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.     

Improved CUSUM charts for monitoring process mean

In this paper, we propose new cumulative sum (CUSUM) and Shewhart-CUSUM (SCUSUM) control charts for monitoring the process mean using ranked-set sampling (RSS) and ordered RSS (ORSS) schemes. The proposed CUSUM charts include the Crosier's CUSUM (CCUSUM) and Shewhart-CCUSUM (SCCUSUM) charts using RSS, and the CUSUM, CCUSUM, SCUSUM and SCCUSUM charts using ORSS. Moreover, fast initial response features are also attached with these CUSUM charts to improve their sensitivities for an initial out-of-control situation. Monte Carlo simulations are used to compute the run length characteristics of the proposed CUSUM charts. Upon comparing the run length performances of the CUSUM charts, it turns out that the proposed CUSUM charts are more sensitive than their existing counterparts. A real dataset is used to explain the implementation of the proposed CUSUM charts.     

Effect of measurement error on exponentially weighted moving average control charts under ranked set sampling schemes

Control charts are a powerful statistical process monitoring tool often used to monitor the stability of manufacturing processes. In quality control applications, measurement errors adversely affect the performance of control charts. In this paper, we study the effect of measurement error on the detection abilities of the exponentially weighted moving average (EWMA) control charts for monitoring process mean based on ranked set sampling (RSS), median RSS (MRSS), imperfect RSS (IRSS) and imperfect MRSS (IMRSS) schemes. We also study the effect of multiple measurements and non-constant error variance on the performances of the EWMA control charts. The EWMA control chart based on simple random sampling is compared with the EWMA control charts based on RSS, MRSS, IRSS and IMRSS schemes. The performances of the EWMA control charts are evaluated in terms of out-of-control average run length and standard deviation of run lengths. It turns out that the EWMA control charts based on MRSS and IMRSS schemes are better than their counterparts for all measurement error cases considered here.     

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.     

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.     

Improved quality control charts for monitoring the process mean,using double-ranked set sampling methods

Statistical control charts are widely used in the manufacturing industry. The Shewhart-type control charts are developed to improve the monitoring process mean by using the double quartile-ranked set sampling, quartile double-ranked set sampling, and double extreme-ranked set sampling methods. In terms of the average run length, the performance of the proposed control charts are compared with the existing control charts based on simple random sampling, ranked set sampling and extreme-ranked set sampling methods. An application of real data is also considered to investigate the performance of the suggested process mean control charts. The findings of the study revealed that the newly suggested control charts are superior to the existing counterparts.     

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.     

A new sampling method for estimating the population mean

A double L ranked set sampling (DLRSS) method is suggested for estimating the population mean. The DLRSS is compared with the simple random sampling (SRS), ranked set sampling (RSS) and L ranked set sampling (LRSS) methods based on the same number of measured units. The conditions for which the suggested estimator performs better than the other estimators are derived. It is found that, the suggested DLRSS estimator is an unbiased of the population mean, and is more efficient than its counterparts using SRS, RSS, and LRSS methods. Real data sets are used for illustration.     

Quality Control Chart for the Mean using Double Ranked Set Sampling

In this paper, an attempt is made to develop Quality Control Charts for monitoring the process mean based on Double Ranked Set Sampling (DRSS) rather than the traditional Simple Random Sampling (SRS). Considering a normal population and several shift values, the performance of the Average Run Length (ARL) of these new charts was compared with the control charts based on Ranked Set Sampling (RSS) and SRS with the same number of observations. It is shown that the new charts do a better job of detecting changes in process mean compared with SRS and RSS.     

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.     

Ordered partially ordered judgment subset sampling with applications to parametric inference

ABSTRACT

In this paper, we use the idea of order statistics from independent and non-identically distributed random variables to propose ordered partially ordered judgment subset sampling (OPOJSS) and then develop optimal linear parametric inferences. The best linear unbiased and invariant estimators of the location and scale parameters of a location-scale family are developed based on OPOJSS. It is shown that, despite the presence or absence of ranking errors, the proposed estimators with OPOJSS are uniformly better than the existing estimators with simple random sampling (SRS), ranked set sampling (RSS), ordered RSS (ORSS) and partially ordered judgment subset sampling (POJSS). Moreover, we also derive the best linear unbiased estimators (BLUEs) of the unknown parameters of the simple linear regression model with replicated observations using POJSS and OPOJSS. It is found that the BLUEs with OPOJSS are more precise than the BLUEs based on SRS, RSS, ORSS and POJSS.     

Efficiency of ranked set sampling in entropy estimation and goodness-of-fit testing for the inverse Gaussian law

When measuring units are expensive or time consuming, while ranking them is relatively easy and inexpensive, it is known that ranked set sampling (RSS) is preferable to simple random sampling (SRS). Many authors have suggested several extensions of RSS. As a variation, Al-Saleh and Al-Kadiri [Double ranked set sampling, Statist. Probab. Lett. 48 (2000), pp. 205–212] introduced double ranked set sampling (DRSS) and it was extended by Al-Saleh and Al-Omari [Multistage ranked set sampling, J. Statist. Plann. Inference 102 (2002), pp. 273–286] to multistage ranked set sampling (MSRSS). The entropy of a random variable (r.v.) is a measure of its uncertainty. It is a measure of the amount of information required on the average to determine the value of a (discrete) r.v.. In this work, we discuss entropy estimation in RSS design and aforementioned extensions and compare the results with those in SRS design in terms of bias and root mean square error (RMSE). Motivated by the above observed efficiency, we continue to investigate entropy-based goodness-of-fit test for the inverse Gaussian distribution using RSS. Critical values for some sample sizes determined by means of Monte Carlo simulations are presented for each design. A Monte Carlo power analysis is performed under various alternative hypotheses in order to compare the proposed testing procedure with the existing methods. The results indicate that tests based on RSS and its extensions are superior alternatives to the entropy test based on SRS.     

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.     

Paired double-ranked set sampling

Abstract

In environmental monitoring and assessment, the main focus is to achieve observational economy and to collect data with unbiased, efficient and cost-effective sampling methods. Ranked set sampling (RSS) is one traditional method that is mostly used for accomplishing observational economy. In this article, we propose an unbiased sampling scheme, named paired double RSS (PDRSS) for estimating the population mean. We study the performance of the mean estimators under PDRSS based on perfect and imperfect rankings. It is shown that, for perfect ranking, the variance of the mean estimator under PDRSS is always less than the variance of mean estimator based on simple random sampling, paired RSS and RSS. The mean estimators under RSS, median RSS, PDRSS, and double RSS are also compared with the regression estimator of population mean based on SRS. The procedure is also illustrated with a case study using a real data set.     

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.     

Partial groups ranked set sampling and mean estimation

Ranked set sampling (RSS) is an advanced sampling method which is very effective for estimating mean of the population when exact measurement of observation is difficult and/or expensive. Balanced Groups RSS (BGRSS) is one of the modification of RSS where only the lowest, the median and the largest ranked units are taken into account. Although BGRSS is advantageous and useful for some specific cases, it has strict restrictions regarding the set size which could be problematic for sampling plans. In this study, we make an improvement on BGRSS and propose a new design called Partial Groups RSS which offers a more flexible sampling plan providing the independence of the set size and sample size. Partial Groups RSS also has a cost advantage over BGRSS. We construct a Monte Carlo simulation study comparing the performance of the mean estimators of the proposed sampling design and BGRSS according to their sampling costs and mean squared errors for various type of distributions. In addition, we give a biometric data application for investigating the efficiency of Partial Groups RSS in real life applications.     

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.     

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.     

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 new adaptive EWMA control chart using auxiliary information for monitoring the process mean

The adaptive memory-type control charts, including the adaptive exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts, have gained considerable attention because of their excellent speed in providing overall good detection over a range of mean shift sizes. In this paper, we propose a new adaptive EWMA (AEWMA) chart using the auxiliary information for efficiently monitoring the infrequent changes in the process mean. The idea is to first estimate the unknown process mean shift using an auxiliary information based mean estimator, and then adaptively update the smoothing constant of the EWMA chart. Using extensive Monte Carlo simulations, the run length profiles of the AEWMA chart are computed and explored. The AEWMA chart is compared with the existing control charts, including the classical EWMA, CUSUM, synthetic EWMA and synthetic CUSUM charts, in terms of the run length characteristics. It turns out that the AEWMA chart performs uniformly better than these control charts when detecting a range of mean shift sizes. An illustrative example is also presented to demonstrate the working and implementation of the proposed and existing control charts.