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This summary is machine-generated.

The likelihood ratio test (LRT) often fails for latent variable models because standard assumptions are violated. A more general theory provides the correct asymptotic theory for these LRTs, improving model comparison.

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Area of Science:

  • Statistics
  • Econometrics
  • Psychometrics

Background:

  • The likelihood ratio test (LRT) is a standard method for comparing nested statistical models.
  • Wilks' theorem provides the asymptotic distribution for LRT statistics, typically a chi-squared distribution.
  • This approximation often fails for models with latent variables, such as factor analysis and structural equation models.

Purpose of the Study:

  • To demonstrate how Wilks' theorem's regularity conditions are violated in latent variable models.
  • To present a more general asymptotic theory for LRTs in these contexts.
  • To illustrate the application of this general theory with practical examples.

Main Methods:

  • Analysis of regularity conditions for Wilks' theorem.
  • Application of Chernoff's general theory for likelihood ratio tests.
  • Illustrative examples using three distinct latent variable models.

Main Results:

  • Identified specific violations of Wilks' theorem's assumptions in common latent variable models.
  • Provided a correct asymptotic theory for LRTs applicable to these models.
  • Demonstrated the practical utility of the general theory through case studies.

Conclusions:

  • The standard chi-squared approximation for LRTs is unreliable for many latent variable models.
  • A generalized asymptotic theory offers a robust alternative for LRTs in these complex models.
  • This work highlights the importance of applying appropriate statistical theory for accurate model comparison.