## Abstract

Let X_{1}; . . . ;X_{n} be independent observations with X_{i} ∼ N(θ _{i}; 1), where (θ _{1}; . . . ; θ _{n}) is an unknown vector of normal means. Let f_{n} (x) = Σ_{i=1}^{n} (d=dx)P_{n}{X_{i} ≤ x}/n be the average marginal density of observations. We consider the problem of testing H_{0} : f_{n} ∈ F_{0}, where F_{0} is a family of mixture densities. This includes detecting nonzero normal means with F_{0} = {f_{δ0}} and testing homogeneity in mixture models with F_{0} = {f_{δμ}}. 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 ϵ_{n} of the GLRT under the null hypothesis. The inequality implies that the significance level of the test is of equal or smaller order than nϵ_{n}^{2}. We show that the test can detect any alternative that is separated from the null by Hellinger distance ϵ_{n}. 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 language | English (US) |
---|---|

Pages (from-to) | 955-978 |

Number of pages | 24 |

Journal | Statistica Sinica |

Volume | 26 |

Issue number | 3 |

DOIs | |

State | Published - Jul 2016 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

## Keywords

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