Abstract
This work deals with semiparametric kernel estimator of probability mass functions which are assumed to be modified Poisson distributions. This semiparametric approach is based on discrete associated kernel method appropriated for modelling count data; in particular, the famous discrete symmetric triangular kernels are used. Two data-driven bandwidth selection procedures are investigated and an explicit expression of optimal bandwidth not available until now is provided. Moreover, some asymptotic properties of the cross-validation criterion adapted for discrete semiparametric kernel estimation are studied. Finally, to measure the performance of semiparametric estimator according to each type of bandwidth parameter, some applications are realized on three real count data-sets from sociology and biology.
Highlights
The traditional approach for estimating count data distribution has been essentially parametric until recently
An explicit optimal bandwidth is provided like that which is available for continuous kernel density estimation
The expression of hopt proposed depends both on parameter a ∈ N of discrete triangular kernel and sample size n; this new optimal bandwidth goes to 0 when sample size n goes to ∞
Summary
The traditional approach for estimating count data distribution has been essentially parametric until recently. Let us remark that a Bayesian local approach is developed by Zougab et al [8,9] for bandwidth selection in discrete nonparametric associated kernel estimation of p.m.f. concerning count data, the problem of their semiparametric regression is treated by Abdous et al [10]. The second application concerns count data characterizing development of spiralling whitefly, which is an insect pest plant collected in Republic of Congo-Brazzaville [11]. The proofs of mathematical results are postponed to the appendix
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