A NOTE ON OPTIMUM STRATIFICATION IN SAMPLING WITH VARYING PROBABILITIES1

Singh and Sukhatme [4] have considered the problem of optimum stratification on an auxiliary variable x when the units from the different strata are selected with probability proportional to the value of the auxiliary variable and the sample sizes for the different strata are determined by using Neyman allocation method. The present paper considers the same problem for the proportional and equal allocation methods. The rules for finding approximately optimum strata boundaries for these two allocation methods have been given. An investigation into the relative efficiency of these allocation methods with respect to the Neyman allocation has also been made. The performance of equal allocation is found to be better than that of proportional allocation and practically equivalent to the Neyman allocation.     

A NOTE ON OPTIMUM STRATIFICATION FOR EQUAL ALLOCATION WITH RATIO AND REGRESSION METHODS OF ESTIMATION

When the information on a highly positively correlated auxiliary variable x is used to construct stratified regression (or ratio) estimates of the population mean of the study variable y, the paper considers the problem of determining approximately optimum strata boundaries (AOSB) on x when the sample size in each stratum is equal. The form of the conditional variance function V(y/x) is assumed to be known. A numerical investigation into the relative efficiency of equal allocation with respect to the Neyman and proportional allocations has also been made. The relative efficiency of equal allocation with respect to Neyman allocation is found to be nearly equal to one.     

Information Loss in Grade Point Conversion

Some information gets lost when numerical scores evaluating performances are converted into letter grades. We propose to measure this information loss through the proportion of variance lost due to grouping. We study various properties of this measure, including its invariance in location and scale equivariant families. The information loss typically decreases with an increase in the number of levels of letter grades. However, it is not appropriate to have too many levels. The optimum number of levels may be determined, either by visual inspection when the information loss becomes marginal/stable, or by minimizing the sum of the information loss and a penalty term, the latter being taken as linear in the number of levels. We also address the problem of determining the groups, or equivalently, the boundaries so that the information loss is minimized, given a fixed number of groups. Finding these optimal boundaries is a computationally intensive exercise even for moderate size data, unless the number of groups is very small. We recommend an alternative way by fitting an appropriate probability distribution. When the probabilistic nature of the data is known, the boundary points turn out to be the solutions to a system of equations; however these solutions may not necessarily have any closed form. We derive the exact or approximate solutions of these equations when the composite scores follow a probability distribution belonging to the Uniform, Triangular, and Gaussian family.     

Optimal stratification in stratified designs using weibull-distributed auxiliary information

Sampling has evolved into a universally accepted approach for gathering information and data mining as it is widely accepted that a reasonably modest-sized sample can sufficiently characterize a much larger population. In stratified sampling designs, the whole population is divided into homogeneous strata in order to achieve higher precision in the estimation. This paper proposes an efficient method of constructing optimum stratum boundaries (OSB) and determining optimum sample size (OSS) for the survey variable. The survey variable may not be available in practice since the variable of interest is unavailable prior to conducting the survey. Thus, the method is based on the auxiliary variable which is usually readily available from past surveys. To illustrate the application as an example using a real data, the auxiliary variable considered for this problem follows Weibull distribution. The stratification problem is formulated as a Mathematical Programming Problem (MPP) that seeks minimization of the variance of the estimated population parameter under Neyman allocation. The solution procedure employs the dynamic programming technique, which results in substantial gains in the precision of the estimates of the population characteristics.     

Optimum points of stratification for two or more stage designs

In this paper, an attempt is made to obtain optimum points of stratification for two or more stage designs with equal p.s.u.'s and the subsequent units. Stratification on the auxiliary variable when the study variable is closely related to the auxiliary variable has also been obtained. The determination of OPS in these cases have been illustrated with the help of some known specific distributions.     

The enumeration of restricted random walks by Sheffer polynomials with applications to statistics

Sheffer polynomials are solutions of certain systems of operator equations. Difference equations, which frequently occur in path enumeration, belong in that class. To find representations of the solutions, the restriction on the paths has to be in the form of boundaries. Such problems have applications in two-sample tests. We also consider paths with more than two step vectors. The gambler's ruin problem illustrates the method. If paths with a given area underneath are counted, q-binomial coefficients come into play. Eulerian Sheffer sequence solve some of such problems.     

最优层界确定方法综述

回顾了分层抽样中层界最优解确定的解析方法和迭代方法,对比并评析了不同方法在计算繁杂程度、有效性等方面的优劣,针对偏态总体分层界限最优解的确定,对几何分层法提出了一种改进思路的变异系数法,并提出了进一步研究最优层界最优解确定方法的思路。     

Some consequences of model misspecification for t testing in a structural equation

This paper examines the distribution theory relevant to statistics obtained from a single structural equation from a system of simultaneous equations. Attention is focused on t statistics in misspecified models which have been estimated by instrumental variable techniques.     

Modern variable selection for longitudinal semi-parametric models with missing data

Penalized methods for variable selection such as the Smoothly Clipped Absolute Deviation penalty have been increasingly applied to aid variable section in regression analysis. Much of the literature has focused on parametric models, while a few recent studies have shifted the focus and developed their applications for the popular semi-parametric, or distribution-free, generalized estimating equations (GEEs) and weighted GEE (WGEE). However, although the WGEE is composed of one main and one missing-data module, available methods only focus on the main module, with no variable selection for the missing-data module. In this paper, we develop a new approach to further extend the existing methods to enable variable selection for both modules. The approach is illustrated by both real and simulated study data.     

Graphical solutions for structural regression assist errors-in-variables modelling

Structural regression attempts to reveal an underlying relationship by compensating for errors in the variables. Ordinary least-squares regression has an entirely different purpose and provides a relationship between error-included variables. Structural model solutions, also known as the errors-in-variables and measurement-error solutions, use various inputs such as the error-variance ratio and x-error variance. This paper proposes that more accurate structural line gradient (coefficient) solutions will result from using the several solutions together as a system of equations. The known data scatter, as measured by the correlation coefficient, should always be used in choosing legitimate combinations of x- and y-error terms. However, this is difficult using equations. Chart solutions are presented to assist users to understand the structural regression process, to observe the correlation coefficient constraint, to assess the impact of their error estimates and, therefore, to provide better quality estimates of the structural regression gradient.     

Optimum stratification and allocation in inventory sampling: An efficient two stage grid search procedure

Standard Methods of optimal stratification are solving the optimization problem as a function of strata boundaries and sample allocation only. In this paper we show that by means of a flexible two stage grid search procedure strata boundaries, sample allocation and furthermore number of strata can be taken into account in an effective way when optimizing stratification and allocation. By means of a Monte Carlo simulation we show that the proposed procedure is efficient compared to the well known standard procedures.     

Optimum Mixture Design for the Detection of Synergistic Effects

In mixture experiments, optimal designs for the estimation of parameters, both linear and non-linear, have been discussed by several authors. Optimal designs for the estimation of a subset of parameters have also been investigated. However, designs for testing the effects of certain factors and interactions have been studied only in the context of response surface models. In this article, we attempt to find the optimum design for testing the presence of synergistic effects in a mixture model. The classical F-test has been considered and the optimum design has been obtained so as to maximize the power of the test. It is observed that the barycenters are necessarily the support points of the trace-optimal design.     

Local likelihood density estimation for interval censored data

The authors propose a class of procedures for local likelihood estimation from data that are either interval‐censored or that have been aggregated into bins. One such procedure relies on an algorithm that generalizes existing self‐consistency algorithms by introducing kernel smoothing at each step of the iteration. The entire class of procedures yields estimates that are obtained as solutions of fixed point equations. By discretizing and applying numerical integration, the authors use fixed point theory to study convergence of algorithms for the class. Rapid convergence is effected by the implementation of a local EM algorithm as a global Newton iteration. The latter requires an explicit solution of the local likelihood equations which can be found by using the symbolic Newton‐Raphson algorithm, if necessary.     

Estimation of a population mean with two-way stratification using a systematic allocation scheme

A technique of systematically allocating a sample to the strata formed by double stratification is presented. The method can proportionally allocate the sample along each variable of stratification. If there are R strata and C strata for the first and second variable of stratification respectively, the technique requires that the total sample size be at least as large as max(R, C). An unbiased estimator of the population mean is given and its variance is obtained. The technique is compared with a random allocation procedure given by Bryant, Hartley, and Jessen (1960). Numerical examples are given suggesting when one technique is superior to the other.     

Estimation of mean in presence of non-response using two phase sampling scheme

This paper considers the problem of estimating the population mean using information on an auxiliary variable in presence of non-response. Some modified ratio, product and regression estimators in double sampling have been suggested and their properties are studied. It is shown that to the first degree of approximation, estimators based on estimated optimum values have the same variance as that of the optimum estimators. An empirical study is carried to judge the merits of the suggested estimators over conventional unbiased estimator and other known estimators. Both theoretical and empirical study results present the soundness and usefulness of the suggested estimators in practice.     

The Equivalence of Neyman Optimum Allocation for Sampling and Equal Proportions for Apportioning the U.S. House of Representatives

We present a surprising though obvious result that seems to have been unnoticed until now. In particular, we demonstrate the equivalence of two well-known problems—the optimal allocation of the fixed overall sample size n among L strata under stratified random sampling and the optimal allocation of the H = 435 seats among the 50 states for apportionment of the U.S. House of Representatives following each decennial census. In spite of the strong similarity manifest in the statements of the two problems, they have not been linked and they have well-known but different solutions; one solution is not explicitly exact (Neyman allocation), and the other (equal proportions) is exact. We give explicit exact solutions for both and note that the solutions are equivalent. In fact, we conclude by showing that both problems are special cases of a general problem. The result is significant for stratified random sampling in that it explicitly shows how to minimize sampling error when estimating a total T_Y while keeping the final overall sample size fixed at n; this is usually not the case in practice with Neyman allocation where the resulting final overall sample size might be near n + L after rounding. An example reveals that controlled rounding with Neyman allocation does not always lead to the optimum allocation, that is, an allocation that minimizes variance.     

An optimization approach applied to the optimal stratification problem

A new optimization algorithm is presented to solve the stratification problem. Assuming the number L of strata and the total sample size n are fixed, we obtain strata boundaries by using an objective function associated with the variance. In this problem, strata boundaries must be determined so that the elements in each stratum are more homogeneous among themselves. To produce more homogeneous strata, this paper proposes a new algorithm that uses the Greedy Randomized Adaptive Search Procedure (GRASP) methodology. Computational results are presented for a set of problems, with the application of the new algorithm and some algorithms from literature.