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We propose a general purpose confidence interval procedure (CIP) forstatistical functionals constructed using data from a stationary time series.The procedures we propose are based on derived distribution-free analogues ofthe $\chi^2$ and Student's $t$ random variables for the statistical functionalcontext, and hence apply in a wide variety of settings including quantileestimation, gradient estimation, M-estimation, CVAR-estimation, and arrivalprocess rate estimation, apart from more traditional statistical settings. Likethe method of subsampling, we use overlapping batches of time series data toestimate the underlying variance parameter; unlike subsampling and thebootstrap, however, we assume that the implied point estimator of thestatistical functional obeys a central limit theorem (CLT) to help identify theweak asymptotics (called OB-x limits, x=I,II,III) of batched Studentizedstatistics. The OB-x limits, certain functionals of the Wiener processparameterized by the size of the batches and the extent of their overlap, formthe essential machinery for characterizing dependence, and consequently thecorrectness of the proposed CIPs. The message from extensive numericalexperimentation is that in settings where a functional CLT on the pointestimator is in effect, using \emph{large overlapping batches} alongside OB-xcritical values yields confidence intervals that are often of significantlyhigher quality than those obtained from more generic methods like subsamplingor the bootstrap. We illustrate using examples from CVaR estimation, ARMAparameter estimation, and NHPP rate estimation; R and MATLAB code for OB-xcritical values is available at~\texttt{web.ics.purdue.edu/~pasupath/}.

Overlapping Batch Confidence Intervals on Statistical Functionals  Constructed from Time Series: Application to Quantiles, Optimization, and  Estimation