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This chapter demonstrates that the numerical methods of earlier chapters are not constrained by statistical philosophy. The theory of Bayesian estimating functions is developed. It is shown that this theory has Bayesian analogues for many of the concepts introduced in earlier chapters. While point estimation is often considered of secondary importance to Bayesians, the Bayesian estimating function methodology does have important applications in areas such as actuarial science.

This chapter gives a survey of the basic concepts of estimating functions, which are used in subsequent chapters. The concept of unbiasedness for estimating functions is introduced as a generalization of the concept of an unbiased estimator. Godambe efficiency, also known as the Godambe optimality criterion, is introduced by generalizing the concept of minimum variance unbiased estimation. Within the class of estimating functions which are unbiased and information unbiased, the score function is characterized as the estimating function with maximal Godambe efficiency. Extensions to the multiparameter case are given, and the connection to the Riesz representation theorem is described briefly. This chapter also discusses a number of examples from semiparametric models, martingale estimating functions for stochastic processes, empirical characteristic function methods and quadrat sampling; the estimating equations in some of these examples have possibly more than one solution.

Two important applications of the SRF concept developed in chapter 2 are point estimation and image simulation. Point estimation considers the SRF Z at an unsampled location, x0, and the goal is to get an estimate for z at x0 which is physically plausible and is optimal in some sense, and to provide a measure of the quality of the estimate. The goal in image simulation is to create an image of Z over the entire domain, one that not only is in agreement with the measurements at their locations, but also captures the correlation pattern of z. We start by considering a family of linear estimators known as kriging. Its appeal is in its simplicity and computational efficiency. We then proceed to discuss Bayesian estimators and will show how to condition estimates on “hard” and “soft” data, and we shall conclude by discussing a couple of simple, easy-to-implement image simulators. One of the simulators presented can be downloaded from the Internet. Linear regression aims at estimating the attribute z at x0: z0 = z(x0), based on a linear combination of n measurements of z: zi = z(xi), i = 1 , . . . ,n. The estimator of z(x0) is z*0, and it is defined by What makes this estimator “linear” is the exclusion of powers and products of measurements. However, nonlinearity may enter the estimation process indirectly, for example, through nonlinear transformation of the attribute. The challenge posed by(3.1) is to determine optimally the n interpolation coefficients λi, and the shift coefficient λ0. The actual estimation error is z*0 - z0; it is unknown, since z0 is unknown, and so no meaningful statement can be made about it. As an alternative, we shall consider the set of all equivalent estimation problems: in this set we maintain the same spatial configuration of measurement locations, but allow for all the possible combinations, or scenarios, of z values at these locations, including x0. We have replaced a single estimation problem with many, but we have improved our situation since now we know the actual z value at x0 and this will allow a systematic approach.