Verallgemeinerte bayes-verfahren bei sequentiellen testproblemen

In this paper, problems of sequential decision theory are taken into consideration by extending the definition of the BAYES rule and treating BAYES rules. This generalisation is quite useful for practice. In many cases only BAYES rules can be calculated. The conditions under which such sequential decision procedures exist are demonstrated, as well as how to construct them on a scheme of backward induction resulting in the conclusion that the existence of BAYES rules needs essentially weaker assumptions than the existence of BAYES rules.Futhermore, methods are searched to simplify the construction of optimal stopping rules. Some illustrative examples are given.     

Bayesian sequential methods for choosing the best multinomial cell: some simulation results

Suboptimal Bayesian sequential methods for choosing the best (i.e. largest probability) multinomial cell are considered and their performance is studied using Monte Carlo simulation. Performance characteristics, such as the probability of correct selection and some other associated with the sample size distribution, are evaluated assuming a maximum sample size. Single observation sequential rules as well as rules, where groups of observations are taken, and fixed sample size rules are discussed.     

An Application of Operational-Subjective Statistical Methods to Rational Expectations

This article presents a sequential scoring analysis of six econometric forecast distributions for the main components of the annual U.S. gross national product (GNP) accounts—nominal GNP, real GNP, and the implicit price deflator. Analysis of sequential forecasts is presented in terms of proper scoring rules. Computations relevant to the calibration and refinement properties of the forecast distributions are discussed. Annual data are studied for the period 1952–1982. The six forecast distributions are distinguished by the different stances they entail with respect to a subjectivist characterization of the rational-expectations hypothesis.     

Asymptotic properties of randomly indexed sequences of random variables

Conditions are given for a randomly indexed sequence of random variables to converge weakly. The key concept employed is the so-called generalized Anscombe condition. The results give a method of determining sequential stopping rules, which have the required accuracy of estimation of an unknown parameter in the case when the observations are not necessarily independent and identically distributed.     

ON EMPIRICAL BAYES STOPPING TIMES

We consider the empirical Bayes decision theory where the component problems are the optimal fixed sample size decision problem and a sequential decision problem. With these components, an empirical Bayes decision procedure selects both a stopping rule function and a terminal decision rule function. Empirical Bayes stopping rules are constructed for each case and the asymptotic behaviours are investigated.     

A global stopping rule for recursive density estimators

In this paper asymptotic sequential fixed-width confidence bounds for an unknown density on the real line, based on integrated squared error, are studied. Using a sequence of Wolverton-Wagner kernel estimators, two classes of stopping rules are established.By the same approach, analogous results can be provided for other types of recursive density estimators.     

Rule generation for categorical time series with Markov assumptions

Several procedures of sequential pattern analysis are designed to detect frequently occurring patterns in a single categorical time series (episode mining). Based on these frequent patterns, rules are generated and evaluated, for example, in terms of their confidence. The confidence value is commonly interpreted as an estimate of a conditional probability, so some kind of stochastic model has to be assumed. The model is identified as a variable length Markov model. With this assumption, the usual confidences are maximum likelihood estimates of the transition probabilities of the Markov model. We discuss possibilities of how to efficiently fit an appropriate model to the data. Based on this model, rules are formulated. It is demonstrated that this new approach generates noticeably less and more reliable rules.     

Fixed size confidence regions for parameters of threshold AR(1) models

For parameters of single and multiple threshold autoregressive models of order one, sequential procedures are proposed for constructing fixed size confidence ellipsoids. Sequential procedures are also proposed for constructing fixed proportional accuracy confidence ellipsoids and fixed width confidence intervals for linear combination of parameters. The confidence ellipsoids and intervals are shown to be asymptotically consistent and the associated stopping rules are shown to be asymptotically efficient as the size/width of the region becomes small.     

Efficient non-parametric sequential procedures for comparing the rates of change of two treatments

Consider a longitudinal experiment where subjects are allocated to one of two treatment arms and are subjected to repeated measurements over time. Two non-parametric group sequential procedures, based on the Wilcoxon rank sum test and fitted with asymptotically efficient allocation rules, are derived to test the equality of the rates of change over time of the two treatments, when the distribution of responses is unknown. The procedures are designed to allow for early stopping to reject the null hypothesis while allocating less subjects to the inferior treatment. Simulations – based on the normal, the logistic and the exponential distributions – showed that the proposed allocation rules substantially reduce allocations to the inferior treatment, but at the expense of a relatively small increase in the total sample size and a moderate decrease in power as compared to the pairwise allocation rule.     

Local information and the design of sequential hypothesis tests

Various results on sequential hypotheses testing are reviewed. Optimal stopping rules are related to a local measure of statistical information. In some cases, local information can be approximated by L-numbers discovered by Lorden, and simple rules based on these approximations are asymptotically optimal to better order than the cost for a single observation.     

Sequential classification on partially ordered sets

Summary. A general theorem on the asymptotically optimal sequential selection of experiments is presented and applied to a Bayesian classification problem when the parameter space is a finite partially ordered set. The main results include establishing conditions under which the posterior probability of the true state converges to 1 almost surely and determining optimal rates of convergence. Properties of a class of experiment selection rules are explored.     

Utilizing auxiliary variables in sequential ratio and regression estimation schemes

Ratio and regression estimators for a mean are considered in conjunction with certain sequential sampling schemes. An auxiliary variable is assumed present and both fixed-cost and fixed- width confidence interval stopping rules are investigated. The asymptotic distributions of the estimators are derived as well as optimal probabilities pertinent to the schemes. Comparisons are made with results of certain double sampling procedures. Estimation of the ratio of two means is also considered and the results of a Monte Carlo simulation are included.     

Development of Adaptive Group Sequential Procedure for Changing Sample Size

In this study, we propose a group sequential procedure that allows the change of necessary sample size at intermediary stage in sequential test. In the procedure, we formulate the conditional power to judge the necessity of the change of sample size in decision rules. Furthermore, we present an integral formula of the power of the test and show how to change the necessary sample size by using the power of the test. In simulation studies, we investigate the characteristics of the change of sample size and the pattern of decision across all stages based on generated normal random numbers.     

On Comparison of Stopping Times in Sequential Procedures for Exponential Families of Stochastic Processes

For curved ( k + 1), k -exponential families of stochastic processes a natural and often studied sequential procedure is to stop observation when a linear combination of the coordinates of the canonical process crosses a prescribed level. For such procedures the model is, approximately or exactly, a non-curved exponential family. Subfamilies of these stopping rules defined by having the same Fisher (expected) information are considered. Within a subfamily the Bartlett correction for a point hypothesis is also constant. Methods for comparing the durations of the sampling periods for the stopping rules in such a subfamily are discussed. It turns out that some stopping times tend to be smaller than others. For exponential families of diffusions and of counting processes the probability that one such stopping time is smaller than another can be given explicity. More generally, an Edgeworth expansion of this probability is given     

Improving sample size recalculation in adaptive clinical trials by resampling

Sample size calculations in clinical trials need to be based on profound parameter assumptions. Wrong parameter choices may lead to too small or too high sample sizes and can have severe ethical and economical consequences. Adaptive group sequential study designs are one solution to deal with planning uncertainties. Here, the sample size can be updated during an ongoing trial based on the observed interim effect. However, the observed interim effect is a random variable and thus does not necessarily correspond to the true effect. One way of dealing with the uncertainty related to this random variable is to include resampling elements in the recalculation strategy. In this paper, we focus on clinical trials with a normally distributed endpoint. We consider resampling of the observed interim test statistic and apply this principle to several established sample size recalculation approaches. The resulting recalculation rules are smoother than the original ones and thus the variability in sample size is lower. In particular, we found that some resampling approaches mimic a group sequential design. In general, incorporating resampling of the interim test statistic in existing sample size recalculation rules results in a substantial performance improvement with respect to a recently published conditional performance score.     

The use of covariate information in the sequential analysis of dichotomous response experiments

A method is presented for the sequential analysis of experiments involving two treatments to which response is dichotomous. Composite hypotheses about the difference in success probabilities are tested, and covariate information is utilized in the analysis. The method is based upon a generalization of Bartlett’s (1946) procedure for using the maximum likelihood estimate of a nuisance parameter in a Sequential Probability Ratio Test (SPRT). Treatment assignment rules studied include pure randomization, randomized blocks, and an adaptive rule which tends to assign the superior treatment to the majority of subjects. It is shown that the use of covariate information can result in important reductions in the expected sample size for specified error probabilities, and that the use of covariate information is essential for the elimination of bias when adaptive assignment rules are employed. Designs of the type presented are easily generated, as the termination criterion is the same as for a Wald SPRT of simple hypotheses.     

The comparison of designs for sequential clinical trials with covariate information

Summary. The paper develops methods for the comparison of randomized rules of the biased coin type for the sequential allocation of treatments in a clinical trial. One important characteristic is the loss , which measures the increase in the variance of parameter estimates due to the imbalance caused by randomization. The other important characteristic is the selection bias measuring the probability of correctly guessing which treatment is to be allocated next. The combination of these two measures leads to the elucidation of admissible designs. Simulations provide clear plots of the behaviour of the designs and make it possible to distinguish good designs from those which are less good.     

A class of simple approximate sequential tests for adaptive comparison of two treatments

We consider the problem of sequentially deciding which of two treatments is superior, A class of simple approximate sequential tests is proposed. These have the probabilities of correct selection approximately independent of the sampling rule and depending on unknown parameters only through the function of interest, such as the difference or ratio of mean responses. The tests are obtained by using a normal approximation, and this is employed to derive approximate expressions for the probabilities of correct selection and the expected sample sizes. A class of data-dependent sampling rules is proposed for minimizing any weighted average of the expected sample sizes on the two treatments, with the weights being allowed to depend on unknown parameters. The tests are studied in the particular cases of exponentially.     

On a sequential subset selection procedure for the least probable multinomial cell

This paper studies a sequential procedure R for selecting a random size subset that contains the multinomial cell which has the smallest cell probability. The stopping rule of the proposed procedure R is the composite of the stopping rules of curtailed sampling, inverse sampling, and the Ramey-Alam sampling. A reslut on the worst configuration is shown and it is employed in computing the procedure parameters that guarantee certain probability requirements. Tables of these procedure parameters, the corresponding probability of correct selection, the expected sample size, and the expected subset size are given for comparison purpose.     

Sequential sampling without recall from a dirichlet process

The problem of sequential sampling without recall from a Dirichlet process is studied. An individual observes a sequential sample p₁, p₂, … . At each stage the observer can either accept the current value p_i or continue sampling. The total cost to the observer is the current value plus a cost for each observation taken after the first. A concrete context for the problem is provided by a shopper who must buy a particular item and can elicit price quotations sequentially but must pay for each quotation. Qualitative properties of optimal search rules are derived which establish that the problem is well behaved. In particular, the reservation-price property is shown to hold. The results extend those in Christensen (1986).