Calculation Of Statistical Error Research Articles

USE OF THE BOOTSTRAP FOR ERROR PROPAGATION IN ESTIMATING ZOOPLANKTON PRODUCTION

Error propagation is the calculation of statistical error in a quantity that comprises multiple components each with associated error. Such quantities constitute the end point of many ecological studies, but composite errors are rarely incorporated so that uncertainty in the final estimate is either unknown or underestimated. In this study we present the use of both parametric (observations are resampled from standard probability distributions) and nonparametric (raw observations are resampled) bootstrap techniques to propagate errors through the many steps involved in the egg-ratio estimation of seasonal production for the freshwater zooplankter Bythotrephes longimanus. We first compute parametric and nonparametric bootstrap estimates of the standard deviation in seasonal production of B. longimanus, showing that it ranges from 21% to 27% of observed seasonal production. We demonstrate that our bootstrapping procedures are robust by developing a theoretical model and showing that the true variance of a parameter is included in 96% of the simulated 95% confidence intervals. We also show that the choice of probability distribution in the parametric bootstrap can change the standard deviation of seasonal production by up to 90%. We argue that ecologists should use such error propagation techniques more routinely than is currently the case.

On confidence limits and statistical convergence of Monte Carlo point-flux estimators with unbounded variance

The problem of analyzing the statistical error of Monte Carlo point-flux estimators with unbounded variance is considered. The results are based on the generalization of the classical limit theorem for summands of identically-distributed random variables on the unbounded-variance case. The Lévy representation of the characteristic functions of stable attracting distributions is implemented. The explicit expression of the stable attracting probability density function for the uncollided point-flux estimates is obtained and the corresponding cumulative probability function is tabulated. It is shown that a 1 x singularity of an unbounded-variance point-flux estimator in the vicinity of the detector point leads to 1 N 1 2 convergence to the normal law. The procedure of estimation of confidence limits and statistical accuracy of unbounded-variance point-flux estimators for intermediate sample size is discussed. The results may be utilized for the calculation of statistical error of Monte Carlo calculations of flux at a point.

Errors in the gran addition method: Part I. Theoretical calculation of statistical errors

The precision in the determination of unknown concentration C 0 by the Gran addition technique is governed by the conditions used for titration. Optimal conditions for which the error in C 0 is a minimum are determined theoretically, by considering the individual errors in each of the parameters. Extrapolation by the least squares fit and by the graphical method are considered.

Accuracy limitations in an unbiased optimum data treatment

Simple explicit formulas are derived for an evaluation of the elements of the covariance matrix needed for the calculation of statistical errors which arise when some optimum linear estimates of some parameters are to be obtained from experimental data. The considered linear estimates are of the Neumann-David type, which represents a generalization of the least squares estimate. These formulas are related to the case of polynomial nonrandom data components with a stationary random component, the correlation function of which may be approximated by an exponential function. The results of this paper can be used during the design stage of the preparation of an experiment for the choice of parameters of experimental facility, such as the observing interval, number of experimental points, etc., as the formulas characterizing errors in results of data treatment are functions of these parameters.