Designing an economically optimal repetitive group-sampling plan based on loss functions

Acceptance sampling is a quality assurance tool, which provides a rule for the producer and the consumer to make acceptance or rejection decision about a lot. This paper attempts to develop a more efficient sampling plan, variables repetitive group sampling plan, based on the total loss to the producer and consumer. To design this model, two constraints are considered to satisfy the opposing priorities and requirements of the producer and the consumer by using Acceptable quality level (AQL) and Limiting quality level (LQL) points on operating characteristic (OC) curve. The objective function of this model is constructed based on the total expected loss. In order to illustrate the application of the proposed model, an example is presented. In addition, the effects of process parameters on the optimal solution and the total expected loss are studied by performing a sensitivity analysis. Finally, the efficiency of the proposed model is compared with the variables single sampling plan, the variables double sampling plan and the repetitive group sampling plan of Balamurali and Jun (2006) in terms of average sample number, total expected loss and its difference with ideal OC curve.     

Variance approximation under balanced sampling

A balanced sampling design has the interesting property that Horvitz–Thompson estimators of totals for a set of balancing variables are equal to the totals we want to estimate, therefore the variance of Horvitz–Thompson estimators of variables of interest are reduced in function of their correlations with the balancing variables. Since it is hard to derive an analytic expression for the joint inclusion probabilities, we derive a general approximation of variance based on a residual technique. This approximation is useful even in the particular case of unequal probability sampling with fixed sample size. Finally, a set of numerical studies with an original methodology allows to validate this approximation.     

Posterior moments of scalar functions of a covariance matrix

Given multivariate normal data and a certain spherically invariant prior distribution on the covariance matrix, it is desired to estimate the moments of the posterior marginal distributions of some scalar functions of the covariance matrix by importance sampling. To this end a family of distributions is defined on the group of orthogonal matrices and a procedure is proposed for selecting one of these distributions for use as a weighting distribution in the importance sampling process. In an example estimates are calculated for the posterior mean and variance of each element in the covariance matrix expressed in the original coordinates, for the posterior mean of each element in the correlation matrix expressed in the original coordinates, and for the posterior mean of each element in the covariance matrix expressed in the coordinates of the principal variables.     

Designing stratified sampling in economic and business surveys

In most economic and business surveys, the target variables (e.g. turnover of enterprises, income of households, etc.) commonly resemble skewed distributions with many small and few large units. In such surveys, if a stratified sampling technique is used as a method of sampling and estimation, the convenient way of stratification such as the use of demographical variables (e.g. gender, socioeconomic class, geographical region, religion, ethnicity, etc.) or other natural criteria, which is widely practiced in economic surveys, may fail to form homogeneous strata and is not much useful in order to increase the precision of the estimates of variables of interest. In this paper, a stratified sampling design for economic surveys based on auxiliary information has been developed, which can be used for constructing optimum stratification and determining optimum sample allocation to maximize the precision in estimate.     

Probit and logistic discriminant functions

Most discriminant functions refer to qualitatively district groups. Talis et al. (1975) introduced the probit discriminant function for distinguishing between two ordered groups. They showed how to estimate this function for mixture sampling and continuous predictor variables. Here an estimation system is given for the more common separate sampling which is applicable to continuous and/or discrete predictor variables. When used solely with continuous variables) this method of estimation is more robust than Tallis!

The relationship of probit and logistic discrimination is discussed.     

SOME THEORY OF SAMPLING ON SUCCESSIVE OCCASIONS1

One of the important theoretical developments in successive sampling has been to provide an optimum estimate by combining two independent estimates (i) a double-sampling regression estimate from the matched portion of the sample using one auxiliary variable with (ii) a mean per unit estimate based on the unmatched portion of the sample. Theory has been generalized in the present paper to provide the optimum estimate by combining a double-sampling multivariate ratio or regression estimate using p auxiliary variables (p≥1) from the matched portion of the sample with a mean per unit estimate from the unmatched portion of the sample. Results have been presented for the more general and practical case when the samples on the two occasions are of unequal size.     

Estimation of finite population mean in simple random sampling and stratified random sampling using two auxiliary variables

In this article, we propose a new class of estimators to estimate the finite population mean by using two auxiliary variables under two different sampling schemes such as simple random sampling and stratified random sampling. The proposed class of estimators gives minimum mean squared error as compared to all other considered estimators. Some real data sets are used to observe the performances of the estimators. We show numerically that the proposed class of estimators performs better as compared to all other competitor estimators.     

Sample-size-restrictive adaptive sampling: an application in estimating localized elements

In sample surveys sometimes one encounters a situation where, for many sampling units, one or more variables of interest are valued zero or negligibly low while for some other units they are substantial because of heavy localization of the high-valued units in certain segments. Estimation may then be inaccurate if a chosen sample fails to capture enough of the high-valued units. In such situations, adaptive sampling, as an extension of the initial sample to capture additional high-valued units, may be more serviceable. However, the size of an adaptive sample may often far exceed that of the initial sample. In this paper we present a method to put desirable constraints on the adaptive sample-size to keep the latter in check. To examine the efficacy of this method, we illustrate its application to estimate total numbers of rural earners through specific vocations in a given district in India simultaneously for several vocations.     

Bayesian estimation in the multidimensional three-parameter logistic model

The Gibbs sampler has a great potential to be an efficient and versatile estimation procedure in item response theory. In this article, based on a data augmentation scheme using the Gibbs sampler, we propose a Bayesian procedure to estimate the multidimensional three-parameter logistic model. With the introduction of the two latent variables, the full conditional distributions are tractable, and consequently the Gibbs sampling is easy to implement. Finally, the technique is illustrated by using simulated and real data, respectively.     

A distribution-free approach to inducing rank correlation among input variables

A method for inducing a desired rank correlation matrix on a multivariate input random variable for use in a simulation study is introduced in this paper. This method is simple to use, is distribution free, preserves the exact form of the marginal distributions on the input variables, and may be used with any type of sampling scheme for which correlation of input variables is a meaningful concept. A Monte Carlo study provides an estimate of the bias and variability associated with the method. Input variables used in a model for study of geologic disposal of radioactive waste provide an example of the usefulness of this procedure. A textbook example shows how the output may be affected by the method presented in this paper.     

A robust estimate of the correlation coefficient for bivariate normal distribution using ranked set sampling

A robust estimate of the correlation coefficient for a bivariate normal distribution using balanced ranked set sampling is studied. We show that this estimate is at least as efficient as the corresponding estimate based on simple random sampling and highly efficient compared to the maximum likelihood estimate using balanced ranked set sampling. The estimate is robust to common ranking errors. Small sample performance of the estimate is studied by simulation under imperfect and perfect ranking. A variance stabilizing transformation for the confidence interval of the correlation coefficient is obtained.     

Population structure and correlation between auxiliary and target variables may affect precision of estimates in forest inventory

Simulations of forest inventory in several populations compared simple random with “quick probability proportional to size” (QPPS) sampling. The latter may be applied in the absence of a list sampling frame and/or prior measurement of the auxiliary variable. The correlation between the auxiliary and target variables required to render QPPS sampling more efficient than simple random sampling varied over the range 0.3–0.6 and was lower when sampling from populations that were skewed to the right. Two possible analytical estimators of the standard error of the estimate of the mean for QPPS sampling were found to be less reliable than bootstrapping.     

A rare event approach to high-dimensional approximate Bayesian computation

Approximate Bayesian computation (ABC) methods permit approximate inference for intractable likelihoods when it is possible to simulate from the model. However, they perform poorly for high-dimensional data and in practice must usually be used in conjunction with dimension reduction methods, resulting in a loss of accuracy which is hard to quantify or control. We propose a new ABC method for high-dimensional data based on rare event methods which we refer to as RE-ABC. This uses a latent variable representation of the model. For a given parameter value, we estimate the probability of the rare event that the latent variables correspond to data roughly consistent with the observations. This is performed using sequential Monte Carlo and slice sampling to systematically search the space of latent variables. In contrast, standard ABC can be viewed as using a more naive Monte Carlo estimate. We use our rare event probability estimator as a likelihood estimate within the pseudo-marginal Metropolis–Hastings algorithm for parameter inference. We provide asymptotics showing that RE-ABC has a lower computational cost for high-dimensional data than standard ABC methods. We also illustrate our approach empirically, on a Gaussian distribution and an application in infectious disease modelling.     

Family of Estimators of Population Mean Using Two Auxiliary Variables in Stratified Random Sampling

A general family of estimators, which use the information of two auxiliary variables in the stratified random sampling, is proposed to estimate the population mean of the variable under study. Under stratified random sampling without replacement scheme, the expressions of bias and mean square error (MSE) up to the first- and second-order approximations are derived. The family of estimators in its optimum case is discussed. Also, an empirical study is carried out to show the properties of the proposed estimators.     

Disentangling the sources of variation in the survival of the European dipper

The population growth rate of the European dipper has been shown to decrease with winter temperature and population size. We examine here the demographic mechanism for this effect by analysing how these factors affect the survival rate. Using more than 20 years of capture-mark-recapture data (1974-1997) based on more than 4000 marked individuals, we perform analyses using open capture-mark-recapture models. This allowed us to estimate the annual apparent survival rates (probability of surviving and staying on the study site from one year to the next one) and the recapture probabilities. We partitioned the variance of the apparent survival rates into sampling variance and process variance using random effects models, and investigated which variables best accounted for temporal process variation. Adult males and females had similar apparent survival rates, with an average of 0.52 and a coefficient of variation of 40%. Chick apparent survival was lower, averaging 0.06 with a coefficient of variation of 42%. Eighty percent of the variance in apparent survival rates was explained by winter temperature and population size for adults and 48% by winter temperature for chicks. The process variance outweighed the sampling variance both for chick and adult survival rates, which explained that shrunk estimates obtained under random effects models were close to MLE estimates. A large proportion of the annual variation in the apparent survival rate of chicks appears to be explained by inter-year differences in dispersal rates.     

Disentangling the sources of variation in the survival of the European dipper

The population growth rate of the European dipper has been shown to decrease with winter temperature and population size. We examine here the demographic mechanism for this effect by analysing how these factors affect the survival rate. Using more than 20 years of capture-mark-recapture data (1974-1997) based on more than 4000 marked individuals, we perform analyses using open capture-mark-recapture models. This allowed us to estimate the annual apparent survival rates (probability of surviving and staying on the study site from one year to the next one) and the recapture probabilities. We partitioned the variance of the apparent survival rates into sampling variance and process variance using random effects models, and investigated which variables best accounted for temporal process variation. Adult males and females had similar apparent survival rates, with an average of 0.52 and a coefficient of variation of 40%. Chick apparent survival was lower, averaging 0.06 with a coefficient of variation of 42%. Eighty percent of the variance in apparent survival rates was explained by winter temperature and population size for adults and 48% by winter temperature for chicks. The process variance outweighed the sampling variance both for chick and adult survival rates, which explained that shrunk estimates obtained under random effects models were close to MLE estimates. A large proportion of the annual variation in the apparent survival rate of chicks appears to be explained by inter-year differences in dispersal rates.     

A Class of Estimators Using Auxiliary Information in Two-Stage Sampling

This paper presents a general class of estimators for the finite population total when the emphasis is laid on the use of two auxiliary variables in a two-stage sampling.     

Importance Sampling for Sums of Lognormal Distributions with Applications to Operational Risk

In this paper we use Importance Sampling to estimate tail probabilities for a finite sum of lognormal distributions. We use a defensive mixture, and develop a method of choosing the parameters via the EM algorithm; we also consider the technique which assumes the importance sampling density to belong to the same parametric family of the random variables to be summed. In both cases, the instrumental density is found by minimizing Cross-Entropy. A comparison based on several simulation experiments shows that the defensive mixture has the best performance. Finally, we study the Poisson-lognormal compound distribution framework and present a real-data application.     

Estimation of unknown parameters in nonlinear and non-Gaussian state-space models

For the last decade, various simulation-based nonlinear and non-Gaussian filters and smoothers have been proposed. In the case where the unknown parameters are included in the nonlinear and non-Gaussian system, however, it is very difficult to estimate the parameters together with the state variables, because the state-space model includes a lot of parameters in general and the simulation-based procedures are subject to the simulation errors or the sampling errors. Therefore, clearly, precise estimates of the parameters cannot be obtained (i.e., the obtained estimates may not be the global optima). In this paper, an attempt is made to estimate the state variables and the unknown parameters simultaneously, where the Monte Carlo optimization procedure is adopted for maximization of the likelihood function.