In typical parameter estimation problems, the signal observation is a function of the parameter set to be estimated as well as some background (environmental/system) parameters assumed known. The assumed background could differ from the true one, leading to biased estimates even at high signal-to-noise ratio (SNR). To analyze this background mismatch problem, a Ziv-Zakai-type lower bound on the mean-square error (MSE) is developed based on the mismatched likelihood ratio test (MLRT). At high SNR, the bound incorporates the increase in MSE due to estimation bias; at low SNR, it includes the threshold effect due to estimation ambiguity. The kernel of the bound's evaluation is the error probability associated with the MLRT. A closed-form expression for this error probability is derived under a random signal model typical of the bearing estimation/passive source localization problem. The mismatch is then analyzed in terms of the related ambiguity functions. Examples of bearing estimation with system (array shape) mismatch demonstrate that the developed bound describes the simulations of the maximum-likelihood estimate well, including the sidelobe-introduced threshold behavior and the bias at high SNR.

Matched field and ocean acoustic tomography concern the estimation of parameters for models of ocean environments using acoustics. Both require full field representations for the observed signals since waveguide effects are important. The authors present Cramer-Rao lower bounds for the attainable accuracy of both methods. These bounds are expressed in terms of the Green's function for the propagation between source and receivers.

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.

The probability density and confidence intervals for the maximum entropy (or regression) method (MEM) of spectral estimation are derived using a Wishart model for the estimated covariance. It is found that the density for the estimated transfer function of the regression filter may be interpreted as a generalization of the student's t distribution. Asymptotic expressions are derived which are the same as those of Akaike. These expressions allow a direct comparison between the performance of the maximum entropy (regression) and maximum likelihood methods under these asymptotic conditions. Confidence intervals are calculated for an example consisting of several closely space tones in a background of white noise. These intervals are compared with those for the maximum likelihood method (MLM). It is demonstrated that, although the MEM has higher peak to background ratios than the MLM, the confidence intervals are correspondingly larger. Generalizations are introduced for frequency wavenumber spectral estimation and for the joint density at different frequencies.

A method of solving Fredholm integral equations by state-variable techniques is presented. A principal feature of this method is that it leads to efficient computer algorithms for calculating numerical solutions. The assumptions made are 1) the kernel of the integral equation is the covariance function of a random process, 2) this random process is the output of a linear system having a white-noise input, 3) this linear system has a finite-dimensional state-variable description. Both the homogeneous and inhomogeneous equations are reduced to two linear first-order differential equations and an associated set of boundary conditions. The coefficients of these differential equations and the boundary conditions are specified directly by the matrices describing the random process that generates the kernel. The eigenvalues of the homogeneous integral equation are found to be solutions of a transcendental equation involving the transition matrix of the vector differential equations. The eigenfunctions follow directly. By using this same transcendental equation, an effective method of calculating the Fredholm determinant is derived. For the inhomogeneous equation, the vector differential equations are identical to those obtained in the state-variable formulation of the optimal linear smoother. Several examples illustrating the methods developed are presented.