Estimation of confidence intervals for the nonlinear parameters of a least-squares fit. Application to cholesteryl nonanoate data

The need for fitting experimental data with a model containing nonlinear parameters occurs frequently. Estimates for the confidence intervals of the nonlinear parameters are occasionally stated, however the method of determination, if any, is rarely explained. Plots of the error surface near the minimum are popular but contain little information. Taylor's series expansion about a good estimate of the solution yields a convergent procedure for nonlinear fitting and a covariance matrix useful for estimation of the parameter confidence intervals. In order to make the advantages of this approach explicit, the data of Pindak and coworkers on the divergence of the cholesteric pitch in cholesteryl nonanoate near the smectic-A phase transition have been reanalyzed. Three-and five-parameter fits yield estimates of 0.674±0.005 and 0.702±0.035, respectively, for the critical exponentv. The results indicate that the validity of the published estimate ofv=0.675±0.025 depends on the independent measurement of two of the five model parameters originally proposed. The influence of weighting factors and highly correlated expansion functions is also demonstrated. A preliminary account of this paper was presented to the Summer Meeting of the AAPT, Boone, N. C., June 20–22, 1974, AAPT Announcer4, No. 2, 30 (1974).