The active quickest detection problem with unknown postchange parameters is studied under the sampling control constraint, where there are p local streams in a system but one is only able to take observations from one and only one of these p local streams at each time instant. The objective is to raise a correct alarm as quickly as possible once the change occurs subject to both false alarm and sampling control constraints. Here we assume that exactly one of the p local streams is affected, and the postchange distribution involves unknown parameters. In this context, we propose an efficient greedy cyclic sampling–based quickest detection algorithm and show that our proposed algorithm is asymptotically optimal in the sense of minimizing the detection delay under both false alarm and sampling control constraints. Numerical studies are conducted to show the effectiveness and applicability of the proposed algorithm.

Levin and Leu (2021) introduced some key inequalities that underlie the lower bound formula for the probability of lattice events when using adaptive members of the Levin-Robbins-Leu family of sequential subset selection procedures for binary outcomes. Here we provide a rigorous proof of the key inequality for each adaptive procedure in the special case of equal odds parameters. We also provide some further insight into why the key inequality holds for arbitrary odds parameters and we present a complete proof in that case for a simple yet non-trivial prototype example. Two errata in the abovementioned publication are also corrected herein.

In this article, we deal with problems of testing hypotheses in the framework of sequential statistical analysis. The main concern is the optimal design and performance evaluation of sampling plans in the Kiefer-Weiss problems. For the case of observations following a discrete exponential family, we provide algorithms for optimal design in the modified Kiefer-Weiss problem and obtain formulas for evaluating their performance, calculating operating characteristic function, average sample number, and some related characteristics. These formulas cover, as a particular case, sequential probability ratio tests (SPRTs) and their truncated versions, as well as optimal finite-horizon sequential tests. On the basis of the developed algorithms we propose a method of construction of optimal tests and their performance evaluation for the original Kiefer-Weiss problem. All algorithms are implemented as functions in the R programming language and can be downloaded from https://github.com/tosinabase/Kiefer-Weiss, where the code for the binomial, Poisson, and negative binomial distributions is readily available. Finally, we make numerical comparisons of the Kiefer-Weiss solution with the SPRT and the fixed-sample-size test having the same levels of the error probabilities.

In this article, we revisit the classic inference problem of minimum risk point estimation for an unknown normal mean when the variance also remains unknown. We propose an alternative three-stage sampling procedure with termination defined via Gini’s mean difference rather than the traditional sample standard deviation. A number of asymptotic properties are investigated both theoretically and empirically. An extensive set of simulations is conducted to demonstrate the remarkable performance of the new procedure. For practical purposes, we also include illustrations using real data sets on the number of days marigold seeds need to flower.

We present a computational approach to the solution of the Kiefer-Weiss problem. Algorithms for construction of the optimal sampling plans and evaluation of their performance are proposed. In the particular case of Bernoulli observations, the proposed algorithms are implemented in the form of R program code. Using the developed computer program, we numerically compare the optimal tests with the respective sequential probability ratio test (SPRT) and the fixed sample size test for a wide range of hypothesized values and type I and type II errors. The results are compared with those of D. Freeman and L. Weiss (Journal of the American Statistical Association, 59, 1964). The R source code for the algorithms of construction of optimal sampling plans and evaluation of their characteristics is available at https://github.com/tosinabase/Kiefer-Weiss.

Assuming normality for the prior and the likelihood, we calculate the rejection region, the power or the conditional power, and the predictive power or the conditional predictive power of one-sided hypotheses with a nonzero threshold that corresponds to a noninferiority test for two-arm trials for five different scenarios, which are nonsequential trials with classical power and Bayesian power and sequential trials with hybrid predictions, Bayesian predictions, and classical predictions. The rejection regions and the powers of one-sided hypotheses with a zero threshold that corresponds to a superiority test for two-arm trials are also obtained. Then the calculations of the various powers are illustrated through two examples. The article can be regarded as a reference manual for researchers interested in power calculations of one-sided hypotheses with a nonzero or zero threshold for the five different scenarios assuming normality for the prior and the likelihood.

The efficient evaluation of high-dimensional integrals is important from both theoretical and practical points of view. In particular, multidimensional integration plays a central role in Bayesian inference, statistical physics, data science, and machine learning. However, due to the curse of dimensionality, deterministic numerical methods are inefficient in the high-dimensional setting. Consequentially, for many practical problems one must resort to approximate estimation techniques such as Monte Carlo methods. In this article, we introduce a novel sequential Monte Carlo algorithm called stratified splitting. The method provides unbiased estimates and can handle various integrand types including indicator functions, which are important for rare-event probability estimation problems. We provide rigorous analysis of the efficiency of the proposed method and present a numerical demonstration of the algorithmic performance when applied to practical application domains. Our numerical experiments suggest that the stratified splitting method is capable of delivering accurate results for a variety of integration problems while requiring reasonable computational effort.

The optimal parameter estimation problem is considered. The optimization problem is solved in the general problem statement. A model-free approach is applied and supposes no knowledge of the model that the parameter to be estimated belongs to. Optimality of the considered estimators in the sense of a special type risk function is established. The considered risk function makes it possible to optimize the asymptotic variances of the estimators and is used for sample size estimation. Applications for optimization of the truncated parameter estimators of heavy-tailed indexes of distributions, such as Pareto type, Cauchy, and log-gamma, are presented. A class of these estimators is introduced having guaranteed accuracy based on a sample of fixed size. Simulation results confirm theoretical results.

In this article, we study sequential testing problems with overlapping hypotheses. We first focus on the simple problem of assessing if the mean μ of a Gaussian distribution is smaller or larger than a fixed if both answers are considered to be correct. Then, we consider probably approximately correct best arm identification in a bandit model: given K probability distributions on with means we derive the asymptotic complexity of identifying, with risk at most δ, an index such that We provide nonasymptotic bounds on the error of a parallel general likelihood ratio test, which can also be used for more general testing problems. We further propose a lower bound on the number of observations needed to identify a correct hypothesis. Those lower bounds rely on information-theoretic arguments, and specifically on two versions of a change of measure lemma (a high-level form and a low-level form) whose relative merits are discussed.

The problem of sequential estimation of the mean with quadratic loss and fixed cost per observation is considered within the Bayesian framework. A robust sequential procedure, not depending on the distributions of outcome variables and the prior, in the Bayes sequential estimation is investigated. In the present article, the second-order approximations to the expected sample size and the Bayes risk of the robust sequential procedure are obtained for the arbitrary distributions of outcome variables and the prior. The second-order efficiency is further discussed in an example.