In this paper, under the stationary α-mixing dependent samples, we develop a novel nonlinear modal regression for time series sequences and establish the consistency and asymptotic property of the proposed nonlinear modal estimator with a shrinking bandwidth h under certain regularity conditions. The asymptotic distribution is shown to be identical to the one derived from the independent observations, whereas the convergence rate ($\sqrt{n{h}^3}$ in which n is the sample size) is slower than that in the nonlinear mean regression. We numerically estimate the proposed nonlinear modal regression model by the use of a modified modal expectation–maximization (MEM) algorithm in conjunction with Taylor expansion. Monte Carlo simulations are presented to demonstrate the good finite sample (prediction) performance of the newly proposed model. We also construct a specified nonlinear modal regression to match the available daily new cases and new deaths data of the COVID-19 outbreak at the state/region level in the United States, and provide forward predictions up to 130 days ahead (from 24 August 2020 to 31 December 2020). In comparison to the traditional nonlinear regressions, the suggested model can fit the COVID-19 data better and produce more precise predictions. The prediction results indicate that there are systematic differences in spreading distributions among states/regions. For most western and eastern states, they have many serious COVID-19 burdens compared to Midwest. We hope that the built nonlinear modal regression can help policymakers to implement fast actions to curb the spread of the infection, avoid overburdening the health system and understand the development of COVID-19 from some points.

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相关数据的非线性模态回归及其在预测 COVID-19 中的应用

在本文中，在平稳α混合相关样本下，我们开发了一种新颖的时间序列序列非线性模态回归，并在一定规律性条件下建立了所提出的具有收缩带宽h的非线性模态估计器的一致性和渐近性质。渐近分布与独立观测得出的渐近分布相同，而收敛率 ($\sqrt{n{H}^3}$其中n是样本大小）比非线性均值回归慢。我们通过使用改进的模态期望最大化（MEM）算法结合泰勒展开来对所提出的非线性模态回归模型进行数值估计。蒙特卡罗模拟证明了新提出的模型良好的有限样本（预测）性能。我们还构建了指定的非线性模态回归，以匹配美国州/地区级别的 COVID-19 疫情每日新增病例和新增死亡数据，并提供最多未来 130 天的前瞻性预测（自 2020 年 8 月 24 日起）至 2020 年 12 月 31 日）。与传统的非线性回归相比，建议的模型可以更好地拟合 COVID-19 数据并产生更精确的预测。预测结果表明各州/地区之间的扩散分布存在系统性差异。对于大多数西部和东部各州来说，与中西部相比，他们面临着许多严重的 COVID-19 负担。我们希望构建的非线性模态回归能够帮助政策制定者采取快速行动来遏制感染的传播，避免卫生系统负担过重，并从某些方面了解COVID-19的发展。