## Abstract

Ascertaining the number of components in a mixture distribution is an interesting and challenging problem for statisticians. Chen, Chen, and Kalbfleisch recently proposed a modified likelihood ratio test (MLRT), which is distribution-free and locally most powerful, asymptotically. In this article we present a new method for testing whether a finite mixture distribution is homogeneous. Our method, the D test, is based on the L ^{2} distance between a fitted homogeneous model and a fitted heterogeneous model. For mixture components from standard parametric families, the D-test statistic has a closed-form expression in terms of parameter estimators, whereas likelihood ratio-type test statistics do not; the latter test statistics are nontrivial functions of both the parameter estimators and the full dataset. The convergence rates of the D-test statistic under a null hypothesis of homogeneity and an alternative hypothesis of heterogeneity are established. The D test is shown to be competitive with the MLRT when the mixture components come from a normal location family. However, in the exponential scale and normal location/scale cases, the relative performances of the D test and the MLRT are mixed. In cases such as these two, we propose to use a weighted D test, in which the measure underlying the L ^{2} distance is changed to accentuate the disparities between the homogeneous and heterogeneous models. Changing the measure is equivalent to computing the D-test statistic using a weighting function or to transforming the data before conducting the D test. Appropriately weighted D tests are competitive in both the exponential scale and normal location/scale cases. After applying the D test to a dataset in which the observations are measurements of firms' financial performances, we conclude with discussion and remarks.

Original language | English |
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Pages (from-to) | 488-498 |

Number of pages | 11 |

Journal | Journal of the American Statistical Association |

Volume | 99 |

Issue number | 466 |

DOIs | |

State | Published - Jun 2004 |

## Keywords

- D test
- Mixtures
- Modified likelihood ratio test
- Weighted L2 measure

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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