Stochastic Processes with Modified Lognormal Distribution Featuring Flexible Upper Tail
具有灵活上尾的修正对数正态分布随机过程AI 解读 ↓
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具有灵活上尾的修正对数正态分布随机过程
- 研究问题
- 如何扩展对数正态分布对上尾及风险函数形状的限制，并构造可用于偏斜数据时间预测和空间插值的随机过程？
- 方法与思路
- 作者利用广义 κ 指数与 κ 对数构造三参数分布，推导统计函数、矩界及似然导数，并将潜在高斯过程变换为新过程。
- 材料陈述的贡献
- 摘要称该族提供可调的较轻右尾及特定参数下的双峰形态，并给出解析表达和基于扭曲高斯过程回归的应用探索。
- 阅读关注点
- 应核对参数空间、尾部与矩性质、过程联合密度的构造，以及不同协方差核下预测和插值的验证设计。
- 阅读边界
- 这是预印本，同行评审及期刊接收未获核实；摘要未提供预测改善的具体证据，也没有金融尾部风险应用的直接说明。
原文摘要
Asymmetric, non-Gaussian probability distributions are often observed in the analysis of natural and engineering datasets. The lognormal distribution is a standard model for data with skewed frequency histograms and fat tails. However, the lognormal law severely restricts the asymptotic dependence of the probability density and the hazard function for high values. Herein we present a family of three-parameter non-Gaussian probability density functions that are based on generalized kappa-exponential and kappa-logarithm functions and investigate its mathematical properties. These kappa-lognormal densities represent continuous deformations of the lognormal with lighter right tails, controlled by the parameter kappa. In addition, bimodal distributions are obtained for certain parameter combinations. We derive closed-form analytic expressions for the main statistical functions of the kappa-lognormal distribution. For the moments, we derive bounds that are based on hypergeometric functions as well as series expansions. Explicit expressions for the gradient and Hessian of the negative log-likelihood are obtained to facilitate numerical maximum-likelihood estimates of the kappa-lognormal parameters from data. We also formulate a joint probability density function for kappa-lognormal stochastic processes by applying Jacobi's multivariate theorem to a latent Gaussian process. Estimation of the kappa-lognormal distribution based on synthetic and real data is explored. Furthermore, we investigate applications of kappa-lognormal processes with different covariance kernels in time series forecasting and spatial interpolation using warped Gaussian process regression. Our results are of practical interest for modeling skewed distributions in various scientific and engineering fields.