A Time Series Analysis of Binary Data

Binary data d 1, d 2, …, dn are assumed to be generated by an underlying real-valued, strictly stationary process, {Xk }, and a response function F. For a given monotone nondecreasing function F from R to [0, 1], Dk takes on 1 with probability F(xk ) and 0 with probability 1 - F(xk ), where Xk = xk. It is shown that all strictly stationary binary processes are characterized by such a procedure. Several approximations to the n-dimensional joint probabilities of Dk are developed when Xk is a Gaussian first-order autoregressive process. Model-building procedures and methods by which to estimate parameters of a given model are discussed. The predictor of d n + 1 that minimizes probability of error among all randomized rules is determined and for certain cases a bound for this probability is found.