Confidence interval determination for spectral estimates using "Tilted densities"

Determining the confidence intervals for a spectral estimate requires knowledge of its probability density function. Except when the spectral estimate is the sum of the magnitude squared of independent and identically distributed Gaussian random variables for which the Chi-squared distribution is applicable, expressions for this probability density are not available. Tapering of the original time series, unequal weighing in the spectral averaging and overlapping estimates from different time segments, all of which are routinely done in practice, introduce dependencies and differences in the distribution of the random variables used to form the spectral estimate. These effects make the Chi-squared model inapplicable in many situations. In most approaches, density is simply approximated as being Chi-squared with an effective number of degrees of freedom calculated from the variance of the estimate. Unfortunately, this can be inaccurate in approximating the tails of the density where confidence interval calculations are usually done. Determining confidence intervals is similar to calculating the performance probabilities, e.g., false alarm and miss probabilities in detection and communication theory. For this, the approach of tilted densities has led to very accurate approximations. In this approach the probabilities are calculated using the semi-invariant function and its derivatives for which analytic expression can easily be derived. The tilted density approach has been applied to determine the confidence intervals for spectral estimates in the general case where tapering, unequal weighing and/or segment overlapping are done. Finally, the accuracy of the equivalent degrees of freedom approach is evaluated.