Minimum mean-square error quadrature

Minimum mean squared error linear estimators of the area under a curve are considered for cases when the observations are observed with error. The underlying functional form giving rise to the observations is left unspecified, leading to use of quadrature estimators for the true area. The optimal estimator is calculated as a shrinkage of some preliminary estimator (based on, e.g., the trapezoidal rule). Applications to selected exponential functions demonstrate that savings in mean squared error varies with the level of underlying variance. For cases where variance at each time point is large, the proposed rule can bring about savings in mean squared error of as much as 30%. For experiments with small underlying variance at each time point, squared bias is of greater importance than variance in contributing to mean squared error, and the value of higher-order quadrature routines that focus on minimizing approximation error is noted.