1 A note on estimating autocovariance from short-time observations Yoel Shkolnisky, Fred J. Sigworth and Amit Singer Abstract We revisit the classical problem of estimating the autocovariance function and power spectrum of a stochastic process. In the typical setting for this problem, one observes a long sequence of samples of the process, from which the autocovariance needs to be estimated. It is well known how to construct consistent estimators of the autocovariance function in this case, and how to trade bias for variance. In physical settings such as cryo-electron microscopy (EM) we are required to estimate the response of the physical instrument through the observation of many short noise sequences, each with a different mean. In this setting, the known estimators are significantly biased, and unlike the typical case, this bias does not disappear as the number of observations increases. The bias originates from replacing the unknown true mean by the sample average. We analyze and demonstrate this bias, derive an unbiased estimator, and examine its performance for various noise processes. EDICS Category: SSP-SSAN Statistical signal analysis I. I NTRODUCTION The autocovariance function (ACF) of a discrete stationary ergodic time series {X t } ∞ t=-∞ with expected value E[X t ]= µ at lag h is defined as C (h)= E[(X t − µ)(X t+h − µ)]. (1) Yoel Shkolnisky is with the Department of Mathematics, Program in Applied Mathematics, Yale University, 10 Hillhouse Ave. PO Box 208283, New Haven, CT 06520-8283 USA. Fred J. Sigworth is with the Department of Cellular and Molecular Physiology, Yale University School of Medicine, 333 Cedar Street, New Haven, CT 06520 USA. Amit Singer is with the Department of Mathematics and PACM, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544-1000 USA. emails: yoel.shkolnisky@yale.edu, fred.sigworth@yale.edu, amits@math.princeton.edu July 14, 2008 DRAFT