Estimation of Population Mean in the Presence of Linear Trend

The present article deals with some methods for estimation of finite populations means in the presence of linear trend among the population values. As a result, we provided a strategy for the selection of sampling interval k for the case of circular systematic sampling, which ensures better estimator for the population mean compared to other choices of the sampling interval. This has been established based on empirical studies. Further we more, applied multiple random starts methods for selecting random samples for the case of linear systematic sampling and diagonal systematic sampling schemes. We also derived the explicit expressions for the variances and their estimates. The relative performances of simple random sampling, linear systematic sampling and diagonal systematic sampling schemes with single and multiple random starts are also assessed based on numerical examples.     

Estimation under systematic sampling schemes for parabolic populations

In this paper, three sampling-estimating strategies involving linear, balanced and modified systematic sampling are considered for the estimation of a finite population total in the presence of parabolic trend. Using appropriate super-population models, their performances are evaluated. For super-population models with constant variance, Yates corrected estimator under linear systematic sampling is shown to perform well. Choices of variance functions under which modified and balanced systematic sampling perform well are also identified based on extensive numerical studies.     

Remainder modified systematic sampling in the presence of linear trend

If the population size is not a multiple of the sample size, then the usual linear systematic sampling design is unattractive, since the sample size obtained will either vary, or be constant and different to the required sample size. Only a few modified systematic sampling designs are known to deal with this problem and in the presence of linear trend, most of these designs do not provide favorable results. In this paper, a modified systematic sampling design, known as remainder modified systematic sampling (RMSS), is introduced. There are seven cases of RMSS and the results in this paper suggest that the proposed design is favorable, regardless of each case, while providing linear trend-free sampling results for three of the seven cases. To obtain linear trend-free sampling for the other cases and thus improve results, an end corrections estimator is constructed.     

Multiple-start balanced modified systematic sampling in the presence of linear trend

ABSTRACT

In this paper, we propose a sampling design termed as multiple-start balanced modified systematic sampling (MBMSS), which involves the supplementation of two or more balanced modified systematic samples, thus permitting us to obtain an unbiased estimate of the associated sampling variance. There are five cases for this design and in the presence of linear trend only one of these cases is optimal. To further improve results for the other cases, we propose an estimator that removes linear trend by applying weights to the first and last sampling units of the selected balanced modified systematic samples and is thus termed as the MBMSS with end corrections (MBMSSEC) estimator. By assuming a linear trend model averaged over a super-population model, we will compare the expected mean square errors (MSEs) of the proposed sample means, to that of simple random sampling (SRS), linear systematic sampling (LSS), stratified random sampling (STR), multiple-start linear systematic sampling (MLSS), and other modified MLSS estimators. As a result, MBMSS is optimal for one of the five possible cases, while the MBMSSEC estimator is preferred for three of the other four cases.     

Remainder Markov systematic sampling

Systematic sampling is the simplest and easiest of the most common sampling methods. However, when the population size N cannot be evenly divided by the sampling size n, systematic sampling cannot be performed. Not only is it difficult to determine the sampling interval k equivalent to the sampling probability of the sampling unit, but also the sample size will be inconstant and the sample mean will be a biased estimator of the population mean. To solve this problem, this paper introduces an improved method for systematic sampling: the remainder Markov systematic sampling method. This new method involves separately finding the first-order and second-order inclusion probabilities. This approach uses the Horvitz-Thompson estimator as an unbiased estimator of the population mean to find the variance of the estimator. This study examines the effectiveness of the proposed method for different super-populations.     

Design effect of randomized systematic sampling

In statistical practice, systematic sampling (SYS) is used in many modifications due to its simple handling. In addition, SYS may provide efficiency gains if it is well adjusted to the structure of the population under study. However, if SYS is based on an inappropriate picture of the population a high decrease of efficiency, i.e. a high increase in variance may result by changing from simple random sampling to SYS. In the context of two-stage designs SYS so far seems often in use for subsampling within the primary units. As an alternative to this practice, we propose to randomize the order of the primary units, then to select systematically a number of primary units and, thereafter, to draw secondary units by simple random sampling without replacement within the primary units selected. This procedure is more efficient than simple random sampling with replacement from the whole population of all secondary units, i.e. the variance of an adequate estimator for a total is never increased by changing from simple random sampling to randomized SYS whatever be the values associated by a characteristic with the secondary units, while there are values for which the variance decreases for the change mentioned. This result should hold generally, even if our proof, so far, is not complete for general sample sizes.     

Effect of systematic sampling on arima models

Given a general homogeneous non-stationary autoregressive integrated moving average process ARIMA(p,d,q), the corresponding model for the subseries obtained by a systematic sampling is derived. The article then shows that the sampled subseries approaches approximately to an integrated moving average process IMA(d,l), l≤(d-l), regardless of the autoregressive and moving average structures in the original series. In particular, the sampled subseries from an ARIMA (p,l,q) process approaches approximately to a simple random walk model.     

Two-dimensional balanced sampling plans excluding adjacent units under sharing a border and island adjacency schemes

In Balanced Sampling Plans Excluding Adjacent Units (BSA plans) first order inclusion probabilities are equal and second order inclusion probabilities are zero for pairs of units at a distance less than or equal to m and are constant for pairs of units which are at a distance greater than m. These plans are useful when the adjacent units in the population provide similar information. In this paper, an algorithm based on linear programming approach has been developed for construction of two-dimensional BSA plans under sharing a border and island adjacency schemes for m ≤ 2. Some results on existence of such BSA plans have also been obtained for each adjacency scheme separately.     

Systematic sampling of periodic functions

Using Fourier series expansions of functions of one and two variables, we find the variances of the sample mean for random and systematic sampling methods in one and two dimensions.     

Systematic and nonrandom sampling in the presence of linear trends

In the case of a finite population with a linear trend, improvements in estimation procedures are suggested for several well-known systematic samplingschemes. The use of the least squares regression estimator is examined and a sampling scheme which reduces its expected mean squared error is proposed.     

Double bootstrap estimation of variance under systematic sampling with probability proportional to size

Variance estimation under systematic sampling with probability proportional to size is known to be a difficult problem. We attempt to tackle this problem by the bootstrap resampling method. It is shown that the usual way to bootstrap fails to give satisfactory variance estimates. As a remedy, we propose a double bootstrap method which is based on certain working models and involves two levels of resampling. Unlike existing methods which deal exclusively with the Horvitz–Thompson estimator, the double bootstrap method can be used to estimate the variance of any statistic. We illustrate this within the context of both mean and median estimation. Empirical results based on five natural populations are encouraging.     

Some remarks on an admissibility condition

Summary In this note we deal with some admissibility conditions proved by G. B. Tranquilli to be sufficient in the class of unbiased estimators of finite population parameters and with respect to (w.r.t.) a quadratic loss function. We show that the same conditions:i) are sufficient for the admissibility of an unbiased estimator with any loss function;ii) imply hyperadmissibility with reference to a particular (critical) population of the. From this fact we deduce that, for a fixed critical population, there is at most one estimator, in the class of all unbiased estimator of a finite population parameter, which satisfies Tranquilli condition. This research was partially supported by a M.U.R.S.T. grant ?Metodi inferenziali basati sul ricampionamento?.     

Equally spaced design points in polynomial regression: A comparison of systematic sampling methods with the optimal design of experiments

In some situations an experimenter may desire to have equally spaced design points. Three methods of obtaining such points on the interval [—1,1]—namely systematic random sampling, centrally located systematic sampling, and a purposive systematic sampling method which includes the endpoints - 1 and 1 as two of the design points-are evaluated under the D-optimal and G-optimal criteria. These methods are also compared to the optimal designs in polynomial regression and to the limiting designs of Kiefer and Studden (1976).     

Control charts for location based on different sampling schemes

Control charts are the most important statistical process control tool for monitoring variations in a process. A number of articles are available in the literature for the X? control chart based on simple random sampling, ranked set sampling, median-ranked set sampling (MRSS), extreme-ranked set sampling, double-ranked set sampling, double median-ranked set sampling and median double-ranked set sampling. In this study, we highlight some limitations of the existing ranked set charting structures. Besides, we propose different runs rules-based control charting structures under a variety of sampling strategies. We evaluate the performance of the control charting structures using power curves as a performance criterion. We observe that the proposed merger of varying runs rules schemes with different sampling strategies improve significantly the detection ability of location control charting structures. More specifically, the MRSS performs the best under both single- and double-ranked set strategies with varying runs rules schemes. We also include a real-life example to explain the proposal and highlight its significance for practical data sets.     

Some remarks on a ridge-type-estimator and good prior means

Pliskin (1987) compared modified ridge regression estimators based on prior information with respect to their mean square error matrices. A further characterization of good prior mean is given here, and the case of different ridge parameters is also considered.     

Circular versions of systematic sampling in the presence of linear trend

In this paper, an alternative approach was employed and derived the explicit expressions of sample mean and its variance for the circular and the diagonal circular systematic schemes since these expressions were unavailable in the existing literature. The study also found the corrected estimator of population mean and average variance under super-population model for these scheme. It further revealed that the circular systematic sampling almost performed better than that of the diagonal circular systematic sampling in both linear and auto-correlated super-population models.     

A review of optimal designs in survey sampling

Results in five areas of survey sampling dealing with the choice of the sampling design are reviewed. In Section 2, the results and discussions surrounding the purposive selection methods suggested by linear regression superpopulation models are reviewed. In Section 3, similar models to those in the previous section are considered; however, random sampling designs are considered and attention is focused on the optimal choice of π_j. Then in Section 4, systematic sampling methods obtained under autocorrelated superpopulation models are reviewed. The next section examines minimax sampling designs. The work in the final section is based solely on the randomization. In Section 6 methods of sample selection which yield inclusion probabilities π_j = n/N and π_ij = n(n - 1)/N(N - 1), but for which there are fewer than _NC_n possible samples, are mentioned briefly.     

Properties of designs for sampling continuous spatial resources from a triangular grid

Precision of systematic designs for sampling continuous response variables defined over a continuous spatial region have been extensively investigated. Estimation of variance, particularly for triangular grids, has been given less attention. Two of the designs proposed for sampling environmental resources in the Environmental Protection Agency's Environmental Monitoring and Assessment Program (EMAP) are based on a triangular grid. Implementation of either design requires derivation and evaluation of a variance estimator adequate for the purposes of EMAP. The performance of the proposed variance estimator was assessed under various surface model representations of the continuous response variable. Of the designs examined, a tessellation-stratified design permitted better estimation of variance, and was generally more precise, than a strict systematic design.     

Efficiency of schemes of sampling from the inverse gaussian-weibull mixture model

The objective of this paper is to study the efficiency of sampling schemes suggested by Hosmer(1973), termed models Ml and M2, relative to the regular random sampling, termed model MO, when samples are drawn from a population having the Inverse Gaussian-Weibull (IG-W) mixture distribution.

It has been shown that whether the efficiency is based on relative variances of the maximum likelihood estimates (ML,E's) of the components of the vector of parameters or on the generalized variances of the MLE's of that vector, Hosmer's models Ml or M2 perform better than model MO.     

Some count-based nonparametric tests for circular symmetry of a bivariate distribution

In this paper, we revisit the problem of testing of the hypothesis of circular symmetry of a bivariate distribution. We propose some nonparametric tests based on sector counts. These include tests based on chi-square goodness-of-fit test, the classical likelihood ratio, mean deviation, and the range. The proposed tests are easy to implement and the exact null distributions for small sample sizes of the test statistics are obtained. Two examples with small and large data sets are given to illustrate the application of the tests proposed. For small and moderate sample sizes, the performances of the proposed tests are evaluated using empirical powers (empirical sizes are also reported). Also, we evaluate the performance of these count-based tests with adaptations of several well-known tests such as the Kolmogorov–Smirnov-type tests, tests based on kernel density estimator, and the Wilcoxon-type tests. It is observed that among the count-based tests the likelihood ratio test performs better.