Kernel smoother and a time-histogram are classical tools for estimating an instantaneous rate of spike occurrences. We recently established a method for selecting the bin width of the time-histogram, based on the principle of minimizing the mean integrated square error (MISE) between the estimated rate and unknown underlying rate. Here we apply the same optimization principle to the kernel density estimation in selecting the width or "bandwidth" of the kernel, and further extend the algorithm to allow a variable bandwidth, in conformity with data. The variable kernel has the potential to accurately grasp non-stationary phenomena, such as abrupt changes in the firing rate, which we often encounter in neuroscience. In order to avoid possible overfitting that may take place due to excessive freedom, we introduced a stiffness constant for bandwidth variability. Our method automatically adjusts the stiffness constant, thereby adapting to the entire set of spike data. It is revealed that the classical kernel smoother may exhibit goodness-of-fit comparable to, or even better than, that of modern sophisticated rate estimation methods, provided that the bandwidth is selected properly for a given set of spike data, according to the optimization methods presented here.