Innovative application of set theory, Cartesian product, and multinomial theorem in demographic research

The bootstrap technique has been widely used to estimate the variances, standard errors, and confidence intervals (CI) of life table parameters, while the paired bootstrap test has been used to compare life table parameters between treatments by assessing the CI of differences. Although a great number of resamplings (B = 1,000 similar to 100,000) has been suggested for the application of the bootstrap technique, each computer simulation will generate different results because the bootstrap sampling with replacement is based on a stochastic process. In order to determine the theoretical and true confidence intervals of population parameters, and thereby, achieve an accurate assessment of differences between treatments, we introduce an innovative application of set theory, Cartesian product, and multinomial theorem for a mathematical formulation of demographic analysis. Moreover, when a bootstrap sample is composed of individuals that cannot produce offspring (i.e., an infertile bootstrap sample), the intrinsic rate of increase (r) and finite rate of increase (lambda) cannot be calculated. Omitting these infertile bootstrap samples will result in biased estimates. This problem of infertile bootstrap samples in demographic research has not been resolved. The integrated application of the set theory, Cartesian products, and multinomial theorem enables the detection of all possible combinations of bootstrap samples, the true CIs of population parameters, and the CIs of differences between treatments; furthermore, the probabilities of both fertile and infertile bootstrap samples can also be calculated. The life table data of the well-known cosmopolitan pest, Myzus persicae (Sulzer) (Hemiptera: Aphididac), were collected and used as examples.