On the estimation bias in first-order bifurcating autoregressive models

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 both the linear-bias-correcting estimator and the bootstrap bias-correcting estimators can be quite effective in reducing the bias of the LS estimator and that the bootstrap estimators are more effective near the boundaries of the range of the autoregressive parameter. We also discuss the extension of these methods to higher order BAR models.