Generalized likelihood ratio test for normal mixtures

Wenhua Jiang, Cun Hui Zhang

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


Let X1; . . . ;Xn be independent observations with Xi ∼ N(θ i; 1), where (θ 1; . . . ; θ n) is an unknown vector of normal means. Let fn (x) = Σi=1n (d=dx)Pn{Xi ≤ x}/n be the average marginal density of observations. We consider the problem of testing H0 : fn ∈ F0, where F0 is a family of mixture densities. This includes detecting nonzero normal means with F0 = {fδ0} and testing homogeneity in mixture models with F0 = {fδμ}. We study a generalized likelihood ratio test (GLRT) based on the generalized maximum likelihood estimator (GMLE, Robbins (1950); Kiefer and Wolfowitz (1956)). We establish a large deviation inequality that provides a divergence rate ϵn of the GLRT under the null hypothesis. The inequality implies that the significance level of the test is of equal or smaller order than nϵn2. We show that the test can detect any alternative that is separated from the null by Hellinger distance ϵn. For the two-component Gaussian mixture, it turns out that the GLRT has full power asymptotically throughout the same region of amplitude sparsity where the Neyman-Pearson likelihood ratio test separates the two hypotheses completely (Donoho and Jin (2004)). We demonstrate the power of the GLRT for moderate samples with numerical experiments.

Original languageEnglish (US)
Pages (from-to)955-978
Number of pages24
JournalStatistica Sinica
Issue number3
StatePublished - Jul 2016

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • Detection boundary
  • Generalized likelihood ratio test
  • Generalized maximum likelihood estimator
  • Normal mixture
  • Sparse normal means


Dive into the research topics of 'Generalized likelihood ratio test for normal mixtures'. Together they form a unique fingerprint.

Cite this