Multidimensional Poverty with Missing Attributes

Suppose that n quantifiable attributes of an individual - e.g. income, life expectancy, education etc. - are important for deciding whether or not that individual is living in poverty. If his profile of attributes falls below a certain poverty frontier then the individual is considered poor. Consider the problem of a policy-implementer who can only observe k < n of these attributes. The bureaucrat wishes to choose a poverty frontier only in those attributes that are observable, so as to minimize the expected number of misclassified individuals. We characterize the solution to this minimization problem