Wald tests of singular hypotheses

Mathias Drton, Han Xiao

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


Motivated by the problem of testing tetrad constraints in factor analysis, we study the large-sample distribution of Wald statistics at parameter points at which the gradient of the tested constraint vanishes. When based on an asymptotically normal estimator, the Wald statistic converges to a rational function of a normal random vector. The rational function is determined by a homogeneous polynomial and a covariance matrix. For quadratic forms and bivariate monomials of arbitrary degree, we show unexpected relationships to chi-square distributions that explain conservative behavior of certain Wald tests. For general monomials, we offer a conjecture according to which the reciprocal of a certain quadratic form in the reciprocals of dependent normal random variables is chi-square distributed.

Original languageEnglish (US)
Pages (from-to)38-59
Number of pages22
Issue number1
StatePublished - Feb 2016

All Science Journal Classification (ASJC) codes

  • Statistics and Probability


  • Asymptotic distribution
  • Factor analysis
  • Large-sample theory
  • Singular parameter point
  • Tetrad
  • Wald statistic


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