Uncertainty bounds for parameter identification with small sample sizes

Consider the problem of determining uncertainty bounds for an M-estimate of a parameter vector from (generally) non-i.i.d. data (M-estimates are those obtained as the solution of a set of equations; maximum likelihood estimates are perhaps the most common type). Calculating uncertainty bounds requires information about the distribution of the estimate. It is well known that M-estimates typically have an asymptotic normal distribution. However, because of their generally complex nonlinear (and implicitly defined) structure, very little is usually known about the finite-sample distribution. This paper presents a method for characterizing the distribution of an M-estimate when the sample size is small. The approach works by comparing the actual (unknown) distribution of the estimate with a closely related known distribution. Some discussion and analysis are included that compare the approach here with the well-known bootstrap and saddlepoint methods. Theoretical justification and an illustration of the approach in a signal-plus-noise estimation problem are presented. This illustrative problem arises in many contexts, including random effects modeling ("unbalanced" case), the problem of combining several independent estimates, Kalman filter-based modeling, small area survey analysis, and quantile calculation for projectile accuracy analysis.