THE GENERALIZATION OF STUDENTS PROBLEM WHEN SEVERAL DIFFERENT POPULATION VARIANCES ARE INVOLVED
THE GENERALIZATION OF STUDENTS PROBLEM WHEN SEVERAL DIFFERENT POPULATION VARIANCES ARE INVOLVED
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DOI：
10.1093/biomet/34.1-2.28
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发表时间：
1947-01-01
期刊：
影响因子：
2.7
通讯作者：
WELCH, BL
中科院分区：
文献类型：
--
作者：
WELCH, BL
总体参数[eta]由统计量[gamma]估计，该统计量关于[eta]正态分布，方差[image]，其中[sigma]i是正数，[sigma]2是未知方差。如果观测数据提供了这些方差的独立估计值，则可以对[gamma]做出类似于“Student”对单个样本平均值给出的概率陈述。给出了精确解的形式，并给出了级数解。该系列收敛迅速为大样本，因此可能是有用的，在这种情况下构建表。还给出了一个近似解，表明[gamma]近似分布为“Student's”t。与精确解的比较表明，对于大样本和小样本的实质性一致性具有极好的一致性。
A population parameter [eta] is estimated by a statistic [gamma], normally distributed about [eta] with variance [image], where the [sigma]i are positive numbers and the [sigma]2 are unknown variances. If independent estimates of these variances are provided by the observed data, it is possible to make probability statements about [gamma] similar to those "Student" gave for the mean of a single sample. The form of the exact solution is given and a series solution is developed. The series converges rapidly for large samples and thus may be useful in constructing tables for this situation. An approximate solution is also given, indicating that [gamma] is approximately distributed as "Student''s" t. Comparison with the exact solution shows excellent agreement for large and substantial agreement for small samples.