A note on estimating autocovariance from short-time observations
Yoel Shkolnisky, Fred J. Sigworth, Amit Singer · 2008
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 th is 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 bias ed, 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.