General Properties and Estimation of Conditional Bernoulli Models

Conditional Bernoulli (in short “CB”) models have been recently applied to many statistical fields including survey sampling, logistic regression, case-control studies, lottery, signal processing and Poisson-Binomial distributions. In this paper, we present several general properties of CB models that are necessary for the applications above. We also show the existence and uniqueness of MLE of parameters in CB models and give two efficient algorithms for computing the MLE. General properties of CB models include: (1) mappings between three characterizations of CB models are homeomorphism modulo rescaling and order-preserving; (2) CB variables are unconditionally independent and conditionally negatively correlated; (3) a simple formula relating inclusion probabilities of adjacent orders can be used to ease computational burden and provide important implication on odds-ratio. Asymptotic properties of CB models are also examined. We show that under a mild condition, (1) CB variables are asymptotically independent; (2) covariances of CB variables are asymptotically on a smaller scale than variances of CB variables; and (3) a CB model can be approximated by a multinomial distribution with the same coverage probabilities. The use and implication of each property are illustrated with related statistical applications.