The maximum entropy method of spectral analysis is examined in terms of the probability density of a finite random sequence. The distribution is found which has maximum entropy within constraints imposed by the stationarity of the sequence. The population power spectral density of the sequence is expressed in terms of the parameters of this distribution. The distribution is shown to be related to a linear regression model for which the parameters may be estimated by the established methods of regression theory. Confidence limits for the spectral estimator are derived, infinite confidence limits being interpreted in terms of cyclic behavior of the ensemble means of the sequence. The null hypothesis that the members of a subset of regression coefficients are all zero is used to estimate the order of the autoregression. Estimated spectra of a sequence of annual mean sunspot numbers are computed using this method and found to have a single significant peak.