Abstract Using maximum likelihood (ML) estimation for discrete choice modeling of small datasets causes two problems. The first problem is that the data may exhibit separation, in which case the ML estimates do not exist. Also, provided they exist, the ML estimates are biased. In this paper, we show how to adapt Firth's penalized likelihood estimation for use in discrete choice modeling. A powerful advantage of Firth's estimation is that, unlike ML estimation, it provides useful estimates in the case of data separation. For aggregates of six or more respondents, Firth estimates have negligible bias. For preference estimates on an individual level, Firth estimates show little bias as long as each person evaluates a sufficient number of choice sets. Additionally, Firth's individual-level estimation makes it possible to construct an empirical distribution of the respondents' preferences without imposing any a priori population distribution and to effectively predict people's choices and detect market segments. Segment recovery may even be better when individual-level estimates are obtained using Firth's method instead of hierarchical Bayes estimation under a normal prior. We base all findings on data from a stated choice study on various forms of employee compensation.

中文翻译：

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使用Firth方法基于选择数据进行模型估计和市场细分

摘要将最大似然（ML）估计用于小型数据集的离散选择建模会导致两个问题。第一个问题是数据可能表现出分离，在这种情况下，ML估计不存在。而且，如果存在，则ML估计值会有偏差。在本文中，我们展示了如何使Firth的惩罚似然估计适用于离散选择建模。Firth估计的强大优势在于，与ML估计不同，它在数据分离的情况下提供了有用的估计。对于六个或更多受访者的总数，Firth估计的偏差可以忽略不计。对于个人的偏好估计，只要每个人评估足够数量的选择集，Firth估计就几乎没有偏差。另外，Firth' 个人水平的估计使构建受访者偏好的经验分布而无需施加任何先验人口分布，并有效地预测了人们的选择并发现了市场细分。当使用Firth方法而不是在正常先验条件下进行分层贝叶斯估计获得个人级别的估计时，分段恢复甚至会更好。我们的所有调查结果均基于对各种形式的员工薪酬的既定选择研究得出的数据。s方法代替常规先验条件下的分层贝叶斯估计。我们的所有调查结果均基于对各种形式的员工薪酬的既定选择研究得出的数据。s方法代替常规先验条件下的分层贝叶斯估计。我们的所有调查结果均基于对各种形式的员工薪酬的既定选择研究得出的数据。