Chen Rui, Hyrien Ollivier
Department of Biostatistics and Computational Biology, University of Rochester Medical Center.
J Stat Plan Inference. 2011 Jul 1;141(7):2209-2227. doi: 10.1016/j.jspi.2011.01.016.
This article deals with quasi- and pseudo-likelihood estimation in a class of continuous-time multi-type Markov branching processes observed at discrete points in time. "Conventional" and conditional estimation are discussed for both approaches. We compare their properties and identify situations where they lead to asymptotically equivalent estimators. Both approaches possess robustness properties, and coincide with maximum likelihood estimation in some cases. Quasi-likelihood functions involving only linear combinations of the data may be unable to estimate all model parameters. Remedial measures exist, including the resort either to non-linear functions of the data or to conditioning the moments on appropriate sigma-algebras. The method of pseudo-likelihood may also resolve this issue. We investigate the properties of these approaches in three examples: the pure birth process, the linear birth-and-death process, and a two-type process that generalizes the previous two examples. Simulations studies are conducted to evaluate performance in finite samples.
本文探讨了在离散时间点观测的一类连续时间多类型马尔可夫分支过程中的拟似然估计和伪似然估计。针对这两种方法，讨论了“常规”估计和条件估计。我们比较了它们的性质，并确定了它们导致渐近等价估计量的情形。这两种方法都具有稳健性，并且在某些情况下与最大似然估计一致。仅涉及数据线性组合的拟似然函数可能无法估计所有模型参数。存在补救措施，包括采用数据的非线性函数或对适当的σ-代数上的矩进行条件设定。伪似然方法也可以解决这个问题。我们在三个例子中研究了这些方法的性质：纯生过程、线性生死过程以及一个推广了前两个例子的两类型过程。进行了模拟研究以评估有限样本中的性能。
