Estimation of autocorrelation in a binary time series
Let \(Y_ t\) be a stationary Gaussian time series with zero mean. Define \(X_ t=1\) for \(Y_ t>0\) and \(X_ t=0\) for \(Y_ t<0\). The lag k transition probability \(\lambda_ k=P(X_ t=1| X_{t-k}=1)\) can be estimated by \[ {\hat \lambda}_ k=1-(n-k)^{- 1}\sum^{n}_{t=k+1}(X_ t+X_{t-k}-2X_ tX_{t-k}). \] The estimator \({\hat \lambda}{}_ k\) is unbiased. The authors derive a formula for var \({\hat \lambda}{}_ 1\), suggest three approximations to var \({\hat \lambda}{}_ 1\) in the AR(1) case and give an explicit formula for var \({\hat \lambda}{}_ 1\) in the MA(1) and MA(2) cases. In an appendix the authors investigate the probability that four dependent normal variables have the same sign. They reduce the general formula to a two-dimensional integral. If the series is an AR(1) process, then a good approximation to the probability is found.
scientific article; zbMATH DE number 4084804
scientific article