姜文华^*

（  苏州大学数学科学学院,苏州　邮编: 215006；  ）

摘要： 设${X_i,ile n}$是服从$X_isim N( heta_i, au_i^2)$的异方差正态观测,其中均值和方差都未知,且$sigma>0$是未知标准差的一个已知下界。记$f_n(x)=sum_{i=1}^n(d/dx)P_n{X_ilex}/n$是平均边缘密度。我们考虑检验原假设$H_0colon f_nincalM_0$, 其中$calM_0$是一族位置—形状混合密度。我们研究一个基于广义极大似然估计的广义似然比检验。我们给出异方差和同方差高斯混合模型的一个联系。这个联系极大地简化了异方差高斯模型中广义极大似然估计的计算。我们通过数值模拟研究广义似然比检验的功效。
关键词： 异方差高斯混合,广义似然比检验,广义极大似然估计

Wenhua Jiang^*

（  School of Mathematical Sciences, Soochow University, Suzhou 215006 ；  ）

Abstract： Let ${X_i,ile n}$ be independent heteroscedastic observationswith $X_isim N( heta_i, au_i^2)$, where means and variances areall unknown, and $sigma>0$ is a known lower bound for the unknownstandard deviations. Let $f_n(x)=sum_{i=1}^n(d/dx)P_n{X_ilex}/n$ be the average marginal density of observations. We considerthe problem of testing $H_0colon f_nincalM_0$, where $calM_0$ isa family of location-scale mixture densities. We study a generalizedlikelihood ratio test (GLRT) based on the generalized maximumlikelihood estimator (GMLE). We state an equivalence between the heteroscedastic andhomoscedastic Gaussian models, which greatly facilitates the computationof the GLRT for the heteroscedastic Gaussian model. We demonstrate the power of the GLRT for moderate samples withnumerical experiments.
Keywords： Heteroscedastic Gaussian mixtures,generalized likelihood ratio test, generalizedmaximum likelihood estimator

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