Skew normal model suffers from inferential drawbacks, namely singular Fisher information in the vicinity of symmetry and diverging of maximum likelihood estimation. To address the above drawbacks, Azzalini and Arellano-Valle (2013) introduced maximum penalised likelihood estimation (MPLE) by subtracting a penalty function from the log-likelihood function with a pre-specified penalty coefficient. Here, we propose a cross-validated MPLE to improve its performance when the underlying model is close to symmetry. We develop a theory for MPLE, where an asymptotic rate for the cross-validated penalty coefficient is derived. We further show that the proposed cross-validated MPLE is asymptotically efficient under certain conditions. In simulation studies and a real data application, we demonstrate that the proposed estimator can outperform the conventional MPLE when the model is close to symmetry.

翻译：偏态正态模型存在推论上的缺陷，即在对称性附近费希尔信息奇异以及极大似然估计发散。针对上述问题，Azzalini和Arellano-Valle (2013) 通过从对数似然函数中减去一个具有预设惩罚系数的惩罚函数，引入了极大惩罚似然估计 (MPLE)。本文提出一种交叉验证MPLE，以在底层模型接近对称性时提升其性能。我们发展了MPLE的理论，推导了交叉验证惩罚系数的渐近速率。进一步证明，在特定条件下，所提出的交叉验证MPLE具有渐近有效性。通过模拟研究和实际数据应用，我们证明当模型接近对称性时，所提估计量能优于传统MPLE。