断代缺失数据下受控分枝过程的极大似然估计

Maximum Likelihood Estimation for Controlled Branching Processes with Missing Data

受控分枝过程是描述种群进化的一类重要模型,其中后代分布和控制分布决定了种群的进化特征,估计这些分布的参数对于过程的预测和控制至关重要。但在实际中,常常会由于观察的间断性或者资料丢失造成样本数据的断代缺失,这给参数估计带来一定的困难。本文主要是基于断代缺失数据,在一些正则假设条件下,推导了缺失样本的分布函数,基于EM算法,得到具有随机控制函数的受控分枝过程中若干参数的极大似然估计,并通过数值模拟验证了该方法的有效性。最后,我们利用此方法对2020年1月23日-2月16日杭州市COVID-19数据进行了实证分析,探索了COVID-19病毒在杭州市的传播机制,评价了疫情防控政策的实施效果。

Controlled branching process with random control distribution is an important model to describe population evolution.The estimation of offspring distribution and control distribution is very important,which determine the evolutionary characteristics of the population.However,in practical problems,the data is missing for some generations which is often caused by the discontinuity of observation or improper preservation of data.In this paper,we focus on the this case.Under some regular assumptions,we obtain the conditional distribution function of missing samples,design the estimation method based on EM algorithm,and verify the effectiveness of this method through numerical simulation.Finally,we use this method to empirically analyze the COVID-19 data of Hangzhou from January 23 to February 16,2020,explore the transmission mechanism of COVID-19 virus in Hangzhou,and evaluate the implementation effect of epidemic prevention and control policy.