The problem of estimating the growth rate of a birth and death processes based on the coalescence times of a sample of $n$ individuals has been considered by several authors (\cite{stadler2009incomplete, williams2022life, mitchell2022clonal, Johnson2023}). This problem has applications, for example, to cancer research, when one is interested in determining the growth rate of a clone. Recently, \cite{Johnson2023} proposed an analytical method for estimating the growth rate using the theory of coalescent point processes, which has comparable accuracy to more computationally intensive methods when the sample size $n$ is large. We use a similar approach to obtain an estimate of the growth rate that is not based on the assumption that $n$ is large. We demonstrate, through simulations using the R package \texttt{cloneRate}, that our estimator of the growth rate performs well in comparison with previous approaches when $n$ is small.

翻译：基于$n$个个体的溯祖时间来估计生灭过程增长率的问题已被多位学者研究（\cite{stadler2009incomplete, williams2022life, mitchell2022clonal, Johnson2023}）。该问题在例如癌症研究中具有应用价值，当研究者关注于确定克隆的增长率时。最近，\cite{Johnson2023} 提出了一种利用溯祖点过程理论来估计增长率的解析方法，当样本量$n$较大时，其精度与计算量更大的方法相当。我们采用类似的方法，得到了一种不基于$n$较大假设的增长率估计量。通过使用R包 \texttt{cloneRate} 进行模拟，我们证明，当$n$较小时，我们的增长率估计量相较于先前方法表现良好。