An omnibus two-sample test for ranked-set sampling data

We develop an omnibus two-sample test for ranked-set sampling (RSS) data. The test statistic is the conditional probability of seeing the observed sequence of ranks in the combined sample, given the observed sequences within the separate samples. We compare the test to existing tests under perfect rankings, finding that it can outperform existing tests in terms of power, particularly when the set size is large. The test does not maintain its level under imperfect rankings. However, one can create a permutation version of the test that is comparable in power to the basic test under perfect rankings and also maintains its level under imperfect rankings. Both tests extend naturally to judgment post-stratification, unbalanced RSS, and even RSS with multiple set sizes. Interestingly, the tests have no simple random sampling analog.     

Estimation of the population mean and median using truncation-based ranked set samples

In this paper, a new sampling method is suggested, namely truncation-based ranked set samples (TBRSS) for estimating the population mean and median. The suggested method is compared with the simple random sampling (SRS), ranked set sampling (RSS), extreme ranked set sampling (ERSS) and median-ranked set sampling (MRSS) methods. It is shown that for estimating the population mean when the underlying distribution is symmetric, TBRSS estimator is unbiased and it is more efficient than the SRS estimator based on the same number of measured units. For asymmetric distributions considered in this study, TBRSS estimator is more efficient than the SRS for all considered distributions except for exponential distribution when the selection coefficient gets large. When compared with ERSS and MRSS methods, TBRSS performs well with respect to ERSS for all considered distributions except for U(0, 1) distribution, while TBRSS efficiency is higher than that of MRSS for U(0, 1) distribution. For estimating the population median, the TBRSS estimators have higher efficiencies when compared with SRS and ERSS. A real data set is used to illustrate the suggested method.     

Efficiency bounds for a generalization of ranked-set sampling

Abstract

Partially rank-ordered set sampling (PROSS) is a generalization of ranked-set sampling (RSS) in which the ranker is not required to give a full ranking in each set. In this paper, we compare the efficiency of the sample mean as an estimator of the population mean under PROSS, RSS, and simple random sampling (SRS). We find that for fixed set size and total sample size, the efficiency of PROSS falls between that of SRS and that of RSS. We also develop a method for finding a sharp upper bound on the efficiency of PROSS relative to SRS for a particular design.     

Best linear unbiased estimates in ranked-set sampling with particular reference to imperfect ordering

SUMMARY Ranked-set sampling is a widely used sampling procedure when sample observations are expensive or difficult to obtain. It departs from simple random sampling by seeking to spread the observations in the sample widely over the distribution or population. This is achieved by ranking methods which may need to employ concomitant information. The ranked-set sample mean is known to be more efficient than the corresponding simple random sample mean. Instead of the ranked-set sample mean, this paper considers the corresponding optimal estimator: the ranked-set best linear unbiased estimator. This is shown to be more efficient, even for normal data, but particularly for skew data, such as from an exponential distribution. The corresponding forms of the estimators are quite distinct from the ranked-set sample mean. Improvement holds where the ordering is perfect or imperfect, with this prospect of improper ordering being explored through the use of concomitants. In addition, the corresponding optimal linear estimator of a scale parameter is also discussed. The results are applied to a biological problem that involves the estimation of root weights for experimental plants, where the expense of measurement implies the need to minimize the number of observations taken.     

Ranked-Set-Sample-Based Tests for Normal and Exponential Means

We consider the problems of testing a normal mean and an exponential mean under the median ranked set sampling (MRSS) scheme. The tests based on MRSS outperform those based on the usual ranked-set samples. In addition, through heuristic arguments we postulate working formulas for computing cut-off points for RSS-based and MRSS-based tests, respectively. Our simulation studies indicate these formulas perform surprisingly well, even for small samples, and solve the task of computation for large samples.     

An EWMA chart for monitoring process mean

In the statistical process control literature, there exists several improved quality control charts based on cost-effective sampling schemes, including the ranked set sampling (RSS) and median RSS (MRSS). A generalized cost-effective RSS scheme has been recently introduced for efficiently estimating the population mean, namely varied L RSS (VLRSS). In this article, we propose a new exponentially weighted moving average (EWMA) control chart for monitoring the process mean using VLRSS, named the EWMA-VLRSS chart, under both perfect and imperfect rankings. The EWMA-VLRSS chart encompasses the existing EWMA charts based on RSS and MRSS (named the EWMA-RSS and EWMA-MRSS charts). We use extensive Monte Carlo simulations to compute the run length characteristics of the EWMA-VLRSS chart. The proposed chart is then compared with the existing EWMA charts. It is found that, with either perfect or imperfect rankings, the EWMA-VLRSS chart is more sensitive than the EWMA-RSS and EWMA-MRSS charts in detecting small to large shifts in the process mean. A real dataset is also used to explain the working of the EWMA-VLRSS chart.     

Bootstrap confidence bands for the CDF using ranked-set sampling

In ranked-set sampling (RSS), a stratification by ranks is used to obtain a sample that tends to be more informative than a simple random sample of the same size. Previous work has shown that if the rankings are perfect, then one can use RSS to obtain Kolmogorov–Smirnov type confidence bands for the CDF that are narrower than those obtained under simple random sampling. Here we develop Kolmogorov–Smirnov type confidence bands that work well whether the rankings are perfect or not. These confidence bands are obtained by using a smoothed bootstrap procedure that takes advantage of special features of RSS. We show through a simulation study that the coverage probabilities are close to nominal even for samples with just two or three observations. A new algorithm allows us to avoid the bootstrap simulation step when sample sizes are relatively small.     

Statistical quality control based on ranked set sampling

Different quality control charts for the sample mean are developed using ranked set sampling (RSS), and two of its modifications, namely median ranked set sampling (MRSS) and extreme ranked set sampling (ERSS). These new charts are compared to the usual control charts based on simple random sampling (SRS) data. The charts based on RSS or one of its modifications are shown to have smaller average run length (ARL) than the classical chart when there is a sustained shift in the process mean. The MRSS and ERSS methods are compared with RSS and SRS data, it turns out that MRSS dominates all other methods in terms of the out-of-control ARL performance. Real data are collected using the RSS, MRSS, and ERSS in cases of perfect and imperfect ranking. These data sets are used to construct the corresponding control charts. These charts are compared to usual SRS chart. Throughout this study we are assuming that the underlying distribution is normal. A check of the normality for our example data set indicated that the normality assumption is reasonable.     

Improved exact confidence intervals for a proportion using ranked-set sampling

We develop new exact confidence intervals for a proportion using ranked-set sampling (RSS). The existing intervals arise from applying the method of Clopper and Pearson (1934) to the total number of successes. We improve on the existing intervals by using the method of Blaker (2000) and by replacing the total number of successes with the maximum likelihood estimator of the proportion. The new intervals outperform the existing intervals in terms of average expected length, and they are also good in an absolute sense, as they come within a few percentage points of a new theoretical bound on the average expected length. Like the existing intervals, the new intervals use a perfect rankings assumption. They are no longer exact under imperfect rankings, but provide coverage close to nominal for mild departures from perfect rankings.     

A Note on Fisher Information and Imperfect Ranked-Set Sampling

We show by example that the Fisher information in an imperfect ranked-set sample may be higher than the Fisher information in a perfect ranked-set sample. This corrects certain misconceptions in the literature. The example also provides an additional counterexample to a common claim about the relationship between imperfect rankings and perfect rankings.     

An algorithm with applications in ranked-set sampling

A mixture of order statistics is a random variable whose distribution is a finite mixture of the distributions for order statistics. Such mixtures show up in the literature on ranked-set sampling and related sampling schemes as models for imperfect rankings. In this paper, we derive an algorithm for computing the probability that independent mixtures of order statistics come in a particular order. The algorithm is far faster than previous proposals from the literature. As an application, we show that the algorithm can be used to create Kolmogorov–Smirnov-type confidence bands that adjust for the presence of imperfect rankings.     

More powerful sign test using median ranked set sample: Finite sample power comparison

Sign test using median ranked set samples (MRSS) is introduced and investigated. We show that, this test is more powerful than the sign tests based on simple random sample (SRS) and ranked set sample (RSS) for finite sample size. It is found that, when the set size of MRSS is odd, the null distribution of the MRSS sign test is the same as the sign test obtained by using SRS. The exact null distributions and the power functions, in case of finite sample sizes, of these tests are derived. Also, the asymptotic distribution of the MRSS sign tests are derived. Numerical comparison of the MRSS sign test power with the power of the SRS sign test and the RSS sign test is given. Illustration of the procedure, using real data set of bilirubin level in Jaundice babies who stay in neonatal intensive care is introduced.     

Estimating the population proportion in modified ranked set sampling methods

In this paper, proportion estimators and associated variance estimators are proposed for a binary variable with a concomitant variable based on modified ranked set sampling methods, which are extreme ranked set sampling (ERSS), median ranked set sampling (MRSS), percentile ranked set sampling (Per-RSS) and L ranked set sampling (LRSS) methods. The Monte Carlo simulation study is performed to compare the performance of the estimators based on bias, mean squared error, and relative efficiency for different levels of correlation coefficient, set and cycle sizes under normal and log-normal distributions. Moreover, the study is supported with real data application.     

A review of nonparametric ranked-set sampling methodology

Ranked-set sampling is an alternative to random sampling for settings in which measurements are difficult or costly. Ranked-set sampling utilizes information gained without measurement to structure the eventual measured sample. This additional information yields improved properties for ranked-set sample procedures relative to their simple random sample counterparts. We review the available nonparametric procedures for data from ranked-set samples. Estimation of the distribution function was the first nonparametric setting to which ranked-set sampling methodology was applied. Since the first paper on the ranked-set sample empirical distribution function, the two-sample location setting, the sign test, and the signed-rank test have all been examined for ranked-set samples. In addition, estimation of the distribution function has been considered in a more general setting. We discuss the similarities and differences in the properties of the ranked-set sample procedures for the various settings     

Ranked set sampling with unequal samples for skew distributions

A ranked set sampling procedure with unequal samples for positively skew distributions (RSSUS) is proposed and used to estimate the population mean. The estimators based on RSSUS are compared with the estimators based on ranked set sampling (RSS) and median ranked set sampling (MRSS) procedures. It is observed that the relative precisions of the estimators based on RSSUS are higher than those of the estimators based on RSS and MRSS procedures.     

Best linear unbiased and invariant estimation in location-scale families based on double-ranked set sampling

Abstract

In this article, we propose the best linear unbiased estimators (BLUEs) and best linear invariant estimators (BLIEs) for the unknown parameters of location-scale family of distributions based on double-ranked set sampling (DRSS) using perfect and imperfect rankings. These estimators are then compared with the BLUEs and BLIEs based on ranked set sampling (RSS). It is shown that under perfect ranking, the proposed estimators are uniformly better than the BLUEs and BLIEs obtained via RSS. We also propose the best linear unbiased quantile (BLUQ) and the best linear invariant quantile (BLIQ) estimators for normal distribution under DRSS. It is observed that the proposed quantile estimators are more efficient than the BLUQ and BLIQ estimators based on RSS for both perfect and imperfect orderings.     

Multistage ranked set sampling

The superiority of using ranked set sampling, for estimating the mean of a population, over simple random sampling, is well established. This technique is useful when visual ordering of a small set of size (m) can be done easily and fairly accurately, but exact measurement of an observation is difficult and expensive. It is noted that for many distributions, an increase in the efficiency of ranked set sampling can be achieved by increasing the set size m. However, in practice, m should be kept very small so that visual ranking errors will not destroy the gain in efficiency. In this paper, multistage ranked set sampling is considered as a generalization of ranked set sampling, that results in an increase of the efficiency for fixed value of m. Steady state efficiency, the limiting efficiency as the number of stages approaches infinity, varies from one distribution to another. It is shown that this efficiency is always larger than 1, close to m² for symmetric distributions and equal to m² for the uniform distribution. Some real applications of the technique are discussed. Data on olive yield of olive trees is collected to illustrate the technique.     

Recursive computation of inclusion probabilities in ranked-set sampling

We derive recursive algorithms for computing first-order and second-order inclusion probabilities for ranked-set sampling from a finite population. These algorithms make it practical to compute inclusion probabilities even for relatively large sample and population sizes. As an application, we use the inclusion probabilities to examine the performance of Horvitz-Thompson estimators under different varieties of balanced ranked-set sampling. We find that it is only for balanced Level 2 sampling that the Horvitz-Thompson estimator can be relied upon to outperform the simple random sampling mean estimator.     

Modified Maximum Likelihood Estimator of Scale Parameter Using Moving Extremes Ranked Set Sampling

A modified maximum likelihood estimator (MMLE) of scale parameter is considered under moving extremes ranked set sampling (MERSS), and its properties are obtained. For some usual scale distributions, we obtain explicit form of the MMLE and prove the MMLE is an unbiased estimator under MERSS. The simulation results show that the MMLE using MERSS is always more efficient than the MLE using simple random sampling, when the same sample size is used. The simulation results also show that the loss of efficiency in using the MMLE instead of the MLE is very small for small sample.     

A NONPARAMETRIC TEST OF SYMMETRY VERSUS ASYMMETRY FOR RANKED-SET SAMPLES

This paper introduces a nonparametric test of symmetry for ranked-set samples to test the asymmetry of the underlying distribution. The test statistic is constructed from the Cramér-von Mises distance function which measures the distance between two probability models. The null distribution of the test statistic is established by constructing symmetric bootstrap samples from a given ranked-set sample. It is shown that the type I error probabilities are stable across all practical symmetric distributions and the test has high power for asymmetric distributions.