing in Multidimensional Contingency Tables (EJ762706)
Author(s):
Maydeu-Olivares, Albert; Joe, Harry
Source:
Psychometrika, v71 n4 p713-732 Dec 2006
Pub Date:
2006-12-00
Pub Type(s):
Journal Articles; Reports - Research
Peer-Reviewed:
Yes
Descriptors:
Testing; Statistical Analysis; Item Response Theory; Goodness of Fit; Psychometrics; Models
Abstract:
We introduce a family of goodness-of-fit statistics for testing composite null hypotheses in multidimensional contingency tables. These statistics are quadratic forms in marginal residuals up to order "r." They are asymptotically chi-square under the null hypothesis when parameters are estimated using any asymptotically normal consistent estimator. For a widely used item response model, when "r" is small and multidimensional tables are sparse, the proposed statistics have accurate empirical Type I errors, unlike Pearson's X[superscript 2]. For this model in nonsparse situations, the proposed statistics are also more powerful than X[superscript 2]. In addition, the proposed statistics are asymptotically chi-square when applied to subtables, and can be used for a piecewise goodness-of-fit assessment to determine the source of misfit in poorly fitting models
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A Bayesian Approach to Nonlinear Latent Variable Models Using the Gibbs Sampler and the Metropolis-Hastings Algorithm. (EJ574592)
Author(s):
Arminger, Gerhard; Muthen, Bengt O.
Source:
Psychometrika, v63 n3 p271-300 Sep 1998
Pub Date:
1998-00-00
Pub Type(s):
Journal Articles; Reports - Descriptive
Peer-Reviewed:
N/A
Descriptors:
Algorithms; Bayesian Statistics; Estimation (Mathematics); Mathematical Models; Simulation
Abstract:
Nonlinear latent variable models are specified that include quadratic forms and interactions of latent regressor variable as special cases. To estimate the parameters, the models are put in a Bayesian framework with conjugate priors for the parameters. The proposed estimation methods are illustrated by two simulation studies. (SLD
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5. Maximization of Sums of Quotients of Quadratic Forms and Some Generalizations. (EJ511058)
Author(s):
Kiers, Henk A. L.
Source:
Psychometrika, v60 n2 p221-45 Jun 1995
Pub Date:
1995-00-00
Pub Type(s):
Reports - Evaluative; Journal Articles
Peer-Reviewed:
N/A
Descriptors:
Algorithms; Equations (Mathematics); Matrices; Multivariate Analysis
Abstract:
Monotonically convergent algorithms are described for maximizing sums of quotients of quadratic forms. Six (constrained) functions are investigated. The general formulation of the functions and the algorithms allow for application of the algorithms in various situations in multivariate analysis
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