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This paper extends regression modeling of positive count data to deal with excessive proportion of one counts. In particular, we propose one-inflated positive (OIP) regression models and present some of their properties. Also, the stochastic hierarchical representation of one-inflated positive poisson and negative binomial regression models are achieved. It is illustrated that the standard OIP model may be inadequate in the presence of one inflation and the lack of independence. Thus, to take into account the inherent correlation of responses, a class of two-level OIP regression models with subjects heterogeneity effects is introduced. A simulation study is conducted to highlight theoretical aspects. Results show that when one-inflation or over-dispersion in the data generating process is ignored, parameter estimates are inefficient and statistically reliable findings are missed. Finally, we analyze a real data set taken from a length of hospital stay study to illustrate the usefulness of our proposed models.

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The analysis of circular data is the main subject in many disciplines, such as meteorology and oceanography. In this article, we introduce a new multimodal skew‐circular model as an extension of the circular beta distribution. We propose a truncated power smoothing spline for modeling the skewness parameter and identifying significant factors of the asymmetry. A Markov chain Monte Carlo scheme is provided to perform statistical inference from a Bayesian perspective. Then, the performance of our modeling methodology to analyze specific circular responses is assessed through several simulation studies. To illustrate the usefulness of the new model in practical applications, we analyze measurements on the wind and wave directions in Norway. We also fit various regression models to show that the cubic smoothing spline approach performs better than competitive models in practical applications. Findings, based on prediction values, confirm that the proposed model can reasonably fit multimodal skewed‐circular responses.

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