Discretized Trigonometric Burr–Hatke Distributions: Properties, INAR(1) Process, and Population Size Estimation

This study presents a novel suite of discrete trigonometric Burr–Hatke distributions, encompassing the discrete tangent Burr–Hatke (DTBH), discrete sine Burr–Hatke (DSBH), discrete cosine Burr–Hatke (DCBH), and discrete type II tangent Burr–Hatke (DTIITBH) distributions to model count data with overdispersion and right-skewed characteristics commonly observed in count data. Empirical analyses using three real-world datasets show that the DTBH and DCBH models, respectively, outperform established alternatives according to model evaluation metrics. This research further proposes their integration into time series and zero-truncated frameworks. A first-order integer-valued autoregressive process (INAR(1)) utilizing negative binomial thinning and DCBH innovations is formulated, termed the NBINAR (1)-DCBHD model. The proposed NBINAR (1)-DCBHD model delivers superior fit and predictive accuracy, validated by lowest AIC, BIC, RMSE, and satisfactory diagnostic tests. Additionally, to address zero-truncated count scenarios, the zero-truncated discrete cosine Burr–Hatke distribution (ZTDCBHD) is proposed and shown to outperform other existing distributions. Estimating population size with the Horvitz–Thompson estimator under the ZTDCBHD yields improved estimates for hidden populations across two different datasets and further validated using a dataset with known population size. The study establishes that the proposed discrete trigonometric Burr–Hatke models present a flexible option for modeling count data in time series and population estimation settings.