Optimal,nearly optimal and robust procedures for the exponential distribution with relative linear exponential loss

In this paper, the problem of sequentially estimating the mean of the exponential distribution with relative linear exponential loss and fixed cost for each observation is considered within the Bayesian framework. An optimal procedure with a deterministic stopping rule is derived. Since the corresponding value of the optimal deterministic stopping rule cannot be obtained directly, an approximate optimal deterministic stopping rule and an asymptotically pointwise optimal rule are proposed. In addition, we propose a robust procedure with a deterministic stopping rule, which does not depend on the parameters of the prior distribution. All of the proposed procedures are shown to be asymptotically optimal. Some numerical studies are conducted to investigate the performances of the proposed procedures. A real data set is provided to illustrate the use of the proposed procedures.     

Approximating distributions of random functionals of Ferguson-Dirichlet priors

We explore the possibility of approximating the Ferguson-Dirichlet prior and the distributions of its random functionals through the simulation of random probability measures. The proposed procedure is based on the constructive definition illustrated in Sethuraman (1994) in conjunction with the use of a random stopping rule. This allows us to set in advance the closeness to the distributions of interest. The distribution of the stopping rule is derived, and the practicability of the simulating procedure is discussed. Sufficient conditions for convergence of random functionals are provided. The numerical applications provided just sketch the idea of the variety of nonparametric procedures that can be easily and safely implemented in a Bayesian setting.     

Pivotal Quantities Based on Sequential Data: A Bootstrap Approach

A bootstrap algorithm is provided for obtaining a confidence interval for the mean of a probability distribution when sequential data are considered. For this kind of data the empirical distribution can be biased but its bias is bounded by the coefficient of variation of the stopping rule associated with the sequential procedure. When using this distribution for resampling the validity of the bootstrap approach is established by means of a series expansion of the corresponding pivotal quantity. A simulation study is carried out using Wang and Tsiatis type tests and considering the normal and exponential distributions to generate the data. This study confirms that for moderate coefficients of variation of the stopping rule, the bootstrap method allows adequate confidence intervals for the parameters to be obtained, whichever is the distribution of data.     

The risks for usual sequential estimates and stopping times of multivariate normal mean for conjugate distribution

The exact formulas of optimal stopping times for usual problems are often difficult to derive. Biekej and Yahav (1965) had provided the large sample approximation known as the asymptotically pointwise optimal (A. P.O.) rule. In Nagao (1997a.b). he has derived the asymptotic formulas for Bayes stopping times for the problems of the mean of a multivariate normal distribution when a covariance matrix is completely unknown and has some structure, respectively. This paper gives the risks for estimate and stopping times which we use in common for some problems. From this result, we find that its increasing amount shows the deficiency of estimate and stopping usually used from the view of the Bayes risk.     

Monitoring parameter changes for random coefficient autoregressive models

In this paper, we develop a monitoring procedure for an early detection of parameter changes in random coefficient autoregressive models. It is shown that the stopping rule signaling a parameter change satisfies the desired asymptotic property as seen in Lee, Lee, and Na (submitted for publication). Simulation results are provided for illustration.     

Optimal designs for both model discrimination and parameter estimation

The KL-optimality criterion has been recently proposed to discriminate between any two statistical models. However, designs which are optimal for model discrimination may be inadequate for parameter estimation. In this paper, the DKL-optimality criterion is proposed which is useful for the dual problem of model discrimination and parameter estimation. An equivalence theorem and a stopping rule for the corresponding iterative algorithms are provided. A pharmacokinetics application and a bioassay example are given to show the good properties of a DKL-optimum design.     

On empirical bayes sequential estimation

We study the empirical Bayes approach to the sequential estimation problem. An empirical Bayes sequential decision procedure, which consists of a stopping rule and a terminal decision rule, is constructed for use in the component. Asymptotic behaviors of the empirical Bayes risk and the empirical Bayes stopping times are investigated as the number of components increase.     

Sequential shrinkage estimation of independent normal means with unkown variances

The Paper considers estimation of the p(> 3)-variate normal mean when the variance-covariance matrix is diagonal with unknown diagonal elements. A class of James-Stein estimators is developed, and is compared with the sample mean under an empirical minimax stopping rule. Asymptotic risk expansions are provided for both the sequential sample mean and the sequential James-Stein estimators. It is shown that the James-Stein estimators dominate the sample mean in a certain asymptotic sense.     

A two‐stage phase II clinical trial design with nested criteria for early stopping and efficacy

We propose a two‐stage design for a single arm clinical trial with an early stopping rule for futility. This design employs different endpoints to assess early stopping and efficacy. The early stopping rule is based on a criteria determined more quickly than that for efficacy. These separate criteria are also nested in the sense that efficacy is a special case of, but usually not identical to, the early stopping endpoint. The design readily allows for planning in terms of statistical significance, power, expected sample size, and expected duration. This method is illustrated with a phase II design comparing rates of disease progression in elderly patients treated for lung cancer to rates found using a historical control. In this example, the early stopping rule is based on the number of patients who exhibit progression‐free survival (PFS) at 2 months post treatment follow‐up. Efficacy is judged by the number of patients who have PFS at 6 months. We demonstrate our design has expected sample size and power comparable with the Simon two‐stage design but exhibits shorter expected duration under a range of useful parameter values.     

Early stopping in seamless phase I/II clinical trials

In recent years, seamless phase I/II clinical trials have drawn much attention, as they consider both toxicity and efficacy endpoints in finding an optimal dose (OD). Engaging an appropriate number of patients in a trial is a challenging task. This paper attempts a dynamic stopping rule to save resources in phase I/II trials. That is, the stopping rule aims to save patients from unnecessary toxic or subtherapeutic doses. We allow a trial to stop early when widths of the confidence intervals for the dose-response parameters become narrower or when the sample size is equal to a predefined size, whichever comes first. The simulation study of dose-response scenarios in various settings demonstrates that the proposed stopping rule can engage an appropriate number of patients. Therefore, we suggest its use in clinical trials.     

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 Bayesian sequential design with binary outcome

Several researchers have proposed solutions to control type I error rate in sequential designs. The use of Bayesian sequential design becomes more common; however, these designs are subject to inflation of the type I error rate. We propose a Bayesian sequential design for binary outcome using an alpha‐spending function to control the overall type I error rate. Algorithms are presented for calculating critical values and power for the proposed designs. We also propose a new stopping rule for futility. Sensitivity analysis is implemented for assessing the effects of varying the parameters of the prior distribution and maximum total sample size on critical values. Alpha‐spending functions are compared using power and actual sample size through simulations. Further simulations show that, when total sample size is fixed, the proposed design has greater power than the traditional Bayesian sequential design, which sets equal stopping bounds at all interim analyses. We also find that the proposed design with the new stopping for futility rule results in greater power and can stop earlier with a smaller actual sample size, compared with the traditional stopping rule for futility when all other conditions are held constant. Finally, we apply the proposed method to a real data set and compare the results with traditional designs.     

A new procedure for steepest ascent

The procedure of steepest ascent consists of performing a sequence of sets of trials. Each set of trials is obtained as a result of proceeding sequentially along the path of maximum increase in response. Until now there has been no formal stopping rule, When response values are subject to random error, the decision to stop can be premature due to a “false” drop in the observed response.

A new stopping rule procedure for steepest ascent is intro-duced that takes into account the random error variation in response values. The new procedure protects against taking too many observations when the true mean response is decreasing, it also protects against stopping. prematurely when the true mean response is increasing, A numerical example is given which illus-trates the method.     

Bayes procedures for detecting a shift in the probability of success in a series of Bernoulli trials

The determination of a stopping rule for the detection of the time of an increase in the success probability of a sequence of independent Bernoulli trials is discussed. Both success probabilities are assumed unknown. A Bayesian approach is applied; the distribution of the location of the shift in the success probability is assumed geometric and the success probabilities are assumed to have known joint prior distribution. The costs involved are penalties for late or early stoppings. The nature of the optimal dynamic programming solution is discussed and a procedure for obtaining a suboptimal stopping rule is determined. The results indicate that the detection procedure is quite effective.     

A new search procedure of steepest ascent in response surface exploration

The method of steepest ascent direction has been widely accepted for process optimization in the applications of response surface methodology (RSM). The procedure of steepest ascent direction is performed on experiments run along the gradient of a fitted linear model. Therefore, the RSM practitioner needs to decide a suitable stopping rule such that the optimum point estimate in the search direction can be determined. However, the details of how to deflect and then halt a search in the steepest ascent direction are not thoroughly described in the literature. In common practice, it is convenient to use the simple stopping rules after one to three response deteriorations in a row after a series of fitted linear models used for exploration. In the literature, there are two formal stopping rules proposed, that is, Myers and Khuri's [A new procedure for steepest ascent, Comm. Statist. Theory Methods A 8(14) (1979), pp. 1359–1376] stopping rule and del Castillo's [Stopping rules for steepest ascent in experimental optimization, Comm. Statist. Simul. Comput. 26(4) (1997), pp. 1599–1615] stopping rule. This paper develops a new procedure for determining how to adjust and then when to stop a steepest ascent search in response surface exploration. This proposal wishes to provide the RSM practitioner with a clear-cut and easy-to-implement procedure that can attain the optimum mean response more accurately than the existing procedures. Through the study of simulation optimization, it shows that the average optimum point and response returned by using the new search procedure are considerably improved when compared with two existing stopping rules. The number of experimental trials required for convergence is greatly reduced as well.     

Another look at subset selection using linear least squares

The QR-factorization provides a set of orthogonal variables which has advantages over other orthogonal representations, such as principal components and the singular-value decomposition, in selecting subsets of regression variables by least squares methods. Stopping rules, in particular, are easily understood. A new stopping rule is derived for prediction. This is derived by approximately minimizing the mean squared error in estimating the squared error of prediction. A clear distinction is made between the kind of stopping rule which is relevant when the objective is prediction, and when the objective is asymptotic consistency. Progress with reducing the bias due to the model selection procedure is briefly summarized.     

Sequential estimation for covariate-adjusted response-adaptive designs

In clinical trials, a covariate-adjusted response-adaptive (CARA) design allows a subject newly entering a trial a better chance of being allocated to a superior treatment regimen based on cumulative information from previous subjects, and adjusts the allocation according to individual covariate information. Since this design allocates subjects sequentially, it is natural to apply a sequential method for estimating the treatment effect in order to make the data analysis more efficient. In this paper, we study the sequential estimation of treatment effect for a general CARA design. A stopping criterion is proposed such that the estimates satisfy a prescribed precision when the sampling is stopped. The properties of estimates and stopping time are obtained under the proposed stopping rule. In addition, we show that the asymptotic properties of the allocation function, under the proposed stopping rule, are the same as those obtained in the non-sequential/fixed sample size counterpart. We then illustrate the performance of the proposed procedure with some simulation results using logistic models. The properties, such as the coverage probability of treatment effect, correct allocation proportion and average sample size, for diverse combinations of initial sample sizes and tuning parameters in the utility function are discussed.     

Minimax optimality of the Shiryayev–Roberts change-point detection rule

The result of Pollak [1985. Optimal detection of a change in distribution. Ann. Statist. 13, 206–227] proving the asymptotic optimality in sequential change-point detection of a suitable Shirayayev–Roberts stopping rule up to terms that vanish in the limit is generalized from the case of two completely specified distributions to that of a composite alternative hypothesis in a multidimensional exponential family. An explicit asymptotic lower bound on the expected Kullback–Leibler information required to detect a change-point is derived and is shown to be attained by a Shirayayev–Roberts stopping rule.     

An adaptive rule for stopping a discovery process under time censorship

Consider a finite population of large but unknown size of hidden objects. Consider searching for these objects for a period of time, at a certain cost, and receiving a reward depending on the sizes of the objects found. Suppose that the size and discovery time of the objects both have unknown distributions, but the conditional distribution of time given size is exponential with an unknown non-negative and non-decreasing function of the size as failure rate. The goal is to find an optimal way to stop the discovery process. Assuming that the above parameters are known, an optimal stopping time is derived and its asymptotic properties are studied. Then, an adaptive rule based on order restricted estimates of the distributions from truncated data is presented. This adaptive rule is shown to perform nearly as well as the optimal stopping time for large population size.     

Bayesian sequential analysis for multiple-arm clinical trials

Use of full Bayesian decision-theoretic approaches to obtain optimal stopping rules for clinical trial designs typically requires the use of Backward Induction. However, the implementation of Backward Induction, apart from simple trial designs, is generally impossible due to analytical and computational difficulties. In this paper we present a numerical approximation of Backward Induction in a multiple-arm clinical trial design comparing k experimental treatments with a standard treatment where patient response is binary. We propose a novel stopping rule, denoted by τ ^p , as an approximation of the optimal stopping rule, using the optimal stopping rule of a single-arm clinical trial obtained by Backward Induction. We then present an example of a double-arm (k=2) clinical trial where we use a simulation-based algorithm together with τ ^p to estimate the expected utility of continuing and compare our estimates with exact values obtained by an implementation of Backward Induction. For trials with more than two treatment arms, we evaluate τ ^p by studying its operating characteristics in a three-arm trial example. Results from these examples show that our approximate trial design has attractive properties and hence offers a relevant solution to the problem posed by Backward Induction.