We analyze a modification of the Richards growth model by introducing a time-dependent perturbation in the growth rate. This modification becomes effective at a special switching time, which represents the first-crossing-time of the Richards growth curve through a given constant boundary. The relevant features of the modified growth model are studied and compared with those of the original one. A sensitivity analysis on the switching time is also performed. Then, we define two different stochastic processes, i.e. a non-homogeneous linear birth–death process and a lognormal diffusion process, such that their means identify to the growth curve under investigation. For the diffusion process, we address the problem of parameters estimation through the maximum likelihood method. The estimates are obtained via meta-heuristic algorithms (namely, Simulated Annealing and Ant Lion Optimizer). A simulation study to validate the estimation procedure is also presented, together with a real application to oil production in France. Special attention is devoted to the approximation of switching time density, viewed as the first-passage-time density for the lognormal process.

中文翻译：

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受时间相关模式、统计分析和应用扰动的 Bertalanffy-Richards 增长模型


我们通过引入增长率的时间相关扰动来分析理查兹增长模型的修改。这种修改在特定的切换时间生效，该切换时间表示理查兹生长曲线首次穿过给定常数边界的时间。研究了修改后的增长模型的相关特征，并与原始增长模型进行了比较。还对切换时间进行了敏感性分析。然后，我们定义两个不同的随机过程，即非齐次线性生灭过程和对数正态扩散过程，使得它们的均值与所研究的生长曲线一致。对于扩散过程，我们通过最大似然法解决参数估计问题。估计是通过元启发式算法（即模拟退火和 Ant Lion 优化器）获得的。还提出了验证估计程序的模拟研究，以及在法国石油生产中的实际应用。特别关注转换时间密度的近似，被视为对数正态过程的首次通过时间密度。