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1.
In a multivariate stratified sample survey with L strata and p > 1 characteristics, defined on each unit of the population, let the estimation of all the p-population means be of interest. As discussed by Cochran ( 1977), since the optimum allocation for one characteristic will not in general be optimum for other characteristics some compromise
must be reached in a multiple characteristics stratified surveys. Various authors worked out allocations that are based on
a compromise criterion. The resulting allocations are optimal for all characteristics in some sense, for example an allocation
that minimizes the trace of the variance-covariance matrix of the estimators of the population means or an allocation that
minimizes the weighted average of the variances or an allocation that maximizes the total relative efficiency of the estimators
as compared to the corresponding individual optimum allocations. In the present paper the problem of optimum allocation in
multivariate stratified random sampling in the presence of nonresponse has been formulated as a multiobjective integer nonlinear
programming problem and a solution procedure is developed using goal programming technique. Three numerical examples are worked
out to illustrate the computational details. A comparison of the proposed method with some well known methods is also carried
out to show the practical utility of the proposed method. 相似文献
2.
The method of choosing the best boundaries that make strata internally homogeneous, given some sample allocation, is known
as optimum stratification. In order to make the strata internally homogeneous, the strata are constructed in such a way that
the strata variances should be as small as possible for the characteristic under study. In this paper the problem of determining
optimum strata boundaries (OSB) is discussed when strata are formed based on a single auxiliary variable with a varying measurement
cost per units across strata. The auxiliary variable considered in the problem is a size variable that holds a common model
for a whole population. The OSB are achieved effectively by assuming a suitable distribution of the auxiliary variable and
creating strata by cutting the range of the distribution at optimum points. The problem of finding the OSB, which minimizes
the variance of the estimated population mean under a weighted stratified balanced sampling, is formulated as a mathematical
programming problem (MPP). Treating the formulated MPP as a multistage decision problem, a solution procedure using dynamic
programming technique is developed. A numerical example using a hospital population data is presented to illustrate the computational
details of the solution procedure. A software program coded in JAVA is written to carry out the computation. The distribution
of the auxiliary variable in this example is considered to be continuous with an exponential density function. 相似文献
3.
When more than one (say p) characteristics in multivariate stratified population are defined on each unit of the population, the individual optimum allocations may differ widely and can not be used practically. Moreover, there may be a situation such that no standard allocation is advisable to all the strata, for one reason or another. In such a situation, Clark and Steel (J R Stat Soc, Ser D Stat 49(2):197–207, 2000) suggested that different allocations may be used for different groups of strata having some common characteristics for double sampling in stratification. Later on, Ahsan et al. (Aligarh J Stat 25:87–97, 2005) used the same concept in univariate stratified sampling. They minimized the variance of the stratified sample mean for a fixed cost to obtain an allocation and called this allocation “mixed allocation”. In the present paper, a “compromise mixed allocation” is worked out for the fixed precisions of the estimates of the p-population means of a multivariate stratified population. A numerical example is also presented. 相似文献
4.
A stratified random sampling plan is one in which the elements of the population are first divided into nonoverlapping groups, and then a simple random sample is selected from each group. In this paper, we focus on determining the optimal sample size of each group. We show that various versions of this problem can be transformed into a particular nonlinear program with a convex objective function, a single linear constraint, and bounded variables. Two branch and bound algorithms are presented for solving the problem. The first algorithm solves the transformed subproblems in the branch and bound tree using a variable pegging procedure. The second algorithm solves the subproblems by performing a search to identify the optimal Lagrange multiplier of the single constraint. We also present linearization and dynamic programming methods that can be used for solving the stratified sampling problem. Computational testing indicates that the pegging branch and bound algorithm is fastest for some classes of problems, and the linearization method is fastest for other classes of problems. 相似文献
5.
An LP is considered where the technology coefficients are unknown and random samples are taken to estimate them. A stochastic programming problem is formulated to find the optimal sample sizes where it is required that a confidence interval should cover the unknown deterministic optimum value by a given probability and the cost of sampling be minimum. 相似文献
6.
This work considers the allocation problem for multivariate stratified random sampling as a problem of integer non-linear stochastic multiobjective mathematical programming. With this goal in mind the asymptotic distribution of the vector of sample variances is studied. Two alternative approaches are suggested for solving the allocation problem for multivariate stratified random sampling. An example is presented by applying the different proposed techniques. 相似文献
7.
Summary The paper considers the problem of optimum stratification on an auxiliary variable x when the information on the auxiliary variable x is also used to estimate the population mean
using ratio or regression methods of estimation. Assuming the form of the regression of the estimation variable y on the auxiliary variable x as also the form of the conditional variance function V( y/x), the problem of determining optimum strata boundaries (OSB) is shown to be a particular case of optimum stratification on
the auxiliary variable for stratified simple random sampling estimate. A numerical investigation has also been made to study
the amount of gain in efficiency that can be brough about by stratifying the population. 相似文献
8.
In a multivariate stratified sampling more than one characteristic are defined on every unit of the population. An optimum
allocation which is optimum for one characteristic will generally be far from optimum for others. A compromise criterion is
needed to work out a usable allocation which is optimum, in some sense, for all the characteristics. When auxiliary information
is also available the precision of the estimates of the parameters can be increased by using it. Furthermore, if the travel
cost within the strata to approach the units selected in the sample is significant the cost function remains no more linear.
In this paper an attempt has been made to obtain a compromise allocation based on minimization of individual coefficients
of variation of the estimates of various characteristics, using auxiliary information and a nonlinear cost function with fixed
budget. A new compromise criterion is suggested. The problem is formulated as a multiobjective all integer nonlinear programming
problem. A solution procedure is also developed using goal programming technique. 相似文献
9.
A general model for the randomized response (RR) method was introduced by Warner (J. Am. Stat. Assoc. 60:63–69, 1965) when a single-sensitive question is under study. However, since social surveys are often based on questionnaires containing more than one sensitive question, the analysis of multiple RR data is of considerable interest. In multivariate stratified surveys with multiple RR data the choice of optimum sample sizes from various strata may be viewed as a multiobjective nonlinear programming problem. The allocation thus obtained may be called a “compromise allocation” in sampling literature. This paper deals with the two-stage stratified Warner’s RR model applied to multiple sensitive questions. The problems of obtaining compromise allocations are formulated as multi-objective integer non linear programming problems with linear and quadratic cost functions as two separate problems. The solution to the formulated problems are achieved through goal programming technique. Numerical examples are presented to illustrate the computational details. 相似文献
10.
A comparison is made between the variance of the estimator of the total of a variable obtained from both a simple and a stratified random sampling, in which the sample sizes of some strata are equal to the stratum population size.It is shown that in this case, the advantage of the stratified sample could depend on the sample size. The paper presents inequalities that determine, as a function of the sample size, when the variance of the estimator obtained with simple sampling is lower than the variance obtained with the stratified sampling. The results give insight in order to prevent overstratification. 相似文献
11.
We develop a weighted local likelihood estimate for the parameters that govern the local spatial dependency of a locally stationary random field. The advantage of this local likelihood estimate is that it smoothly downweights the influence of faraway observations, works for irregular sampling locations, and when designed appropriately, can trade bias and variance for reducing estimation error. This paper starts with an exposition of our technique on the problem of estimating an unknown positive function when multiplied by a stationary random field. This example gives concrete evidence of the benefits of our local likelihood as compared to unweighted local likelihoods. We then discuss the difficult problem of estimating a bandwidth parameter that controls the amount of influence from distant observations. Finally we present a simulation experiment for estimating the local smoothness of a local Matérn random field when observing the field at random sampling locations in [0,1] 2. The local Matérn is a fully nonstationary random field, has a closed form covariance, can attain any degree of differentiability or Hölder smoothness and behaves locally like a stationary Matérn. We include an appendix that proves the positive definiteness of this covariance function. 相似文献
12.
Summary Murthy and Nanjamma [4] studied the problem of construction of almost unbiased ratio estimators for any sampling design using
the technique of interpenetrating subsamples. Subsequently, Rao [7], [8] has given a general method of constructing unbiased
ratio estimators by considering linear combinations of the two simple estimators based on the ratio of means and the mean
of ratios. However, it is difficult to choose an optimum weight (Rao [9]) which minimizes the variance of the combined estimator
since the weights are random in certain cases. In this note, we consider a different method of combining these estimators
and obtain a general class of almost unbiased ratio estimators of which Murthy and Nanjamma's is a particular case and derive
an optimum in this class. The case of simple random sampling where a similar class of almost unbiased ratio estimators can
be developed is briefly discussed. The results are illustrated by means of simple numerical examples. 相似文献
13.
In the multiple-output regression context, Hallin et al. (Ann Statist 38:635–669, 2010) introduced a powerful data-analytical tool based on regression quantile regions. However, the computation of these regions, that are obtained by considering in all directions an original concept of directional
regression quantiles, is a very challenging problem. Paindaveine and Šiman (Comput Stat Data Anal 2011b) described a first elegant solution relying on linear programming techniques. The present paper provides another solution
based on the fact that the quantile regions can also be computed from a competing concept of projection regression quantiles, elaborated in Kong and Mizera (Quantile tomography: using quantiles with multivariate data 2008) and Paindaveine and Šiman (J Multivar Anal 2011a). As a by-product, this alternative solution further provides various characteristics useful for statistical inference. We
describe in detail the algorithm solving the parametric programming problem involved, and illustrate the resulting procedure
on simulated data. We show through simulations that the Matlab implementation of the algorithm proposed in this paper is faster than that from Paindaveine and Šiman (Comput Stat Data Anal
2011b) in various cases. 相似文献
14.
We study a stratified multisite cluster‐sampling panel time series approach in order to analyse and evaluate the quality and reliability of produced items, motivated by the problem to sample and analyse multisite outdoor measurements from photovoltaic systems. The specific stratified sampling in spatial clusters reduces sampling costs and allows for heterogeneity as well as for the analysis of spatial correlations due to defects and damages that tend to occur in clusters. The analysis is based on weighted least squares using data‐dependent weights. We show that this does not affect consistency and asymptotic normality of the least squares estimator under the proposed sampling design under general conditions. The estimation of the relevant variance–covariance matrices is discussed in detail for various models including nested designs and random effects. The strata corresponding to damages or manufacturers are modelled via a quality feature by means of a threshold approach. The analysis of outdoor electroluminescence images shows that spatial correlations and local clusters may arise in such photovoltaic data. Further, relevant statistics such as the mean pixel intensity cannot be assumed to follow a Gaussian law. We investigate the proposed inferential tools in detail by simulations in order to assess the influence of spatial cluster correlations and serial correlations on the test's size and power. ©2016 The Authors. Applied Stochastic Models in Business and Industry published by John Wiley & Sons, Ltd. 相似文献
15.
Stochastic linear programs can be solved approximately by drawing a subset of all possible random scenarios and solving the
problem based on this subset, an approach known as sample average approximation (SAA). The value of the objective function
at the optimal solution obtained via SAA provides an estimate of the true optimal objective function value. This estimator
is known to be optimistically biased; the expected optimal objective function value for the sampled problem is lower (for
minimization problems) than the optimal objective function value for the true problem. We investigate how two alternative
sampling methods, antithetic variates (AV) and Latin Hypercube (LH) sampling, affect both the bias and variance, and thus
the mean squared error (MSE), of this estimator. For a simple example, we analytically express the reductions in bias and
variance obtained by these two alternative sampling methods. For eight test problems from the literature, we computationally
investigate the impact of these sampling methods on bias and variance. We find that both sampling methods are effective at
reducing mean squared error, with Latin Hypercube sampling outperforming antithetic variates. For our analytic example and
the eight test problems we derive or estimate the condition number as defined in Shapiro et al. (Math. Program. 94:1–19, 2002). We find that for ill-conditioned problems, bias plays a larger role in MSE, and AV and LH sampling methods are more likely
to reduce bias. 相似文献
16.
The two main and contradicting criteria guiding sampling design are accuracy of estimators and sampling costs. In stratified random sampling, the sample size must be allocated to strata in order to optimize both objectives. 相似文献
17.
In this paper we apply the Dirichlet HC and HD functions to a generalization of the sharing problem in which the population
is finite, and sampling is without replacement. In doing so we extend the Dirichlet HC and HD functions, and associated waiting
time results, from Sobel and Frankowski (Congressus Numerantium 106:171–191, 1995) to handle vector arguments. We also provide Maple procedures for their computation. Our results for the sharing problem
generalize the results for with replacement sampling given in Sobel and Frankowski (Am Math Mon 101:833–847, 1994a). 相似文献
18.
Considering the possible correlation between the characteristics (variables) in multivariate stratified random sampling, a modified Prékopa’s approach is suggested for the problem of optimum allocation in multivariate stratified random sampling. An example is solved by applying the proposed methodology. 相似文献
19.
We study the multi-stage portfolio selection problem where the utility function of an investor is ambiguous. The ambiguity is characterized by dynamic stochastic dominance constraints, which are able to capture the dynamics of the random return sequence during the investment process. We propose a multi-stage dynamic stochastic dominance constrained portfolio selection model, and use a mixed normal distribution with time-varying weights and the K-means clustering technique to generate a scenario tree for the transformation of the proposed model. Based on the scenario tree representation, we derive two linear programming approximation problems, using the sampling approach or the duality theory, which provide an upper bound approximation and a lower bound approximation for the original nonconvex problem. The upper bound is asymptotically tight with infinitely many samples. Numerical results illustrate the practicality and efficiency of the proposed new model and solution techniques. 相似文献
20.
We report a new optimal resolution for the statistical stratification problem under proportional sampling allocation among strata. Consider a finite population of N units, a random sample of n units selected from this population and a number L of strata. Thus, we have to define which units belong to each stratum so as to minimize the variance of a total estimator for one desired variable of interest in each stratum, and consequently reduce the overall variance for such quantity. In order to solve this problem, an optimal algorithm based on the concept of minimal path in a graph is proposed and assessed. Computational results using real data from IBGE (Brazilian Central Statistical Office) are provided. 相似文献
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