Wald tests of singular hypotheses

Mathias Drton and Han Xiao

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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.

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Bernoulli, Volume 22, Number 1 (2016), 38-59.

Received: June 2013
Revised: February 2014
First available in Project Euclid: 30 September 2015

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asymptotic distribution factor analysis large-sample theory singular parameter point tetrad Wald statistic


Drton, Mathias; Xiao, Han. Wald tests of singular hypotheses. Bernoulli 22 (2016), no. 1, 38--59. doi:10.3150/14-BEJ620.

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