The exponentially truncated centered kernel: theoretical properties and applications in discrete distribution estimation

We introduce the Exponentially Truncated Centered (ETC) kernel, a new discrete associated kernel for nonparametric estimation of probability mass functions on count data. The ETC kernel combines exponentially decaying weights with a finite, locally adaptive truncation radius, thereby reducing boundary effects at zero while maintaining second-order accuracy at interior points. We derive closed-form expressions for the bias, variance, and mean integrated squared error, and propose an explicit normalization to enforce the summation-to-one constraint in finite samples. Bandwidth and truncation radius are selected jointly via a least-squares cross-validation criterion. We conduct an extensive simulation study comparing ETC with a broad range of competitors, including classical kernels (Binomial, discrete triangular) and more recent flexible proposals (Conway-Maxwell-Poisson, double Poisson, Gamma-Count, and optimal symmetric). The results indicate that ETC is consistently competitive in integrated squared error, with notable gains in multimodal settings, while its closed-form normalization provides a tangible computational advantage over kernels relying on infinite-series approximations. Real-data applications further illustrate its practical usefulness.