Based on $n$ replicate measurements that require known normalization factors and assuming an underlying normal distribution for individual measurements but with unknown standard deviation, a combined likelihood function is derived that takes the form of a Student's $t$-distribution with $\nu = n-1$ degrees of freedom and $t=(\psi -\overline{Y})/s$, where $\psi $ is the true value of the measurement quantity calculated from the forward model, and $\overline{Y}$ and $s$ are average and standard error of the mean obtained from the $n$ measurements defined with weighting proportional to the inverse of the normalization factor squared. Assuming an underlying triangle distribution rather than a normal distribution does not produce a large change for six replicates. Examples of replicate data from an animal study and sequential occupational urine and fecal monitoring are given. The use of the empirical likelihood function in data modeling is discussed.

中文翻译：

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改进的基于归一化相关复制测量的经验似然函数。

基于需要已知归一化因子的 $n$ 重复测量并假设单个测量的基础正态分布但标准偏差未知，得出组合似然函数，其采用学生的 $t$-分布形式，$\nu = n-1$自由度和$t=(\psi -\overline{Y})/s$，其中$\psi $为正向模型计算出的测量量的真值，$\overline{Y }$ 和 $s$ 是从 $n$ 测量中获得的平均值的平均值和标准误差，其加权与归一化因子平方的倒数成正比。假设基本三角形分布而不是正态分布不会对六个重复产生大的变化。给出了来自动物研究和连续职业尿液和粪便监测的重复数据示例。讨论了经验似然函数在数据建模中的使用。