The search of appropriate models for describing the currency return distribution is one of the main interests not only in finance, but also in the more recent trans-disciplinary econophysics research field. Such a search is recently focusing on cryptocurrencies, due to their proliferation. Although there is no agreement of what theoretical models are the most appropriate, there is a general consensus that the cryptocurrency return distribution is highly-peaked and heavy-tailed, with a large excess kurtosis and the tail distribution decreases as the inverse cubic power-law. With these requirements, the Laplace distribution can be considered as a valid candidate model. However, there are two limitations to be taken into account: 1) the Laplace tail distribution does not decrease as the inverse cubic power-law and 2) the excess kurtosis is fixed at 3. To make the tailedness of the Laplace distribution more flexible, still maintaining its peculiar symmetric shape, we introduce the Laplace scale mixture (LSM) family of distributions. Each member of the family is obtained by dividing the scale parameter of the conditional Laplace distribution by a convenient mixing random variable taking values on all or part of the positive real line and whose distribution depends on a parameter vector $\mathit{\theta }$ governing the tail behavior of the resulting LSM. For illustrative purposes, we consider different mixing distributions; they give rise to LSMs having a closed-form probability density function where the Laplace distribution is obtained as a special case under a convenient choice of $\mathit{\theta }$. We describe an EM algorithm to obtain maximum likelihood estimates of the parameters for all the considered LSMs. Interestingly, we show how the influence of observations associated with large scaled absolute distances is reduced (downweighted), with respect to the nested Laplace distribution, in the estimation phase. We fit these models to the returns of 4 cryptocurrencies, considering several classical symmetric distributions for comparison. The analysis shows how the proposed models represent a valid alternative to the considered competitors in terms of AIC, BIC and likelihood-ratio tests, but also in reproducing the larger empirical excess kurtosis and in resembling the empirical inverse cubic power-law decrease of the tail distribution.

寻找合适的模型来描述货币收益分布不仅是金融领域的主要兴趣之一，也是最近跨学科经济物理学研究领域的主要兴趣之一。由于加密货币的激增，这种搜索最近集中在加密货币上。尽管对于哪种理论模型最合适尚无定论，但普遍共识是，加密货币收益分布高度高峰且尾部很重，峰度过大，尾部分布随三次立方幂律反而减小。根据这些要求，可以将拉普拉斯分布视为有效的候选模型。但是，有两个限制要考虑：1）拉普拉斯的尾部分布不会因立方逆幂律而减小； 2）峰度固定为3。为了使Laplace分布的尾部更加灵活，同时仍保持其特殊的对称形状，我们引入了Laplace比例混合（LSM）系列分布。通过将条件拉普拉斯分布的比例参数除以方便的混合随机变量来获得该族的每个成员，该随机变量采用在全部或部分正实线上的值，并且其分布取决于参数向量$\mathit{\theta }$控制生成的LSM的尾部行为。为了说明的目的，我们考虑不同的混合分布。它们产生了具有封闭形式概率密度函数的LSM，在特殊情况下，通过方便的选择，可以获得Laplace分布$\mathit{\theta }$。我们描述了一种EM算法，以获得所有考虑的LSM的参数的最大似然估计。有趣的是，我们展示了在估计阶段相对于嵌套的拉普拉斯分布，如何减小与大比例缩放的绝对距离相关的观察结果的影响（降低权重）。考虑到几种经典的对称分布进行比较，我们将这些模型拟合为4种加密货币的收益。分析表明，所提出的模型在AIC，BIC和似然比检验方面如何代表考虑的竞争对手的有效替代方案，而且还可以再现较大的经验过量峰度，并且类似于尾部的经验逆立方幂律递减分配。