Spatio-temporal processes with a continuous index in space and time are useful for modeling spatio-temporal data in many scientific disciplines such as environmental and health sciences. However, approaches that enable simultaneous estimation of the mean and covariance functions for such spatio-temporal processes are limited. Here, we propose a flexible spatio-temporal model with partially linear regression in the mean function and local stationarity in the covariance function. We develop a profile likelihood method for estimation and an effective bandwidth selection in the presence of spatio-temporally correlated errors. Specifically, we employ a family of bimodal kernels to alleviate bias, which may be of independent interest for semiparametric spatial statistics. The theoretical properties of our profile likelihood estimation, including consistency and asymptotic normality, are established. A simulation study is conducted and suggests sound empirical properties, while a health hazard data example further illustrates the methodology.

具有时空连续索引的时空过程对于建模许多科学学科（例如环境和健康科学）中的时空数据很有用。但是，能够同时估计此类时空过程的均值和协方差函数的方法受到限制。在这里，我们提出了一种灵活的时空模型，在均值函数中具有部分线性回归，而在协方差函数中具有局部平稳性。我们开发了一种轮廓似然方法，用于估计和时空相关误差的存在下的有效带宽选择。具体而言，我们采用了双峰核族来减轻偏差，这对于半参数空间统计可能是独立感兴趣的。我们的轮廓似然估计的理论特性，包括一致性和渐近正态性。进行了模拟研究，并提出了合理的经验特性，而健康危害数据示例进一步说明了该方法。