关于广义圆投影Cauchy分布
On the generalized circular projected Cauchy distribution
AI总结：
本文研究了广义圆投影Cauchy分布，推导其与环绕Cauchy分布的关系，探讨其单峰性条件，并提供了均值结果长度、Kullback-Leibler散度等的非解析公式及累积概率函数和熵的解析形式，还提出了无需假设浓度参数相等的对数似然比检验方法。
摘要：
[1]提出了广义圆投影Cauchy（GCPC）分布，其特例为环绕Cauchy分布。本文首先推导了其与环绕Cauchy分布的关系，然后试图表征该分布。我们建立了分布呈现单峰性的条件。我们为GCPC分布提供了均值结果长度和Kullback-Leibler散度的非解析公式，以及累积概率函数和熵的解析形式。我们提出了无需假设浓度参数相等的对数似然比检验方法用于一个或两个位置参数。我们重新审视了有和无预测变量的最大似然估计。在回归设置中，我们简要提及了圆形和单纯形预测变量的添加。模拟研究展示了a）当错误假设真实分布为环绕Cauchy分布时对数似然比检验的性能，以及b）回归系数的实证收敛速率。通过实际数据分析示例，我们展示了如何避免对数似然陷入局部最大值，并纠正了回归设置中的错误。
英文摘要：
\cite{tsagris2025a} proposed the generalized circular projected Cauchy (GCPC) distribution, whose special case is the wrapped Cauchy distribution. In this paper we first derive the relationship with the wrapped Cauchy distribution, and then we attempt to characterize the distribution. We establish the conditions under which the distribution exhibits unimodality. We provide non-analytical formulas for the mean resultant length and the Kullback-Leibler divergence, and analytical form for the cumulative probability function and the entropy of the GCPC distribution. We propose log-likelihood ratio tests for one, or two location parameters without assuming equality of the concentration parameters. We revisit maximum likelihood estimation with and without predictors. In the regression setting we briefly mention the addition of circular and simplicial predictors. Simulation studies illustrate a) the performance of the log-likelihood ratio test when one falsely assumes that the true distribution is the wrapped Cauchy distribution, and b) the empirical rate of convergence of the regression coefficients. Using a real data analysis example we show how to avoid the log-likelihood being trapped in a local maximum and we correct a mistake in the regression setting.
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