Peskir and Shiryaev ( 2000, 2002) solved both problems of sequential analysis for Poisson processes in full generality (see also Peskir and Shiryaev 2006, Chap. VI, Sects. 23 and 24). The standard disorder problem for observable Poisson processes with unknown intensities was introduced and solved by Davis ( 1976) given certain restrictions on the model parameters. The first solutions of these problems in the continuous-time setting were obtained in the case of observable Wiener processes with constant drift rates by Shiryaev ( 1978, Chap. IV, Sects. 2 and 4) (see also references to original sources therein). The sequential testing and quickest change-point detection problems were originally formulated and solved for sequences of observable independent identically distributed random variables by Shiryaev ( 1978, Chap. IV, Sects. 1 and 3) (see also references to original sources therein). 1994 Poor and Hadjiliadis 2008 Shiryaev 2019 for an overview). Such problems found applications in many real-world systems in which the amount of observation data is increasing over time (see, e.g. In the classical Bayesian formulation, it is assumed that the random time \(\theta \) takes the value 0 with probability \(\pi \) and is exponentially distributed given that \(\theta > 0\). The problem of quickest change-point (or disorder) detection for an Wiener process is to find a stopping time of alarm \(\tau \) which is as close as possible to the unknown time of change-point \(\theta \) at which the local drift rate of the process changes from one constant value to another. ![]() In the Bayesian formulation of this problem, it is assumed that these alternatives have an a priori given distribution. We also prove the subGaussian supermartingale to be admissible.The problem of sequential testing for two simple hypotheses about the drift rate of an observable Wiener process (or Brownian motion) is to detect the form of its constant drift rate from one of the two given alternatives. We provide several nontrivial examples, with special focus on testing symmetry, where our new constructions render past methods inadmissible. ![]() Informally, if one wishes to perform anytime-valid sequential inference, then any existing approach can be recovered or dominated using nonnegative martingales. Our proofs utilize several modern mathematical tools for composite testing and estimation problems: max-martingales, Snell envelopes, transfinite induction, and new Doob-Lévy martingales make appearances in previously unencountered ways. Thus, informally, nonnegative (super)martingales are known to be sufficient for \emph, we show that all admissible constructions of confidence sequences, p-processes, or e-processes must necessarily utilize nonnegative martingales. ![]() Examining the literature, one finds that at the heart of all these (quite different) approaches has been the identification of nonnegative (super)martingales. Confidence sequences, anytime p-values (called p-processes in this paper), and e-processes all enable sequential inference for composite and nonparametric classes of distributions at arbitrary stopping times.
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