In this article we present a new INteger-valued AutoRegressive (INAR) model with the aim of extracting baseline patterns of cattle fallen stock registered over an 5-year period at a local scale. We introduce HINAR as a generalization of the classical Poisson-based INAR models whose innovations follow a Hermite distribution. In order to assess trends and seasonality in these time series, we fit different models with time-dependent parameters by specifying proper functions. Using real world examples, we illustrate how to estimate parameters by maximum likelihood and validate the fitted models. We also show a detailed method to forecast. Our proposed model supposes a good solution for studying discrete time series when the counts have many zeros, low counts and moderate overdispersion. This model has been applied to the analysis of fallen cattle registered at a local scale as part of the development of a veterinary syndromic surveillance system.

Different critical values deduced by simulation have been proposed that greatly improve Lenth's original proposal. However, these simulations assume that all effects are zerosomething not realistic--producing bigger than desired critical values and thus significance levels lower that intended. This article, in accordance with George Box [2] well known idea that Experimental Design should be about learning and not about testing and based on studying how the presence of a realistic number and size of active effects affects critical values, proposes to use t = 2 for any number of runs equal or greater than 8. And it shows that this solution, in addition of being simpler, provides under reasonable realistic situations better results than those obtained by simulation.

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