A new uniform AR(1) time series model (NUAR(1))

Miroslav M. Risti\'c and Biljana \v C. Popovi\'c

Abstract: We present a new first-order autoregressive time series model (so-called NUAR(1) model) for continuous uniform $(0,1)$ variables, given by $$ X_n=\begin{cases} \alpha X_{n-1}, & \text{ w.p. } \alpha,
\beta X_{n-1}+\varepsilon_n, & \text{ w.p. } 1-\alpha, \end{cases} $$ where $0<\alpha,\beta<1$, $(1-\alpha)/\beta\in\{1,2,\dots\}$ and $\{\varepsilon_n\}$ is the innovation sequence of independent and identically distributed random variables, such that each $X_n$ has continuous uniform $(0,1)$ distribution. The distribution of the innovation sequence and autoregressive structure of NUAR(1) model are discussed. It is shown that this model is partially time-reversible if the parameters are equal. We give also the estimates of the parameters of the model.

Keywords: Autoregressive process, continous uniform (0,1) distribution, time series, estimation, random coefficients, residuals

Classification (MSC2000): 62M10

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