Robustness of The Confidence Interval for At-Risk-Of-Poverty-Rate

Zieliński (2009) constructed a nonparametric interval for At-Risk-of-Poverty-Rate. It appeared that the confidence level of the interval depends on the underlying distribution of the income. For some distributions (e.g. lognormal, gamma, Pareto) the confidence level may be smaller than the nominal one. The question is, what is the largest deviance from the nominal level? In the paper, a more general problem is considered, i.e. the problem of robustness of the confidence level of the confidence interval for binomial probability. The worst distribution is derived as well as the smallest true confidence level is calculated. Some asymptotic remarks (sample size tends to infinity) are also given.