New two-stage sampling designs based on neoteric ranked set sampling

Neoteric ranked set sampling (NRSS) is a recently developed sampling plan, derived from the well-known ranked set sampling (RSS) scheme. It has already been proved that NRSS provides more efficient estimators for population mean and variance compared to RSS and other sampling designs based on ranked sets. In this work, we propose and evaluate the performance of some two-stage sampling designs based on NRSS. Five different sampling schemes are proposed. Through an extensive Monte Carlo simulation study, we verified that all proposed sampling designs outperform RSS, NRSS, and the original double RSS design, producing estimators for the population mean with a lower mean square error. Furthermore, as with NRSS, two-stage NRSS estimators present some bias for asymmetric distributions. We complement the study with a discussion on the relative performance of the proposed estimators. Moreover, an additional simulation based on data of the diameter and height of pine trees is presented.     

Smooth estimation of a reliability function in ranked set sampling

The authors develop a kernel-based estimator of a dynamic reliability measure for use with independent ranked set samples. The estimator is in the form of a ratio, whose numerator and denominator are shown to outperform their rivals based on simple random samples. Some asymptotic properties about the proposed estimator are also established. Simulation studies reveal finite-sample properties of the estimator. The technique is finally applied on an agricultural data set.     

On extropy properties of ranked set sampling

Ranked set sampling is a sampling design that allows the experimenter to span the full range values in the population and it can be used widely in industrial, environmental and ecological studies. In this paper, we consider the information content of ranked set sampling in terms of extropy measure. It is shown that the ranked set sampling performs better than its simple random sample counterpart of the same size. Monotone properties and stochastic orders are investigated. Sharp bounds on the extropy of RSS data based on the projection method in the non-parametric set-up as well as Steffensen inequalities in the parametric context are established. The extropy measure can also be used as a discrimination tool between RSS and SRS data.     

Reliability estimation based on ranked set sampling

In this paper we consider the problem of estimating the reliability of an exponential component based on a Ranked Set Sample (RSS) of size n. Given the first r observations of that sample, 1≤r≤n, we construct an unbiased estimator for this reliability and we show that these n unbiased estimators are the only ones in a certain class of estimators. The variances of some of these estimators are compared. By viewing the observations of the RSS of size n as the lifetimes of n independent k-out-of-n systems, 1≤k≤n, we are able to utilize known properties of these systems in conjunction with the powerful tools of majorization and Schur functions to derive our results.     

Improved estimation of scale parameters of morgenstern type bivariate uniform distribution using ranked set sampling

ABSTRACT

In this article we suggest some improved version of estimators of scale parameter of Morgenstern-type bivariate uniform distribution (MTBUD) based on the observations made on the units of the ranked set sampling regarding the study variable Y which is correlated with the auxiliary variable X, when (X, Y) follows a MTBUD. We also suggest some linear shrinkage estimators of scale parameter of Morgenstern type bivariate uniform distribution (MTBUD). Efficiency comparisons are also made in this work.     

Inference in the presence of ranking error in ranked set sampling

The author proposes inference techniques for ranked set sample data in the presence of judgment ranking errors. He bases his analysis on the models of Bohn & Wolfe (1994) and Frey (2007a, b), of which parameters are estimated by minimizing a distance measure. He then uses the fitted models to calibrate confidence intervals and tests. He shows the validity of his approach through simulation and illustrates its application through the construction of distribution‐free confidence intervals for the median area of apple tree leaves covered by a spray.     

Two-stage cluster sampling with hybrid ranked set sampling in the secondary sampling frame

In surveys of natural resources in agriculture, ecology, fisheries, forestry, environmental management, etc., cost-effective sampling methods are of major concern. In this paper, we propose a two-stage cluster sampling (TSCS) in integration with the hybrid ranked set sampling (HRSS)—named TSCS-HRSS—in the second stage of sampling for estimating the population mean. The TSCS-HRSS scheme encompasses several existing ranked set sampling (RSS) schemes and may help in selecting a smaller number of units to rank. It is shown both theoretically and numerically that the TSCS-HRSS provides an unbiased estimator of the population mean and it is more precise than the mean estimators based on TSCS with SRS and RSS schemes. An unbiased estimator of the variance of the proposed mean estimator is also derived. A similar trend is observed when studying the impact of imperfect rankings on the performance of the TSCS-HRSS based mean estimator.     

Performances of some goodness-of-fit tests for sampling designs in ranked set sampling

In this study, we consider different sampling designs of ranked set sampling (RSS) and give empirical distribution function (EDF) estimators for each sampling designs. We provide comparative graphs for the EDFs. Using these EDFs, power of five goodness-of-fit tests are obtained by Monte Carlo simulations for Tukey's g–h distributions under RSS and simple random sampling (SRS). Performances of these tests are compared with the tests based on the SRS. Also, critical values belong to these tests are obtained for different set and cycle sizes.     

Estimation of the distribution function under hybrid ranked set sampling

Recently, a hybrid ranked set sampling (HRSS) scheme has been proposed in the literature. The HRSS scheme encompasses several existing ranked set sampling (RSS) schemes, and it is a cost-effective alternative to the classical RSS and double RSS schemes. In this paper, we propose an improved estimator for estimating the cumulative distribution function (CDF) using HRSS. It is shown, both theoretically and numerically, that the CDF estimator under HRSS scheme is unbiased and its variance is always less than the variance of the CDF estimator with simple random sampling (SRS). An unbiased estimator of the variance of CDF estimator using HRSS is also derived. Using Monte Carlo simulations, we also study the performances of the proposed and existing CDF estimators under both perfect and imperfect rankings. It turns out that the proposed CDF estimator is by far a superior alternative to the existing CDF estimators with SRS, RSS and L-RSS schemes. For a practical application, a real data set is considered on the bilirubin level of babies in neonatal intensive care.     

Performance of coefficient of variation estimators in ranked set sampling

In this paper, we propose and evaluate the performance of different parametric and nonparametric estimators for the population coefficient of variation considering Ranked Set Sampling (RSS) under normal distribution. The performance of the proposed estimators was assessed based on the bias and relative efficiency provided by a Monte Carlo simulation study. An application in anthropometric measurements data from a human population is also presented. The results showed that the proposed estimators via RSS present an expressively lower mean squared error when compared to the usual estimator, obtained via Simple Random Sampling. Also, it was verified the superiority of the maximum likelihood estimator, given the necessary assumptions of normality and perfect ranking are met.     

Improved best linear unbiased estimators for the simple linear regression model using double ranked set sampling schemes

ABSTRACT

In this paper, we consider the best linear unbiased estimators (BLUEs) based on double ranked set sampling (DRSS) and ordered DRSS (ODRSS) schemes for the simple linear regression model with replicated observations. We assume three symmetric distributions for the random error term, i.e., normal, Laplace and some scale contaminated normal distributions. The proposed BLUEs under DRSS (BLUEs-DRSS) and ODRSS (BLUEs-ODRSS) are compared with the BLUEs based on ordered simple random sampling (OSRS), ranked set sampling (RSS), and ordered RSS (ORSS) schemes. These estimators are compared in terms of relative efficiency (RE), RE of determinant (RED), and RE of trace (RET). It is found that the BLUEs-ODRSS are uniformly better than the BLUEs based on OSRS, RSS, ORSS, and DRSS schemes. We also compare the estimators based on imperfect RSS (IRSS) schemes. It is worth mentioning here that the BLUEs under ordered imperfect DRSS (OIDRSS) are better than their counterparts based on IRSS, ordered IRSS (OIRSS), and imperfect DRSS (IDRSS) methods. Moreover, for sensitivity analysis of the BLUEs, we calculate REs and REDs of the BLUEs under the assumption of normality when in fact the parent distribution follows a non normal symmetric distribution. It turns out that even under violation of normality assumptions, BLUEs of the intercept and the slope parameters are found to be unbiased with equal REs under each sampling scheme. It is also observed that the BLUEs under ODRSS are more efficient than the existing BLUEs.     

A cautionary note on the analysis of randomized block designs with a few missing values

The randomized block design is routinely employed in the social and biopharmaceutical sciences. With no missing values, analysis of variance (AOV) can be used to analyze such experiments. However, if some data are missing, the AOV formulae are no longer applicable, and iterative methods such as restricted maximum likelihood (REML) are recommended, assuming block effects are treated as random. Despite the well-known advantages of REML, methods like AOV based on complete cases (blocks) only (CC-AOV) continue to be used by researchers, particularly in situations where routinely only a few missing values are encountered. Reasons for this appear to include a natural proclivity for non-iterative, summary-statistic-based methods, and a presumption that CC-AOV is only trivially less efficient than REML with only a few missing values (say≤10%). The purpose of this note is two-fold. First, to caution that CC-AOV can be considerably less powerful than REML even with only a few missing values. Second, to offer a summary-statistic-based, pairwise-available-case-estimation (PACE) alternative to CC-AOV. PACE, which is identical to AOV (and REML) with no missing values, outperforms CC-AOV in terms of statistical power. However, it is recommended in lieu of REMLonly if software to implement the latter is unavailable, or the use of a “transparent” formula-based approach is deemed necessary. An example using real data is provided for illustration.     

Estimating variances of strata in ranked set sampling

In RSS, the variance of observations in each ranked set plays an important role in finding an optimal design for unbalanced RSS and in inferring the population mean. The empirical estimator (i.e., the sample variance in a given ranked set) is most commonly used for estimating the variance in the literature. However, the empirical estimator does not use the information in the entire data over different ranked sets. Further, it is highly variable when the sample size is not large enough, as is typical in RSS applications. In this paper, we propose a plug-in estimator for the variance of each set, which is more efficient than the empirical one. The estimator uses a result in order statistics which characterizes the cumulative distribution function (CDF) of the rth order statistics as a function of the population CDF. We analytically prove the asymptotic normality of the proposed estimator. We further apply it to estimate the standard error of the RSS mean estimator. Both our simulation and empirical study show that our estimators consistently outperform existing methods.     

Estimation and prediction for spatial generalized linear mixed models using high order Laplace approximation

Estimation and prediction in generalized linear mixed models are often hampered by intractable high dimensional integrals. This paper provides a framework to solve this intractability, using asymptotic expansions when the number of random effects is large. To that end, we first derive a modified Laplace approximation when the number of random effects is increasing at a lower rate than the sample size. Second, we propose an approximate likelihood method based on the asymptotic expansion of the log-likelihood using the modified Laplace approximation which is maximized using a quasi-Newton algorithm. Finally, we define the second order plug-in predictive density based on a similar expansion to the plug-in predictive density and show that it is a normal density. Our simulations show that in comparison to other approximations, our method has better performance. Our methods are readily applied to non-Gaussian spatial data and as an example, the analysis of the rhizoctonia root rot data is presented.     

A Jonckheere-Terpstra-type test for perfect ranking in balanced ranked set sampling

Many methods based on ranked set sampling (RSS) assume perfect ranking of the samples. Here, by using the data measured by a balanced RSS scheme, we propose a nonparametric test for the assumption of perfect ranking. The test statistic that we use formally corresponds to the Jonckheere-Terpstra-type test statistic. We show formal relations of the proposed test for perfect ranking to other methods proposed recently in the literature. Through an empirical power study, we demonstrate that the proposed method performs favorably compared to many of its competitors.     

Some simple nonparametric methods to test for perfect ranking in ranked set sampling

A lot of research on ranked set sampling (RSS) is based on the assumption that the ranking is perfect. Hence, it is necessary to develop some tests that could be used to validate this assumption of perfect ranking. In this paper, we introduce some simple nonparametric methods for this purpose. We specifically define three test statistics, _Nk,_Sk$N_k,S_k$ and _Ak$A_k$, based on one-cycle RSS, which are all associated with the ordered ranked set sample (ORSS). We then derive the exact null distributions and exact power functions of all these tests. Next, by using the sum or the maximum of each statistic over all cycles, we propose six test statistics for the case of multi-cycle RSS. We compare the performance of all these tests with that of the Kolmogorov–Smirnov test statistic proposed earlier by Stokes and Sager [1988. Characterization of a ranked-set sample with application to estimating distribution functions. J. Amer. Statist. Assoc. 83, 35–42] and display that all proposed test statistics are more powerful. Finally, we present an example to illustrate the test procedures discussed here.     

Efficient estimators with categorical ranked set samples: estimation procedures for osteoporosis

Ranked set sampling (RSS) design as a cost-effective sampling is a powerful tool in situations where measuring the variable of interest is costly and time-consuming; however, ranking information about sampling units can be obtained easily through inexpensive and easy to measure characteristics at little or no cost. In this paper, we study RSS data for analysis of an ordinal population. First, we compare the problem of non-representative extreme samples under RSS and commonly-used simple random sampling. Using RSS data with tie information, we propose non-parametric and maximum likelihood estimators for population parameters. Through extensive numerical studies, we investigate the effect of various factors including ranking ability, tie generating mechanisms, the number of categories and population setting on the performance of the estimators. Finally, we apply the proposed methods to the bone disorder data to estimate the proportions of patients with osteopenia and osteoporosis status.     

Estimation of the parameters of Downton's bivariate exponential distribution using ranked set sampling scheme

The parameters of Downton's bivariate exponential distribution are estimated based on a ranked set sample. Parametric and nonparametric methods are considered. The suggested estimators are compared to the corresponding ones based on simple random sampling. It turns out that some of the suggested estimators are significantly more efficient than the ones based on simple random sampling.     

Estimating the population proportion in pair ranked set sampling with application to air quality monitoring

In this paper, we consider the problem of estimating the population proportion in pair ranked set sampling design. An unbiased estimator for the population proportion is proposed, and its theoretical properties are studied. It is shown that the estimator is more (less) efficient than its counterpart in simple random sampling (ranked set sampling). Asymptotic normality of the estimator is also established. Application of the suggested procedure is illustrated using a data set from an environmental study.