The likelihood ratio test with the Box-Cox transformation for the normal mixture problem: Power and sample size study
The likelihood ratio test with the Box-Cox transformation for the normal mixture problem: Power and sample size study
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DOI：
10.1081/sac-200033328
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发表时间：
2004-01-01
影响因子：
0.9
通讯作者：
Finch, SJ
中科院分区：
文献类型：
--
作者：
Ning, YM;Finch, SJ
通过模拟和回归，我们研究了似然比检验的备择分布，其中零假设数据来自限制Box-Cox变换后的正态分布，备择假设假设数据来自限制（可能不同）Box-Cox变换后的两个正态分布的混合。样本中的观测数称为N。分量之间的标准化距离（转换后）为D =（mu（2）- mu（1））/sigma，其中mu（1）和mu（2）是分量均值，sigma（2）是它们的共同方差。一个分量包含观察到的分数pi，另一个分量包含1 - pi。模拟结果表明，功率的混合比例的依赖性，与功率降低的混合比例从0.5不同。替代分布似乎是非中心卡方分布，自由度约为2.48 + 10 N（-0.75），非中心性参数为0.174 N（D - 1.4）（2）x [pi（1 - pi）]。当D = 2时，至少需要900个观察值才能使5%检验的功效达到95%。对于D、功效和显著性水平的固定值，当pi大于或等于0.90或pi小于或等于0.10时，需要更多的观测值。我们给出了所研究的替代方案的估计功效，以及50%、80%、90%和95%功效所需的样本量表。
Through simulation and regression, we study the alternative distribution of the likelihood ratio test in which the null hypothesis postulates that the data are from a normal distribution after a restricted Box-Cox transformation and the alternative hypothesis postulates that they are from a mixture of two normals after a restricted (possibly different) Box-Cox transformation. The number of observations in the sample is called N. The standardized distance between components (after transformation) is D = (mu(2) - mu(1))/sigma, where mu(1) and mu(2) are the component means and sigma(2) is their common variance. One component contains the fraction pi of observed, and the other 1 - pi. The simulation results demonstrate a dependence of power on the mixing proportion, with power decreasing as the mixing proportion differs from 0.5. The alternative distribution appears to be a non-central chi-squared with approximately 2.48 + 10N(-0.75) degrees of freedom and non-centrality parameter 0.174N(D - 1.4)(2) x [pi(1 - pi)]. At least 900 observations are needed to have power 95% for a 5% test when D = 2. For fixed values of D, power, and significance level, substantially more observations are necessary when pi greater than or equal to 0.90 or pi less than or equal to 0.10. We give the estimated powers for the alternatives studied and a table of sample sizes needed for 50%, 80%, 90%, and 95% power.