Variable sample size method for equality constrained optimization problems

An equality constrained optimization problem with a deterministic objective function and constraints in the form of mathematical expectation is considered. The constraints are transformed into the Sample Average Approximation form resulting in deterministic problem. A method which combines a variable sample size procedure with line search is applied to a penalty reformulation. The method generates a sequence that converges towards first-order critical points. The final stage of the optimization procedure employs the full sample and the SAA problem is eventually solved with significantly smaller cost. Preliminary numerical results show that the proposed method can produce significant savings compared to SAA method and some heuristic sample update counterparts while generating a solution of the same quality.     

The empirical bayes rules with floating optimal sample size for exponential conditional distributions

Summary We consider the empirical Bayes solution in such a situation where the sample size is successively determined by a rule which includes the Bayes risks and the observation costs. The empirical Bayes floating optimal sample size depends on current as well as on previous information assumed to be collected from earlier performances of similar decisions. The sampling is done from an exponential conditional distribution, with a single parameter. The proofs, which show the asymptotic optimality of the empirical Bayes solution, are presented for a hypotheses-testing problem. A straight generalization to a multiple decision problem is also given.     

Addressing the sample size problem in behavioural operational research: simulating the newsvendor problem

Laboratory-based experimental studies with human participants are beneficial for testing hypotheses in behavioural operational research. However, such experiments are not without their problems. One specific problem is obtaining a sufficient sample size, not only in terms of the number of participants but also the time they are willing to devote to an experiment. In this paper, we explore how agent-based simulation (ABS) can be used to address the sample size problem and demonstrate the approach in the newsvendor setting. The decision-making strategies of a small sample of individual decision-makers are determined through laboratory experiments. The interactions of these suppliers and retailers are then simulated using an ABS to generate a large sample set of decisions. With only a small number of participants, we demonstrate that it is possible to produce similar results to previous experimental studies that involved much larger sample sizes. We conclude that ABS provides the potential to extend the scope of experimental research in behavioural operational research.     

Structure of sequential tests minimizing an expected sample size

Summary The testing problem is to decide on the basis of repeated independent observations which of the probability densitiesf andg is true. Given upper bounds on the probabilities of error, the object is to minimize the expected sample size if the densityp is true (allowed to differ fromf andg). A characterization of the structure of optimal tests is obtained which is particularly informative in the case wheref,g, andp belong to a Koopman-Darmois family. Ifp=f org, then the optimal tests are sequential probability ratio tests (SPRT's) and a new proof of the well-known optimality property of these tests is obtained as a corollary.Research supported by National Science Foundation grant MCS 76-08489     

Stratification and sample size of data sources for agricultural mathematical programming models

A comparison is made between the variance of the estimator of the total of a variable obtained from both a simple and a stratified random sampling, in which the sample sizes of some strata are equal to the stratum population size.It is shown that in this case, the advantage of the stratified sample could depend on the sample size. The paper presents inequalities that determine, as a function of the sample size, when the variance of the estimator obtained with simple sampling is lower than the variance obtained with the stratified sampling. The results give insight in order to prevent overstratification.     

On sample size control in sample average approximations for solving smooth stochastic programs

We consider smooth stochastic programs and develop a discrete-time optimal-control problem for adaptively selecting sample sizes in a class of algorithms based on variable sample average approximations (VSAA). The control problem aims to minimize the expected computational cost to obtain a near-optimal solution of a stochastic program and is solved approximately using dynamic programming. The optimal-control problem depends on unknown parameters such as rate of convergence, computational cost per iteration, and sampling error. Hence, we implement the approach within a receding-horizon framework where parameters are estimated and the optimal-control problem is solved repeatedly during the calculations of a VSAA algorithm. The resulting sample-size selection policy consistently produces near-optimal solutions in short computing times as compared to other plausible policies in several numerical examples.     

Adaptive sample size and importance sampling in estimation-based local search for the probabilistic traveling salesman problem

The probabilistic traveling salesman problem is a paradigmatic example of a stochastic combinatorial optimization problem. For this problem, recently an estimation-based local search algorithm using delta evaluation has been proposed. In this paper, we adopt two well-known variance reduction procedures in the estimation-based local search algorithm: the first is an adaptive sampling procedure that selects the appropriate size of the sample to be used in Monte Carlo evaluation; the second is a procedure that adopts importance sampling to reduce the variance involved in the cost estimation. We investigate several possible strategies for applying these procedures to the given problem and we identify the most effective one. Experimental results show that a particular heuristic customization of the two procedures increases significantly the effectiveness of the estimation-based local search.     

Iterated probability distributions and extremes with random sample size

In this paper the possible nondegenerated limit distributions for the n-fold mapping of a given probability distribution are considered. If the mapping used for the iteration procedure is a probability generating function of a positive integer-valued random variable then the results can be applied to the max-stability of distributions of random variables with random sample size.     

On the location parameter confidence intervals based on a random size sample from a partially known population

The problem of constructing confidence intervals of a fixed length for the location parameter based on a random size sample is considered. It is proposed to use the confidence interval $$\theta _p^* - u\sqrt p /\sigma< \theta< \theta _p^* + u\sqrt p /\sigma $$ , whereθ _p ^* is an adaptive estimator,σ ² is the Fisher information, and p^?1 is the mean of the sample size. Nonparametric bounds are given for the limit as p → 0 confidence probability. Bibliography: 5 titles.     

Likelihood and sample size: The understandings of students and their teachers

The aim of this study was to consider the match of student statistical understanding and teacher pedagogical content knowledge in relation to sample size and likelihood. Students were given two contexts within which to compare the likelihood of events for different sample sizes. Teachers were presented with one of the contexts and asked what their students would do and how they would remediate incorrect responses. The data also provided the opportunity for a detailed hierarchical analysis of students’ and teachers’ understandings. Analysis of student solutions revealed a wide range of reasoning, some of which was apparently unfamiliar to teachers.     

Asymptotic efficiency of Kolmogorov-Smirnov type symmetry tests in the case of a sample of random size

One considers Kolmogorov-Smirnov type statistics for testing the symmetry hypothesis for samples of random size. Under certain restrictions, one finds the asymptotic behavior of the probabilities of large deviations of these statistics and, with its aid, their Bahadur asymptotic efficiency.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 153, pp. 122–128, 1986.     

Power, sample size and sampling costs for clustered data

The data collected in epidemiological or clinical studies are frequently clustered. In such settings, appropriate variance adjustments must be made in order to estimate the sufficient sample size correctly. This paper works through the sample size calculations for clustered data. Importantly, our explicit variance expressions also enable us to optimize the design with respect to the number of clusters and number of subjects; the objective could be either to maximize the power or to minimize the costs with given costs on the clusters and on the individuals. In our approach, units on different levels and treatment groups can have different costs, but the members of the same cluster are assumed to belong to the same treatment group. Design considerations in the health coaching project TERVA are used as motivating examples. R-functions for carrying out the computations presented are provided.     

An integral representation of the variance of the number of correctly guessed pairs in the matching problem of a two-dimensional sample

One investigates the matching problem in a two-dimensional sample.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 136, pp. 113–120, 1984.     

Testing multivariate normality in incomplete data of small sample size

In longitudinal studies with small samples and incomplete data, multivariate normal-based models continue to be a powerful tool for analysis. This has included a broad scope of biomedical studies. Testing the assumption of multivariate normality (MVN) is critical. Although many methods are available for testing normality in complete data with large samples, a few deal with the testing in small samples. For example, Liang et al. (J. Statist. Planning and Inference 86 (2000) 129) propose a projection procedure for testing MVN for complete-data with small samples where the sample sizes may be close to the dimension. To our knowledge, no statistical methods for testing MVN in incomplete data with small samples are yet available. This article develops a test procedure in such a setting using multiple imputations and the projection test. To utilize the incomplete data structure in multiple imputation, we adopt a noniterative inverse Bayes formulae (IBF) sampling procedure instead of the iterative Gibbs sampling to generate iid samples. Simulations are performed for both complete and incomplete data when the sample size is less than the dimension. The method is illustrated with a real study on an anticancer drug.     

On unlinking of continuous statistics of normal sample

This paper contains a theorem connected with Y. V. Linnik’s conjecture on conditions of independence of a pair of normal sample statistics, which are continuous and homogeneous. These properties are significant for the problem, as it is seen from the proof.     

Middle school students' understanding of the role sample size plays in experimental probability

In this study, we examined the impact of an instructional program on sixth-grade students' understanding of experimental probability as it relates to sample size. As the number of trials in an experiment increases, the experimental probability is more likely to reflect the parent distribution; thus, smaller samples are more likely to yield unusual results. Results of this study indicate that, while typical middle school students are seemingly unaware of the relationship between experimental probability and sample size, appropriate cognitive activity focused on results of simulations of random phenomena can foster conceptual development. We witnessed growth that occurred as a result of key instructional tasks and concomitant mental activity.     

Asymptotic properties of sample quantiles from a finite population

In this paper we consider the problem of estimating quantiles of a finite population of size N on the basis of a finite sample of size n selected without replacement. We prove the asymptotic normality of the sample quantile and show that the scaled variance of the sample quantile converges to the asymptotic variance under a slight moment condition. We also consider the performance of the bootstrap in this case, and find that the usual (Efron’s) bootstrap method fails to be consistent, but a suitably modified version of the bootstrapped quantile converges to the same asymptotic distribution as the sample quantile. Consistency of the modified bootstrap variance estimate is also proved under the same moment conditions.     

Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications

Inspired by a recent work by Alexander et al. (J Bank Finance 30:583–605, 2006) which proposes a smoothing method to deal with nonsmoothness in a conditional value-at-risk problem, we consider a smoothing scheme for a general class of nonsmooth stochastic problems. Assuming that a smoothed problem is solved by a sample average approximation method, we investigate the convergence of stationary points of the smoothed sample average approximation problem as sample size increases and show that w.p.1 accumulation points of the stationary points of the approximation problem are weak stationary points of their counterparts of the true problem. Moreover, under some metric regularity conditions, we obtain an error bound on approximate stationary points. The convergence result is applied to a conditional value-at-risk problem and an inventory control problem.      

Optimization of sample size and order size in an inventory model with quality inspection and return of defective items

To ensure all products as perfect, inspection is essential, even though it is not possible to inspect all products after producing them like some special type products as plastic joint for the water pipe. In this direction, this paper develops an inventory model with lot inspection policy. With the help of lot inspection, all products need not to be verified still the retailer can decide the quality of products during inspection. If retailer founds products as imperfect quality, the products are sent back to supplier. As it is lot inspection, mis-clarification errors (Type-I error and Type-II error) are introduced to model the problem. Two possible cases are discussed for sending back products as defective lots are immediately withdrawn from the system and send back to supplier with retailer’s payment and for second case, retailer sends defective products during receiving next lot from supplier with supplier’s investment, like in food industry or in hygiene product industry. The model is solved analytically and results indicate that optimal order size and sample size are intrinsically linked and maximize the total profit. Numerical examples, graphical representations, and sensitivity analysis are given to illustrate the model. The results suggest that sending defective products maintaining the first case is the more profitable than the second case.     

Asymptotics of the necessary sample size under small error probabilities

The asymptotics of the necessary sample size is considered in testing close hypotheses when the error probabilities vanish. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part III.