This paper presents a method to specify a strictly stationary univariate time series model with particular emphasis on the marginal characteristics (fat tailedness, skewness etc.). It is the first time in time series models with specified marginal distribution, a non-parametric specification is used. Through a Copula distribution, the marginal aspect are separated and the information contained within the order statistics allow to efficiently model a discretely-varied time series. The estimation is done through Bayesian method. The method is invariant to any copula family and for any level of heterogeneity in the random variable. Using count times series of weekly firearm homicides in Cape Town, South Africa, we show our method efficiently estimates the copula parameter representing the first-order Markov chain transition density.