Small-sample performance of a modified least-failures sampling procedure for bernoulli subset selection

We describe two sequential sampling procedures for Bernoulli subset selection which were shown to exhibit desirable behavior for large-sample problems. These procedures have identical performance characteristics in terms of the number of observations taken from any one of the populations under investigation, but one of the procedures employs one-at-a-time sampling while theother allows observations to be taken in blocks during early stages of experimentation. In this paper, a simulation study of their behavior for small-sample cases (n > 25) reveals that they canresult in a savings (sometimes substantial) in the expected total number of observations requiredto terminate the experiment as compared to single-stage procedures. Hence they may be quite usefulto a practitioner for screening purposes when sampling is limited.     

Unbiased estimation following selection for bernoulli populations

Although a large number of selection procedures have been published in the statistics literature, the selection approach has received only limited use in applications. One drawback to the use of such procedures has been the lack of parameter estimates, which prevents quantitative comparisons among the treatments. To partially address this criticism, we present a general method for constructing unbiased estimators of the success probabilities after the termination of a sequential experiment involving two or more Bernoulli populations. Some theoretical properties are presented, and examples are provided for several different selection procedures.     

A subset selection procedure for multinomial distributions

A subset selection procedure is developed for selecting a subset containing the multinomial population that has the highest value of a certain linear combination of the multinomial cell probabilities; such population is called the ‘best’. The multivariate normal large sample approximation to the multinomial distribution is used to derive expressions for the probability of a correct selection, and for the threshold constant involved in the procedure. The procedure guarantees that the probability of a correct selection is at least at a pre-assigned level. The proposed procedure is an extension of Gupta and Sobel's [14] selection procedure for binomials and of Bakir's [2] restrictive selection procedure for multinomials. One illustration of the procedure concerns population income mobility in four countries: Peru, Russia, South Africa and the USA. Analysis indicates that Russia and Peru fall in the selected subset containing the best population with respect to income mobility from poverty to a higher-income status. The procedure is also applied to data concerning grade distribution for students in a certain freshman class.     

Equal probability of correct selection for bernoulli selection procedures

The problem of selecting the Bernoulli population which has the highest "success" probability is considered. It has been noted in several articles that the probability of a correct selection is the same, uniformly in the Bernoulli p-vector (P1,P2,….,Pk), for two or more different selection procedures. We give a general theorem which explains this phenomenon.

An application of particular interest arises when "strong" curtailment of a single-stage procedure (as introduced by Bechhofer and Kulkarni (1982a) )is employed; the corresponding result for "weak" curtailment of a single-stage procedure needs no proof. The use of strong curtailment in place of weak curtailment requires no more (and usually many less) observations to achieve the same.     

A restricted subset selection procedure for Weibull populations

Consider k(k ≥ 2) two-parameter Weibull populations. We want to select a subset of the populations not exceeding m in size such that the subset contains at least ? of the t best populations. We have proposed a procedure which uses either the maximum likelihood estimators or ‘simplified’ linear estimators of the parameters. The estimators are based on type II censored data. The ranking of the populations is done by comparing their reliabilities at a certain fixed time. In selected cases the constants for the procedure are tabulated using Monte Carlo methods.     

On sequentially rejective subset selection procedures

A technique for selection procedures, called sequential rejection, is investigated. It is shown that this technique is posssible to apply to certain selection goals of the "all or nothing" type, i.e. "selecting a subset containing all good populations" or "selecting a subset containing no bad population". The analogy with existing sequential techniques in the general theory of simultaneous statistical inference is pointed out.     

A model-based sample selection procedure for one-stage sampling

The problem of sample selection, when a one-stage superpopulation model-based approach is used to predict individual variate values for each unit in a finite population based on a sample of only some of the units, is investigated. The model framework is discussed and a sample selection scheme based on the model is derived. The sample selection scheme is evaluated using actual data. Future research topics including multiple predictions per unit are suggested.     

A procedure for selection of a gamma-Poisson single sampling plan by attributes

Design and evaluation of sampling plans by attributes and by variables are important aspects in the area of acceptance sampling research. Various procedures for the selection of conventional single sampling by attributes have been developed and are available in the literature. This paper presents a design methodology and tables for the selection of parameters of single sampling plans for specified requirements (strengths) under the conditions of a gamma prior and Poisson sampling distribution. The relative efficiency of gamma-Poisson single sampling plans over conventional plans is discussed through empirical illustrations.     

A generalized procedure for selection of attribute double sampling plan

A compact table, yielding attribute double sampling plan indexed by AQL, AOQL and LQL and double sampling scheme indexed by AQL is presented. Method of selecting the plans are indicated through examples.     

A decision-theoretic approach to subset selection

The multiple decision problem of selecting a random non-empty subset of populations, out of k populations, that are close in some sense to the best population is considered in a decision-theoretic framework. Uniformly optimal procedures for non-negative semi-additive loss are derived. A class of likelihood-ratio type of procedures is shown to be admissible for monotone additive loss.     

A simple test for equally likely random responses in a sequence of bernoulli trials

A simple versatile combinatorial test is given for the null hypothesis that the two kinds of responses in a sequence of binary random variables are both equiprobable and randomly distributed. It is sensitive to a preponderance of either kind of response in either of two pre-selected halves of the positions of the sequence, e.g., the right half vs. the left, or the center vs. both ends. It combines the nice features of being, on the one hand, relatively assumption free and easy to apply and, on the other hand, sensitive to basic patterns underlying a poor fit of the empirical distribution to a theoretical model.     

Distribution-free subset selection for incompletely ranked data

In market research and some other areas, it is common that a sample of n judges (consumers, evaluators, etc.) are asked to independently rank a series of k objects or candidates. It is usually difficult to obtain the judges' full cooperation to completely rank all k objects. A practical way to overcome this difficulty is to give each judge the freedom to choose the number of top candidates he is willing to rank. A frequently encountered question in this type of survey is how to select the best object or candidate from the incompletely ranked data. This paper proposes a subset selection procedure which constructs a random subset of all the k objects involved in the survey such that the best object is included in the subset with a prespecified confidence. It is shown that the proposed subset selection procedure is distribution-free over a very broad class of underlying distributions. An example from a market research study is used to illustrate the proposed procedure.     

Waiting time paradox and size biased sampling

Feller's (1971) 'Waiting time for the bus paradox1 is explained by the fact that the long Interarrival times have a better chance of covering an arbitrary point in time than do short interarrival times, This gives rise to size biased sampling and the length of the Interarrival time that contains the arbitrary point has a moment distribution, This paper deals with the characterization of the exponential distribution based on this 'Waiting time paradox'. Similar characterizations of the binomial, the negative binomial, the Poisson and the geometric distributions are obtained.     

A modified bootstrap procedure for cluster sampling variance estimation of species richness

Variance estimators for probability sample-based predictions of species richness (S) are typically conditional on the sample (expected variance). In practical applications, sample sizes are typically small, and the variance of input parameters to a richness estimator should not be ignored. We propose a modified bootstrap variance estimator that attempts to capture the sampling variance by generating B replications of the richness prediction from stochastically resampled data of species incidence. The variance estimator is demonstrated for the observed richness (SO), five richness estimators, and with simulated cluster sampling (without replacement) in 11 finite populations of forest tree species. A key feature of the bootstrap procedure is a probabilistic augmentation of a species incidence matrix by the number of species expected to be ‘lost’ in a conventional bootstrap resampling scheme. In Monte-Carlo (MC) simulations, the modified bootstrap procedure performed well in terms of tracking the average MC estimates of richness and standard errors. Bootstrap-based estimates of standard errors were as a rule conservative. Extensions to other sampling designs, estimators of species richness and diversity, and estimates of change are possible.     

A class of confidence intervals for sequential phase II clinical trials with binary outcome

The phase II clinical trials often use the binary outcome. Thus, accessing the success rate of the treatment is a primary objective for the phase II clinical trials. Reporting confidence intervals is a common practice for clinical trials. Due to the group sequence design and relatively small sample size, many existing confidence intervals for phase II trials are much conservative. In this paper, we propose a class of confidence intervals for binary outcomes. We also provide a general theory to assess the coverage of confidence intervals for discrete distributions, and hence make recommendations for choosing the parameter in calculating the confidence interval. The proposed method is applied to Simon's [14] optimal two-stage design with numerical studies. The proposed method can be viewed as an alternative approach for the confidence interval for discrete distributions in general.     

Two new goals for selection based on proportions

Procedures are derived for selecting, with controlled probability of error, (1) a subset of populations which contains all populations better than a dual probability/proportion standard and (2) a subset of populations which both contains all populations better than an upper probability/proportion standard and also contains no populations worse than a lower probability/proportion standard. The procedures are motivated by current investigations in the area of computer performance evaluation.     

A two-sample parametric group sequential procedure for monitoring clinical trials

A group sequential procedure is presented which allows for staggered entry of patients, random loss to followup, and utilizes the flexible boundary approach of Lan and DeMets. The proposed procedure assumes that response times are nearly exponentially distributed. An example as well as simulation studies comparing the performance of the proposed procedure with the group sequential logrank are also presented.     

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.