Weibull Sine Generalized Distribution Family: Fundamental Properties, Sub-model, Simulations, with Biomedical Applications

This study explores the integration of trigonometric functions into traditional statistical models, focusing on the development of the Weibull Sine Generalized (WSG-G) family of distributions. A special case was formulated name Weibull Sine generalized exponential (WSG-E) distribution. This new distribution extends the baseline exponential distribution, accommodating heavier tails and outliers, thereby effectively modeling positively skewed data. Key statistics such as mean, variance, skewness, and kurtosis indicate the distribution's capacity to handle clustered data. A simulation study demonstrates the performance of Maximum Likelihood Estimation (MLE), revealing convergence in the mean squared error and root mean squared error for the parameter α with increasing sample sizes, although convergence is less evident for other parameters. The WSG-E distribution's applicability is further illustrated through its fitting of medical datasets on bladder cancer remission times and growth hormone deficiency in children, both characterized by extreme values. Overall, the WSG-E distribution proves to be a robust model for skewed data, and future research could extend this framework to additional continuous distributions.

本研究探讨了将三角函数融入传统统计模型的研究方向，重点聚焦于威布尔正弦广义分布（Weibull Sine Generalized, WSG-G）族的构建。本研究推导得到该分布族的一个特例——威布尔正弦广义指数分布（Weibull Sine Generalized Exponential, WSG-E）。该新分布以基线指数分布为基础进行拓展，可适配厚尾与异常值场景，能够有效对正偏态数据进行建模。诸如均值、方差、偏度与峰度等关键统计量均表明，该分布具备处理成簇数据的能力。一项模拟研究验证了极大似然估计（Maximum Likelihood Estimation, MLE）的性能：随着样本量增加，参数α的均方误差与均方根误差均呈现收敛趋势，但其余参数的收敛性则相对不显著。通过对膀胱癌缓解时长与儿童生长激素缺乏症两类均含极端值的医学数据集进行拟合，进一步验证了WSG-E分布的适用性。综上，WSG-E分布可作为偏态数据的稳健建模工具，未来研究可将该框架拓展至更多连续分布的构建中。