Approximate optimal allocation in repeated sampling from a finite population

Samples of size n are drawn from a finite population on each of two occasions. On the first occasion a variate x is measured, and on the second a variate y. In estimating the population mean of y, the variance of the best linear unbiased combination of means for matched and unmatched samples is itself minimized, with respect to the sampling design on the second occasion, by a certain degree of matching. This optimal allocation depends on the population correlation coefficient, which previous authors have assumed known. We estimate the correlation from an initial matched sample, then an approximately optimal allocation is completed and an estimator formed which, under a bivariate normal superpopulation model, has model expected mean square error equal, apart from an error of order n^-2, to the minimum enjoyed by any linear, unbiased estimator.     

Implementing optimal allocation for sequential continuous responses with multiple treatments

In practice, it is important to find optimal allocation strategies for continuous response with multiple treatments under some optimization criteria. In this article, we focus on exponential responses. For a multivariate test of homogeneity, we obtain the optimal allocation strategies to maximize power while (1) fixing sample size and (2) fixing expected total responses. Then the doubly adaptive biased coin design [Hu, F., Zhang, L.-X., 2004. Asymptotic properties of doubly adaptive biased coin designs for multi-treatment clinical trials. The Annals of Statistics 21, 268–301] is used to implement the optimal allocation strategies. Simulation results show that the proposed procedures have advantages over complete randomization with respect to both inferential (power) and ethical standpoints on average. It is important to note that one can usually implement optimal allocation strategies numerically for other continuous responses, though it is usually not easy to get the closed form of the optimal allocation theoretically.     

Admissible two-stage designs for phase II cancer clinical trials that incorporate the expected sample size under the alternative hypothesis

two‐stage studies may be chosen optimally by minimising a single characteristic like the maximum sample size. However, given that an investigator will initially select a null treatment e?ect and the clinically relevant di?erence, it is better to choose a design that also considers the expected sample size for each of these values. The maximum sample size and the two expected sample sizes are here combined to produce an expected loss function to ?nd designs that are admissible. Given the prior odds of success and the importance of the total sample size, minimising the expected loss gives the optimal design for this situation. A novel triangular graph to represent the admissible designs helps guide the decision‐making process. The H ₀‐optimal, H ₁‐optimal, H ₀‐minimax and H ₁‐minimax designs are all particular cases of admissible designs. The commonly used H ₀‐optimal design is rarely good when allowing stopping for e?cacy. Additionally, the δ‐minimax design, which minimises the maximum expected sample size, is sometimes admissible under the loss function. However, the results can be varied and each situation will require the evaluation of all the admissible designs. Software to do this is provided. Copyright © 2012 John Wiley & Sons, Ltd.     

The Optimal Allocation Combination for the One-Sided Sequential Screening Procedure Based on the Individual Misclassification Error

In this article, the expected total costs of three kinds of quality cost functions for the one-sided sequential screening procedure based on the individual misclassification error are obtained, where the expected total cost is the sum of the expected cost of inspection, the expected cost of rejection, and the expected cost of quality. The computational formulas for three kinds of expected total costs are derived when k screening variables are allocated into r stages. The optimal allocation combination is determined based on the criterion of minimum expected total cost. At last, we give one example to illustrate the selection of the optimal allocation combination for the sequential screening procedure.     

The optimal allocation combination for the two-sided sequential screening procedure based on the individual misclassification error

In this paper, the expected total costs (ETCs) of three kinds of quality cost functions for the two-sided sequential screening procedure (SQSP) based on the individual misclassification error are obtained, where the ETC is the sum of the expected cost of inspection, the expected cost of rejection and the expected cost of quality. The general formulas for all the desired probabilities and three ETCs when k screening variables are allocated into r-stages are derived. The optimal allocation combination for each ETC is determined based on the criterion of minimum ETC. Finally, we give two examples to illustrate the selection of the optimal allocation combination for the SQSP.     

Convergence properties of the expected improvement algorithm with fixed mean and covariance functions

This paper deals with the convergence of the expected improvement algorithm, a popular global optimization algorithm based on a Gaussian process model of the function to be optimized. The first result is that under some mild hypotheses on the covariance function k of the Gaussian process, the expected improvement algorithm produces a dense sequence of evaluation points in the search domain, when the function to be optimized is in the reproducing kernel Hilbert space generated by k  . The second result states that the density property also holds for P-almost$\mathsf{P}\text{-almost}$ all continuous functions, where P is the (prior) probability distribution induced by the Gaussian process.     

Optimal U-type design for Bayesian nonparametric multiresponse prediction

This paper presents an extension of the work of Yue and Chatterjee (2010) about U-type designs for Bayesian nonparametric response prediction. We consider nonparametric Bayesian regression model with p responses. We use U-type designs with n runs, m factors and q levels for the nonparametric multiresponse prediction based on the asymptotic Bayesian criterion. A lower bound for the proposed criterion is established, and some optimal and nearly optimal designs for the illustrative models are given.     

Optimality of random allocation design for the control of accidental bias in sequential experiments

In comparing two treatments, suppose the suitable subjects arrive sequentially and must be treated at once. Known or unknown to the experimenter there may be nuisance factors systematically affecting the subjects. Accidental bias is a measure of the influence of these factors in the analysis of data. We show in this paper that the random allocation design minimizes the accidental bias among all designs that allocate n, out of 2n, subjects to each treatment and do not prefer either treatment in the assignment. When the final imbalance is allowed to be nonzero, optimal and efficient designs are given. In particular the random allocation design is shown to be very efficient in this broader setup.     

A new economic scheme for CCC charts with run rules based on average number of inspected items

New statistical techniques and procedures have been developed to control high-yield processes along with looking for process improvement opportunities and minimizing production cost. Cumulative count of conforming control chart is generally a technique for high-quality processes, when nonconforming items are rarely produced. The objective of this study is to design control chart based on cumulative count of conforming items and run rules that develops an economic model based on the average number of inspected items to design m-of-m CCC chart in order to facilitate minimum average cost per item produced. The optimal design parameters for different values of nonconforming fraction and different cost parameters in each scenario are determined. Finally, to analyze the behavior of optimal economic solutions, sensitivity analysis of the model parameters is performed.     

Identifying the determinants of foreign direct investment: a data-specific model selection approach

In this paper, we examine the potential determinants of foreign direct investment. For this purpose, we apply new exact subset selection procedures, which are based on idealized assumptions, as well as their possibly more plausible empirical counterparts to an international data set to select the optimal set of predictors. Unlike the standard model selection procedures AIC and BIC, which penalize only the number of variables included in a model, and the subset selection procedures RIC and MRIC, which consider also the total number of available candidate variables, our data-specific procedures even take the correlation structure of all candidate variables into account. Our main focus is on a new procedure, which we have designed for situations where some of the potential predictors are certain to be included in the model. For a sample of 73 developing countries, this procedure selects only four variables, namely imports, net income from abroad, gross capital formation, and GDP per capita. An important secondary finding of our study is that the data-specific procedures, which are based on extensive simulations and are therefore very time-consuming, can be approximated reasonably well by the much simpler exact methods.     

The construction of optimal designs for dose-escalation studies

Methods for the construction of A-, MV-, D- and E-optimal designs for dose-escalation studies are presented. Algebraic results proved elusive and explicit expressions for the requisite optimal designs are only given for a restricted class of traditional designs. Recourse to numerical procedures and heuristics is therefore made. Complete enumeration of all possible designs is discussed but is, as expected, highly computer intensive. Two exchange algorithms, one based on block exchanges and termed the Block Exchange Algorithm and the other a candidate-set-free algorithm based on individual exchanges and termed the Best Move Algorithm, are therefore introduced. Of these the latter is the most computationally effective. The methodology is illustrated by means of a range of carefully selected examples.     

The equivalence of the mid p-value and the expected p-value for testing equality of two balanced Binomial proportions

This paper considers the problem of testing equality between two independent binomial proportions. Hwang and Yang (Statist. Sinica 11 (2001) 807) apply the Neyman–Pearson fundamental lemma and the estimated truth approach to derive optimal procedures, named expected p-values. This p-value has been shown to be identical to the mid p-value in Lancaster (J. Amer. Statist. Assoc. (1961) 223) for the one-sided test. For the two-sided test, the paper proves the usual two-sided mid p-value is identical to the expected p-value in the balanced sample case.     

Optimal investment and premium control for insurers with ambiguity

Abstract

In this paper, we consider the optimal investment and premium control problem for insurers who worry about model ambiguity. Different from previous works, we assume that the insurer’s surplus process is described by a non-homogeneous compound Poisson model and the insurer has ambiguity on both the financial market and the insurance market. Our purpose is to find the impacts of model ambiguity on optimal policies. With the objective of maximizing the expected utility of terminal wealth, the closed-form solutions of the optimal investment and premium policies are obtained by solving HJB equations. Finally, numerical examples are also given to illustrate the results.     

D-optimal designs for combined polynomial and trigonometric regression on a partial circle

Consider the D-optimal designs for a combined polynomial and trigonometric regression on a partial circle. It is shown that the optimal design is equally supported and the structure of the optimal design depends only on the length of the design interval and the support points are analytic functions of this parameter. Moreover, the Taylor expansion of the optimal support points can be determined efficiently by a recursive procedure. Examples are presented to illustrate the procedures for computing the optimal designs.     

An extension of the ranked set sampling theory

This paper is concerned with ranked set sampling theory which is useful to estimate the population mean when the order of a sample of small size can be found without measurements or with rough methods. Consider n sets of elements each set having size m. All elements of each set are ranked but only one is selected and quantified. The average of the quantified elements is adopted as the estimator. In this paper we introduce the notion of selective probability which is a generalization of a notion from Yanagawa and Shirahata (1976). Uniformly optimal unbiased procedures are found for some (n,m). Furthermore, procedures which are unbiased for all distributions and are good for symmetric distributions are studied for (n,m) which do not allow uniformly optimal unbiased procedures.     

Optimum treatment allocation for dual-objective clinical trials with binary outcomes

A common problem in randomized controlled clinical trials is the optimal assignment of patients to treatment protocols, The traditional optimal design assumes a single criterion, although in reality, there are usually more than one objective in a clinical trial. In this paper, optimal treatment allocation schemes are found for a dual-objective clinical trial with a binary response. A graphical method for finding the optimal strategy is proposed and illustrative examples are discussed.     

A modified least-failures sampling procedure for bernoulli subset selection

We restrict attention to a class of Bernoulli subset selection procedures which take observations one-at-a-time and can be compared directly to the Gupta-Sobel single-stage procedure. For the criterion of minimizing the expected total number of observations required to terminate experimentation, we show that optimal sampling rules within this class are not of practical interest. We thus turn to procedures which, although not optimal, exhibit desirable behavior with regard to this criterion. A procedure which employs a modification of the so-called least-failures sampling rule is proposed, and is shown to possess many desirable properties among a restricted class of Bernoulli subset selection procedures. Within this class, it is optimal for minimizing the number of observations taken from populations excluded from consideration following a subset selection experiment, and asymptotically optimal for minimizing the expected total number of observations required. In addition, it can result in substantial savings in the expected total num¬ber of observations required as compared to a single-stage procedure, thus it may be de¬sirable to a practitioner if sampling is costly or the sample size is limited.     

Optimal design of experimental epidemics

We consider the optimal design of controlled experimental epidemics or transmission experiments, whose purpose is to inform the practitioner about disease transmission and recovery rates. Our methodology employs Gaussian diffusion approximations, applicable to epidemics that can be modeled as density-dependent Markov processes and involving relatively large numbers of organisms. We focus on finding (i) the optimal times at which to collect data about the state of the system for a small number of discrete observations, (ii) the optimal numbers of susceptible and infective individuals to begin an experiment with, and (iii) the optimal number of replicate epidemics to use. We adopt the popular D-optimality criterion as providing an appropriate objective function for designing our experiments, since this leads to estimates with maximum precision, subject to valid assumptions about parameter values. We demonstrate the broad applicability of our methodology using a diverse array of compartmental epidemic models: a time-homogeneous SIS epidemic, a time-inhomogeneous SI epidemic with exponentially decreasing transmission rates and a partially observed SIR epidemic where the infectious period for an individual has a gamma distribution.     

Optimal Strategies for a Long-Term Static Investor

The optimal strategies for a long-term static investor are studied. Given a portfolio of a stock and a bond, we derive the optimal allocation of the capitals to maximize the expected long-term growth rate of a utility function of the wealth. When the bond has a constant interest rate, three models for the underlying stock price processes are studied: Heston model, 3/2 model, and jump diffusion model. We also study the optimal strategies for a portfolio in which the stock price process follows a Black-Scholes model and the bond process has a Vasicek interest rate that is correlated to the stock price.