Designing and selection of two-plan variables scheme indexed by crossover point

In this paper, the scheme of the inspection plan, namely the tightened normal tightened (n_T, n_N; k) is considered and procedures and necessary tables are developed for the selection of the variables sampling scheme, indexed through crossover point (COP). The importance of COP, the properties and advantages of the operating characteristic curve with respect to COP are studied.     

Sequential sampling in the search for new shared species

In microbial sciences, as well as other disciplines, it is often valuable to sample communities in a sequential or group sequential manner, in order to determine their structure or their similarity. We develop sequential sampling procedures to accomplish this by first assuming that one observation is drawn with replacement from each population at a time. Suppose that the sampling is terminated after n pairs of observations and k shared species were discovered, and assume that we receive payoff h(k)−cn, where h(k) is non-decreasing and the sampling cost c is non-negative. Similar to Rasmussen and Starr (1979), we show that an optimal stopping rule exists if h(k+1)−h(k) is non-increasing. An analogous result holds for group sequential sampling. This leads to using an estimate of the probability of discovering new shared species as a stopping indicator for comparing two populations with respect to the similarity index. We show by simulation and real examples that this is a feasible approach which can help to reduce the sample size.     

Single-stage sampling procedure of the t best populations under heteroscedasticity

Given k( ? 3) independent normal populations with unknown means and unknown and unequal variances, a single-stage sampling procedure to select the best t out of k populations is proposed and the procedure is completely independent of the unknown means and the unknown variances. For various combinations of k and probability requirement, tables of procedure parameters are provided for practitioners.     

An inequality for random replacement sampling plans

A sample (X₁ …, X_n) is drawn from a population of size N. Karlin (1974) conjectured that for any function ? in a certain class of real-valued functions on the sample space, ? is at least as large for sampling with replacement as for any other random replacement sampling plan. This conjecture is proved under the assumption that ?     

Estimation of a finite population distribution function based on a linear model with unknown heteroscedastic errors

The authors consider a finite population ρ = {(Y_k, x_k), k = 1,…,N} conforming to a linear superpopulation model with unknown heteroscedastic errors, the variances of which are values of a smooth enough function of the auxiliary variable X for their nonparametric estimation. They describe a method of the Chambers‐Dunstan type for estimation of the distribution of {Y_k, k = 1,…, N} from a sample drawn from without replacement, and determine the asymptotic distribution of its estimation error. They also consider estimation of its mean squared error in particular cases, evaluating both the analytical estimator derived by “plugging‐in” the asymptotic variance, and a bootstrap approach that is also applicable to estimation of parameters other than mean squared error. These proposed methods are compared with some common competitors in simulation studies.     

Construction and selection of tightened–normal–tightened variables sampling scheme of type TNTVSS (n 1, n 2; k)

This paper provides tables for the construction and selection of tightened–normal–tightened variables sampling scheme of type TNTVSS (n ₁, n ₂; k). The method of designing the scheme indexed by (AQL, α) and (LQL, β) is indicated. The TNTVSS (n _T , n _N; k) is compared with conventional single sampling plans for variables and with TNT (n ₁, n ₂; c) scheme for attributes, and it is shown that the TNTVSS is more efficient.     

Comparison of Three Sampling Designs

Three sampling designs are considered for estimating the sum of k population means by the sum of the corresponding sample means. These are (a) the optimal design; (b) equal sample sizes from all populations; and (c) sample sizes that render equal variances to all sample means. Designs (b) and (c) are equally inefficient, and may yield a variance up to k times as large as that of (a). Similar results are true when the cost of sampling is introduced, and they depend on the population sampled.     

Particle swarm optimization based Liu-type estimator

In this study, a new method for the estimation of the shrinkage and biasing parameters of Liu-type estimator is proposed. Because k is kept constant and d is optimized in Liu’s method, a (k, d) pair is not guaranteed to be the optimal point in terms of the mean square error of the parameters. The optimum (k, d) pair that minimizes the mean square error, which is a function of the parameters k and d, should be estimated through a simultaneous optimization process rather than through a two-stage process. In this study, by utilizing a different objective function, the parameters k and d are optimized simultaneously with the particle swarm optimization technique.     

On bootstrapping the number of components in finite mixtures of Poisson distributions

Finite mixture models arise in a natural way in that they are modeling unobserved population heterogeneity. It is assumed that the population consists of an unknown number k of subpopulations with parameters λ₁, ..., λ_k receiving weights p₁, ..., p_k. Because of the irregularity of the parameter space, the log-likelihood-ratio statistic (LRS) does not have a (χ²) limit distribution and therefore it is difficult to use the LRS to test for the number of components. These problems are circumvented by using the nonparametric bootstrap such that the mixture algorithm is applied B times to bootstrap samples obtained from the original sample with replacement. The number of components k is obtained as the mode of the bootstrap distribution of k. This approach is presented using the Times newspaper data and investigated in a simulation study for mixtures of Poisson data.     

A General Model for Repeated Audit Controls Using Monotone Subsampling

Abstract

In categorical repeated audit controls, fallible auditors classify sample elements in order to estimate the population fraction of elements in certain categories. To take possible misclassifications into account, subsequent checks are performed with a decreasing number of observations. In this paper a model is presented for a general repeated audit control system, where k subsequent auditors classify elements into r categories. Two different subsampling procedures will be discussed, named “stratified” and “random” sampling. Although these two sampling methods lead to different probability distributions, it is shown that the likelihood inferences are identical. The MLE are derived and the situations with undefined MLE are examined in detail; it is shown that an unbiased MLE can be obtained by stratified sampling. Three different methods for constructing confidence upper limits are discussed; the Bayesian upper limit seems to be the most satisfactory. Our theoretical results are applied to two cases with r = 2 and k = 2 or 3, respectively.     

A Bivariate Optimal Replacement Policy for a System With Age-dependent Minimal Repair and Cumulative Repair-cost Limit

In this article, a repairable system with age-dependent failure type and minimal repair based on a cumulative repair-cost limit policy is studied, where the information of entire repair-cost history is adopted to decide whether the system is repaired or replaced. As the failures occur, the system has two failure types: (i) a Type-I failure (minor) type that is rectified by a minimal repair, and (ii) a Type-II failure (catastrophic) type that calls for a replacement. We consider a bivariate replacement policy, denoted by (n,T), in which the system is replaced at life age T, or at the n-th Type-I failure, or at the kth Type-I failure (k < n and due to a minor failure at which the accumulated repair cost exceeds the pre-determined limit), or at the first Type-II failure, whichever occurs first. The optimal minimum-cost replacement policy (n,T)* is derived analytically in terms of its existence and uniqueness. Several classical models in maintenance literature could be regard as special cases of the presented model. Finally, a numerical example is given to illustrate the theoretical results.     

Implications of random cut‐points theory for the Mann‐Whitney and binomial tests

Through random cut‐points theory, the author extends inference for ordered categorical data to the unspecified continuum underlying the ordered categories. He shows that a random cut‐point Mann‐Whitney test yields slightly smaller p‐values than the conventional test for most data. However, when at least P% of the data lie in one of the k categories (with P = 80 for k = 2, P = 67 for k = 3,…, P = 18 for k = 30), he also shows that the conventional test can yield much smaller p‐values, and hence misleadingly liberal inference for the underlying continuum. The author derives formulas for exact tests; for k = 2, the Mann‐Whitney test is but a binomial test.     

Efficient sampling on the simplex with a self-adjusting logit transform proposal

A vector of k positive coordinates lies in the k-dimensional simplex when the sum of all coordinates in the vector is constrained to equal 1. Sampling distributions efficiently on the simplex can be difficult because of this constraint. This paper introduces a transformed logit-scale proposal for Markov Chain Monte Carlo that naturally adjusts step size based on the position in the simplex. This enables efficient sampling on the simplex even when the simplex is high dimensional and/or includes coordinates of differing orders of magnitude. Implementation of this method is shown with the SALTSampler R package and comparisons are made to other simpler sampling schemes to illustrate the improvement in performance this method provides. A simulation of a typical calibration problem also demonstrates the utility of this method.     

Evaluation and prediction of polygon approximations of planar contours for shape analysis

Contours may be viewed as the 2D outline of the image of an object. This type of data arises in medical imaging as well as in computer vision and can be modeled as data on a manifold and can be studied using statistical shape analysis. Practically speaking, each observed contour, while theoretically infinite dimensional, must be discretized for computations. As such, the coordinates for each contour as obtained at k sampling times, resulting in the contour being represented as a k-dimensional complex vector. While choosing large values of k will result in closer approximations to the original contour, this will also result in higher computational costs in the subsequent analysis. The goal of this study is to determine reasonable values for k so as to keep the computational cost low while maintaining accuracy. To do this, we consider two methods for selecting sample points and determine lower bounds for k for obtaining a desired level of approximation error using two different criteria. Because this process is computationally inefficient to perform on a large scale, we then develop models for predicting the lower bounds for k based on simple characteristics of the contours.     

The roles of prior catchability assumptions and sample features on bayes estimates of the number of classes in a population

The object of this paper is to explain the role played by the catchability and sampling in the Bayesian estimation of k, the unknown number of classes in a multinomial population. It is shown that the posterior distribution of k increases as the capture probabilities of the classes become more unequal, and that the posterior distribution of k increases with the number of classes observed in the sample and decreases with the sample size. Moreover, it is shown that the posterior mean of k is consistent.     

Consistency of stochastic approximation algorithm with quasi-associated random errors

ABSTRACT

Many mathematical and physical problems are led to find a root of a real function f. This kind of equation is an inverse problem and it is difficult to solve it. Especially in engineering sciences, the analytical expression of the function f is unknown to the experimenter, but it can be measured at each point x_k with M(x_k) as expected value and induced error ξ_k. The aim is to approximate the unique root θ under some assumptions on the function f and errors ξ_k. We use a stochastic approximation algorithm that constructs a sequence (x_k)_{k ? 1}. We establish the almost complete convergence of the sequence (x_k)_k to the exact root θ by considering the errors (ξ_k)_k quasi-associated and we illustrate the method by numerical examples to show its efficiency.     

Confidence Intervals for the Median of a Finite Population Based on the Sign Test

This article presents methods for constructing confidence intervals for the median of a finite population under simple random sampling without replacement, stratified random sampling, and cluster sampling. The confidence intervals, as well as point estimates and test statistics, are derived from sign estimating functions which are based on the well-known sign test. Therefore, a unified approach for inference about the median of a finite population is given.     

ARL-Design of S Charts with k-of-k Runs Rules

The standard S chart signals an out-of-control condition when one point exceeds a control limit. It can be augmented with runs rules to improve its performance in detecting assignable causes. A commonly used rule signals when k consecutive points exceed a control limit. This rule can be used alone or to supplement the standard chart. In this article we derive ARL expressions for charts with the k-of-k runs rule. We show how to design S charts with this runs rule, compare their ARL performance, and make a control chart recommendation when it is important to monitor for both increases and decreases in process dispersion.     

On grouping sets of exponential populations

Suppose we have k random samples each of size n from a two parameter exponential distribution with location parameters μ _i i=1,…,k, and where each item has the same, unknown scale parameter. A multistage procedure is developed to determine t≦k groups such that in any one group the distributions have μ_i's that are not appreciably different. The method yields a unique grouping and extends the approach of the Kumar and Pate1 test.The emphasis is on the development of a procedure based on the null sampling distribution of the maximum gap of the ordered first order statistics from exponential distributions. The procedure is based on complete ordered samples or censored (of any or of all) samples.     

Planning step-stress test plans under Type-I censoring for the log-location-scale case

In this paper, we consider a k-level step-stress accelerated life-testing (ALT) experiment with unequal duration steps τ=(τ₁, …, τ _k ). Censoring is allowed only at the change-stress point in the final stage. A general log-location-scale lifetime distribution with mean life which is a linear function of stress, along with a cumulative exposure model, is considered as the working model. Under this model, the determination of the optimal choice of τ for both Weibull and lognormal distributions are addressed using the variance–optimality criterion. Numerical results show that for a general log-location-scale distributions, the optimal k-step-stress ALT model with unequal duration steps reduces just to a 2-level step-stress ALT model.