Uncertainty Bounds in Parameter Estimation with Limited Data

Consider the problem of determining uncertainty bounds for parameter estimates with a small sample size of data. Calculating uncertainty bounds requires information about the distribution of the estimate. Although many common parameter estimation methods (e.g., maximum likelihood, least squares, maximum a posteriori, etc.) have an asymptotic normal distribution, very little is usually known about the finite-sample distribution. This paper presents a method for characterizing the distribution of an estimate when the sample size is small. The approach works by comparing the actual (unknown) distribution of the estimate with an “idealized” (known) distribution. Some discussion and analysis are included that compare the approach here with the well-known bootstrap and saddlepoint methods. Example applications of the approach are presented in the areas of signal-plus-noise modeling, nonlinear regression, and time series correlation analysis. The signal-plus-noise problem is treated in greatest deta il; this problem arises in many contexts, including state-space modeling, the problem of combining several independent estimates, and quantile calculation for projectile accuracy analysis.