Maximum Likelihood Estimation of the Bivariate Logistic Regression with Threshold Parameters

In discussing the relationship between the successful probability of a binary random variable and a certain set of explanatory variables, the logistic regression is popular in various fields, especially in medicine researches. For example, (1) using remedy methods to explain the probability of recovery from some disease; (2) building a prediction model for the e-commence customer buying behavior; (3) building a model to decide the cut-point of the life insurance sales hiring; (4) building a credit score function; or (5) deciding the threshold parameter for the extreme human injure. Sometimes a variable must exceed a certain value to have influence on the response variable. For example, the income of a household must exceed some value to influence the successful probability. We call this value "threshold". On the other hand, it is possible to observe paired binary variables in the business study. In this article, we propose a bivariate logistic regression model, and also consider variables that may have threshold parameter. The association between the pair binary variables is considered as a constant odds ratio for demonstration. We used maximum likelihood methodology to find parameter estimates via a modified simplex algorithm, and suggested a parametric bootstrapping method to estimate the standard deviation of the maximum likelihood estimator.