Correlation in the statistical analysis of a reverse fourier neutron time-of-flight experiment: I. Theoretical considerations

Abstract

A thorough theoretical analysis of the methods available for estimation from a reverse Fourier neutron time-of-flight experiment is carried out. Independent variables of the correlated observations are found from the frequency domain by means of the discrete Fourier cosine and sine transforms. The number of independent frequency components is shown to be less than the number of original observatins, so that the distribution and the variance-covariance matrix of the observations are singular. Although estimation can be carried out by employing the independent variables in the frequency domain, use of the original observations is particularly suitable for diffraction spectra analysis. It is pointed out that least-squares estimators for a non-singular or a singular distribution are the same provided that ina singular case the generalized inverse of the variance-covariance matrix is employed instead of the true inverse. Besides the general methods a convenient new approach to least-squares estimation from the correlated time-of-flight observations is discussed. In this method the estimators are especially simple due to the assumption of the resolution function being proportional to the covariance function, which defines the extent of the correlation between the adjacent channels of a spectrum. Finally, the consequences of the correlation are evaluated by omitting the covariances of observations when the sum of squares is minimized. This procedure is shown to yield unbiased estimators which do not have minimum variance among the class of unbiased estimators in consequences of an incorrect weighting scheme for the independent frequency variables of an experiment.