Testing randomness

The hypothesis of randomness is fundamental in statistical machine learning and in many areas of nonparametric statistics; it says that the observations are assumed to be independent and coming from the same unknown probability distribution. This hypothesis is close, in certain respects, to the hypothesis of exchangeability, which postulates that the distribution of the observations is invariant with respect to their permutations. This paper reviews known methods of testing the two hypotheses concentrating on the online mode of testing, when the observations arrive sequentially. All known online methods for testing these hypotheses are based on conformal martingales, which are defined and studied in detail. The paper emphasizes conceptual and practical aspects and states two kinds of results. Validity results limit the probability of a false alarm or the frequency of false alarms for various procedures based on conformal martingales, including conformal versions of the CUSUM and Shiryaev-Roberts procedures. Efficiency results establish connections between randomness, exchangeability, and conformal martingales.