In this paper, we study the bias of the least squares (LS) estimation for the stationary first‐order bifurcating autoregressive (BAR(1)) model, which is commonly used to model binary tree‐structured data that appear in many applications, most famously cell lineage applications. We first show that the LS estimator can have large bias for both small‐ and moderate‐sized samples and that this bias is dependent on the values of both the autoregressive parameter (the target parameter) and the correlation between model errors. We also provide a first‐order approximation to the bias of the LS estimator and show, empirically, that this approximation can accurately describe the bias as a function of the autoregressive parameter and the errors correlation over wide combinations of their values. Then we study two approaches for correcting the bias of the LS estimator, namely, bootstrap bias correction and bias correction through linear bias functions. Both single and double bootstrap bias‐corrected versions of the LS estimator are defined and studied empirically in comparison with a linear‐bias‐correcting estimator as well as the standard LS estimator. The empirical results, based on both simulated and real data, demonstrate that the linear‐bias‐correcting estimator can be quite effective in reducing the bias and that it is more effective than the bootstrap estimators in most cases.

中文翻译：

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一阶分支自回归模型的估计偏差

在本文中，我们研究了固定一阶分叉自回归（BAR（1））模型的最小二乘估计（LS）估计的偏差，该模型通常用于对出现在许多应用中的二进制树结构数据进行建模，大多数著名的细胞谱系应用。我们首先表明，对于小样本和中型样本，LS估计量都可能具有较大的偏差，并且该偏差取决于自回归参数（目标参数）的值以及模型误差之间的相关性。我们还提供了LS估计量的偏差的一阶近似值，并凭经验表明，这种近似值可以准确地将偏差描述为自回归参数的函数，以及它们的值的广泛组合中的误差相关性。然后，我们研究了两种用于校正LS估计器偏差的方法，即自举偏差校正和通过线性偏差函数进行偏差校正。与线性偏置校正估算器和标准LS估算器相比，对LS估算器的单引导和双自举偏置校正版本进行了定义和研究。基于模拟和真实数据的经验结果表明，线性偏置校正估计器可以有效降低偏差，并且在大多数情况下，其效率比自举估计器更有效。与线性偏置校正估算器和标准LS估算器相比，对LS估算器的单引导和双自举偏置校正版本进行了定义和研究。基于模拟和真实数据的经验结果表明，线性偏置校正估计器可以有效降低偏差，并且在大多数情况下，其效率比自举估计器更有效。与线性偏置校正估算器和标准LS估算器相比，对LS估算器的单引导和双自举偏置校正版本进行了定义和研究。基于模拟和真实数据的经验结果表明，线性偏置校正估计器可以有效降低偏差，并且在大多数情况下，其效率比自举估计器更有效。