Zur bestimmung von bayes-lösungen bei messender prüfung

A model of sampling inspection by variables is presented, where the loss function and the a priori distribution of the unknown defective fraction are assumed to be known. The existence of a BAYES rule in the class of relevant decision functions (sampling plans) is proved. Furthermore, it is shown that under certain conditions the decision functions "accept without sampling" or "reject without sampling" are BAYESian decision rules. Some numerical examples are presented to illustrate the theory developed in the paper.     

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.     

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 Some Aspects of Unbiased Estimation of Parameters in Quasi-Binomial Distributions

In this article, an attempt has been made to settle the question of existence of unbiased estimator of the key parameter p of the quasi-binomial distributions of Type I (QBD I) and of Type II (QBD II), with/without any knowledge of the other parameter φ appearing in the expressions for probability functions of the QBD's. This is studied with reference to a single observation, a random sample of finite size m as also with samples drawn by suitably defined sequential sampling rules.     

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.     

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.     

Über das asymptotische verhalten von bayesschen und maximum-likelihood-schätzungen

The paper deals with the asymptotic equivalence of the maximum-likelihood-estimator with sums of independent and identically distributed random variables for a family of distribution functions, the density of which may vanish depending on the parameter. Further the asymptotic equivalence of the maximum-likelihood-estimator with BAYES estimators is shown.     

Sequential estimation of the variance of an unknown distribution

Sequential fixed-width and risk-efficient estimation of the variance of an unspecified distribution is considered. The second-order asymptotic properties of the sequential rules are studied. Extensive simulation studies are carried out in order to study the small sample behavior of the sequential rules for some frequently used distributions.     

On the correspondence between frequentist and Bayesian tests

Modern theory for statistical hypothesis testing can broadly be classified as Bayesian or frequentist. Unfortunately, one can reach divergent conclusions if Bayesian and frequentist approaches are applied in parallel to analyze the same data set. This is a serious impasse since there is a lack of consensus on when to use one approach in detriment of the other. However, this conflict can be resolved. The present paper shows the existence of a perfect equivalence between Bayesian and frequentist methods for testing. Hence, Bayesian and frequentist decision rules can always be calibrated, in both directions, in order to present concordant results.     

An alternative decision rule for quality control in sequential sampling plans

The conventional rule for acceptance sampling based on sequential samples is based on the rationale of hypothesis testing developed by Wald (1947). This type of decision rule tests the hypothesis of P=p₁ versus P=p₂ as a proxy for determining whether P>d or P<d with P₁<d<p₂. It requires a zone of indifference between the rejection and acceptance levels p₂ and p₁. In this note, we propose an alternative rule for making the decision based on the confidence level of a one-sided Bayesian interval estimate of the parameter. This method results in direct determination of whether the proportion of defects P in the population is greater or less than a prespecified level d, rather than test two points as proxy for the decision. We present a numerical illustration of the rule and an example of determining rejection and acceptance numbers. Ue also compare the results with two conventional rules.     

A sequential sampling plan for the inverse gaussian mean

The inverse gaussian distribution is a flexible model which has been extensively applied in the theory of generalized linear models and accelerated life testing where early failure times predominate. More recently it has received attention in areas such as quality control, and as an underlying model that provides an alternative to the analysis of variance. In reliability testing and acceptance sampling data acquisition is often in the face of scarce resources and may be both costly and time-consuming. In such settings it is desirable to reach a statistically sound decision as quickly as possible. Based on sequential probability ratio tests (SPRT), sequential sampling plans provide one method of arriving at a timely, statistically based decision. A sequential sampling plan for the inverse gaussian process mean when the value of the shape parameter of the density is known is presented in this paper.     

Incorporating scientific,ethical and economic considerations into the design of clinical trials in the pharmaceutical industry: a sequential approach

A flexible sequential approach to the design of clinical trials is discussed herein. This approach is based on a “confidence sequence” viewpoint instead of the rigid stopping and terminal decision rules in conventional sequential testing theory. By using an appropriate confidence sequence, one can always ensure a prescribed degree of scientific rigor (confidence) in establishing the drug to be effective. Moreover, one also has the option of terminating the trial early when there is already enough statistical evidence for concluding that the drug is effective, or when the drug shows uniorseen harmful effects, or when the data predict that there is little chance of arriving at a definitive conclusion in favor of the drug by the scheduled end of the trial. We discuss how these and other ethical and economic considerations can be readily incorporated into the stopping criteria of the trial.     

On hsu's model in regression analysis

The paper examplifies with Hsu’s model a general pattern as how to derive results of variance component estimation from well known results on mean estimation, as far as linear model theory is concerned. This ’ dispersion-mean-correspondence‘provides new and short proofs for various theorems from the literature, concerning unbiased invariant quadratic estimators with minimum BAYES risk or minimum variance. For pure variance component models, unbiased non-negative quadratic estimability is characterized in terms of the design matrices.     

Statistical process control and model monitoring

This paper is concerned with model monitoring and quality control schemes, which are founded on a decision theoretic formulation. After identifying unacceptable weaknesses associated with Wald, sequential probability ratio test (SPRT) and Cuscore monitors, the Bayes decision monitor is developed. In particular, the paper focuses on what is termed a 'popular decision scheme' (PDS) for which the monitoring run loss functions are specified simply in terms of two indiff erence qualities. For most applications, the PDS results in forward cumulative sum tests of functions of the observations. For many exponential family applications, the PDS is equivalent to well-used SPRTs and Cusums. In particular, a neat interpretation of V-mask cusum chart settings is derived when simultaneously running two symmetric PDSs. However, apart from providing a decision theoretic basis for monitoring, sensible procedures occur in applications for which SPRTs and Cuscores are particularly unsatisfactory. Average run lengths (ARLs) are given for two special cases, and the inadequacy of the Wald and similar ARL approximations is revealed. Generalizations and applications to normal and dynamic linear models are discussed. The paper concludes by deriving conditions under which sequences of forward and backward sequential or Cusum chart tests are equivalent.     

Sensitivity of Bayesian sampling inspection schemes to a non-normal prior distribution

summary In this paper we derive the predictive density function of a future observation under the assumption of Edgeworth-type non-normal prior distribution for the unknown mean of a normal population. Fixed size single sample and sequential sampling inspection plans, in a decisive prediction framework, are examined for their sensitivity to departures from normality of the prior distribution. Numerical illustrations indicate that the decision to market the remaining items of a given lot for a fixed size plan may be sensitive to the presence of skewness or kurtosis in the prior distribution. However, Bayes'decision based on the sequential plan may not change though expected gains may change with variation in the non-normality of the prior distribution.     

Some Geometric Methods for Constructing Decision Criteria Based On Two-Dimensional Parameters

This paper reviews two types of geometric methods proposed in recent years for defining statistical decision rules based on 2-dimensional parameters that characterize treatment effect in a medical setting. A common example is that of making decisions, such as comparing treatments or selecting a best dose, based on both the probability of efficacy and the probability toxicity. In most applications, the 2-dimensional parameter is defined in terms of a model parameter of higher dimension including effects of treatment and possibly covariates. Each method uses a geometric construct in the 2-dimensional parameter space based on a set of elicited parameter pairs as a basis for defining decision rules. The first construct is a family of contours that partitions the parameter space, with the contours constructed so that all parameter pairs on a given contour are equally desirable. The partition is used to define statistical decision rules that discriminate between parameter pairs in term of their desirabilities. The second construct is a convex 2-dimensional set of desirable parameter pairs, with decisions based on posterior probabilities of this set for given combinations of treatments and covariates under a Bayesian formulation. A general framework for all of these methods is provided, and each method is illustrated by one or more applications.     

Conditional probability of correct selection under the continuum partition with applications

The problem of selecting the best population from among a finite number of populations in the presence of uncertainty is a problem one faces in many scientific investigations, and has been studied extensively, Many selection procedures have been derived for different selection goals. However, most of these selection procedures, being frequentist in nature, don't tell how to incorporate the information in a particular sample to give a data-dependent measure of correct selection achieved for this particular sample. They often assign the same decision and probability of correct selection for two different sample values, one of which actually seems intuitively much more conclusive than the other. The methodology of conditional inference offers an approach which achieves both frequentist interpret ability and a data-dependent measure of conclusiveness. By partitioning the sample space into a family of subsets, the achieved probability of correct selection is computed by conditioning on which subset the sample falls in. In this paper, the partition considered is the so called continuum partition, while the selection rules are both the fixed-size and random-size subset selection rules. Under the distributional assumption of being monotone likelihood ratio, results on least favourable configuration and alpha-correct selection are established. These re-sults are not only useful in themselves, but also are used to design a new sequential procedure with elimination for selecting the best of k Binomial populations. Comparisons between this new procedure and some other se-quential selection procedures with regard to total expected sample size and some risk functions are carried out by simulations.     

An application of a multi‐stage strategy involving confidence intervals to evaluate whether a response rate is favourable or not

In early clinical development of new medicines, a single‐arm study with a limited number of patients is often used to provide a preliminary assessment of a response rate. A multi‐stage design may be indicated, especially when the first stage should only include very few patients so as to enable rapid identification of an ineffective drug. We used decision rules based on several types of nominal confidence intervals to evaluate a three‐stage design for a study that includes at most 30 patients. For each decision rule, we used exact binomial calculations to determine the probability of continuing to further stages as well as to evaluate Type I and Type II error rates. Examples are provided to illustrate the methods for evaluating alternative decision rules and to provide guidance on how to extend the methods to situations with modifications to the number of stages or number of patients per stage in the study design. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

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.