Exponential Tilting for Zero-inflated Interval Regression with Applications to Cyber Security Survey Data

Non-negative ordered survey data often exhibit an unusually high frequency of zeros in the first interval. Zero-inflated ordered probit models handle the excess of zeros by combining a split probit model and an ordered probit model. In the presence of data violating distributional assumptions, standard inference based on the maximum likelihood method gives biased estimates with large standard errors. In this paper, we consider robust inference for the zero-inflated ordered probit model based on the exponential tilting methodology. Exponential tilting selects unequal weights for the observations in such a way as to maximise the likelihood function subject to moving a given distance from equally weighted scores. As a result, observations that are incompatible with the assumed zero-inflated distribution receive a relatively small weight. Our methodology is motivated by the analysis of survey data on cyber security breaches to study the relationship between investments in cyber defences and costs from cyber breaches. Robust estimates obtained via tilting clearly show an e ect of the investments in reducing the amount of the loss from a cyber breach.