Bayesian testing of nonparametric hypotheses and its application to global optimization

Random distribution functions are the basic tool for solving nonparametric decision-theoretic problems. In 1974, Doksum introduced the family of distributions neutral to the right, that is, distributions such thatF(t₁),[F(t₂)–F(t₁)]/[1 –F(t₁)],...,[F(t_k)–F(t_{k – 1})]/[1 –F(t_{k – 1})] are independent whenevert₁ < ... <t_kIn practice, application of distributions neutral to the right has been prevented by the lack of a manageable analytical expression for probabilities of the typeP(F(t)<q) for fixedt andq. A subclass of such distributions can be provided which allows for a close expression of the characteristic function of log[1–F(t)], given the sample. Then, thea posteriori distribution ofF(t) is obtained by numerical evaluation of a Fourier integral. As an application, the global optimization problem is formulated as a problem of inference about the quantiles of the distributionF(y) of the random variableY=f(X), wheref is the objective function andX is a random point in the search domain.The author thanks J. Koronacki and R. Zielinski of the Polish Academy of Sciences for their valuable criticism during the final draft of the paper.