Using Pearson’s System of Curves to Approximate the Distributions of the Difference between Two Correlated Estimates of Signal-to-Noise Ratios: The Cases of Bivariate Normal and Bivariate Lognormal Distributions

Background: The signal-to-noise ratio (SNR) is recognized as an index of measurements reproducibility. We derive the maximum likelihood estimators of SNR and discuss confidence interval construction on the difference between two correlated SNRs when the readings are from bivariate normal and bivariate lognormal distribution. We use the Pearson’s system of curves to approximate the difference between the two estimates and use the bootstrap methods to validate the approximate distributions of the statistic of interest. Methods: The paper uses the delta method to find the first four central moments, and hence the skewness and kurtosis which are important in the determination of the parameters of the Pearson’s distribution. Results: The approach is illustrated in two examples; one from veterinary microbiology and food safety data and the other on data from clinical medicine. We derived the four central moments of the target statistics, together with the bootstrap method to evaluate the parameters of Pearson’s distribution. The fitted Pearson’s curves of Types I and II were recommended based on the available data. The R-codes are also provided to be readily used by the readers.