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.     

Regression estimators in extreme and median ranked set samples

The ranked set sampling (RSS) method as suggested by McIntyre (1952) may be modified to come up with new sampling methods that can be made more efficient than the usual RSS method. Two such modifications, namely extreme and median ranked set sampling methods, are considered in this study. These two methods are generally easier to use in the field and less prone to problems resulting from errors in ranking. Two regression-type estimators based on extreme ranked set sampling (ERSS) and median ranked set sampling (MRSS) for estimating the population mean of the variable of interest are considered in this study and compared with the regression-type estimators based on RSS suggested by Yu & Lam (1997). It turned out that when the variable of interest and the concomitant variable jointly followed a bivariate normal distribution, the regression-type estimator of the population mean based on ERSS dominates all other estimators considered.     

Estimation of mean based on modified robust extreme ranked set sampling

In this paper, double robust extreme ranked set sampling (DRERSS) and its properties for estimating the population mean are considered. It turns out that, when the underlying distribution is symmetric, DRERSS gives unbiased estimators of the population mean. Also, it is found that DRERSS is more efficient than the simple random sampling (SRS), ranked set sampling (RSS), and extreme ranked set sampling (ERSS) methods. For asymmetric distributions considered in this study, the DRERSS has a small bias and it is more efficient than SRS, RSS, and ERSS. A real data set is used to illustrate the DRERSS method.     

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.     

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.     

Control charts for attributes with maxima nominated samples

We develop quality control charts for attributes using the maxima nomination sampling (MNS) method and compare them with the usual control charts based on simple random sampling (SRS) method, using average run length (ARL) performance, the required sample size in detecting quality improvement, and non-existence region for control limits. We study the effect of the sample size, the set size, and nonconformity proportion on the performance of MNS control charts using ARL curve. We show that MNS control chart can be used as a better benchmark for indicating quality improvement or quality deterioration relative to its SRS counterpart. We consider MNS charts from a cost perspective. We also develop MNS attribute control charts using randomized tests. A computer program is designed to determine the optimal control limits for an MNS p-chart such that, assuming known parameter values, the absolute deviation between the ARL and a specific nominal value is minimized. We provide good approximations for the optimal MNS control limits using regression analysis. Theoretical results are augmented with numerical evaluations. These show that MNS based control charts can yield substantial improvement over the usual control charts based on SRS.     

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.     

Robust extreme ranked set sampling

In this paper, a robust extreme ranked set sampling (RERSS) procedure for estimating the population mean is introduced. It is shown that the proposed method gives an unbiased estimator with smaller variance, provided the underlying distribution is symmetric. However, for asymmetric distributions a weighted mean is given, where the optimal weights are computed by using Shannon's entropy. The performance of the population mean estimator is discussed along with its properties. Monte Carlo simulations are used to demonstrate the performance of the RERSS estimator relative to the simple random sample (SRS), ranked set sampling (RSS) and extreme ranked set sampling (ERSS) estimators. The results indicate that the proposed estimator is more efficient than the estimators based on the traditional sampling methods.     

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.     

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.     

Robust extreme double ranked set sampling

As a well-known method for selecting representative samples of populations, ranked set sampling (RSS) has been considered increasingly in recent years. This (RSS) method has proved to be more efficient than the usual simple random sampling (SRS) for estimating most of the population parameters. In order to have a more efficient estimate of the population mean, a new sampling scheme called as robust extreme double ranked set sampling (REDRSS) is introduced and investigated in this paper. A simulation study shows that using REDRSS scheme gives more efficient estimates of population mean with smaller variance than the usual SRS, RSS and most other sampling schemes based on RSS estimators in non-uniform (symmetric or non-symmetric) distributions.     

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.     

Selected Ranked Set Sampling

This paper proposes a sampling procedure called selected ranked set sampling (SRSS), in which only selected observations from a ranked set sample (RSS) are measured. This paper describes the optimal linear estimation of location and scale parameters based on SRSS, and for some distributions it presents the required tables for optimal selections. For these distributions, the optimal SRSS estimators are compared with the other popular simple random sample (SRS) and RSS estimators. In every situation the estimators based on SRSS are found advantageous at least in some respect, compared to those obtained from SRS or RSS. The SRSS method with errors in ranking is also described. The relative precision of the estimator of the population mean is investigated for different degrees of correlations between the actual and erroneous ranking. The paper reports the minimum value of the correlation coefficient between the actual and the erroneous ranking required for achieving better precision with respect to the usual SRS estimator and with respect to the RSS estimator.     

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.     

On the Kaplan–Meier estimator based on ranked set samples

When quantification of all sampling units is expensive but a set of units can be ranked, without formal measurement, ranked set sampling (RSS) is a cost-efficient alternate to simple random sampling (SRS). In this paper, we study the Kaplan–Meier estimator of survival probability based on RSS under random censoring time setup, and propose nonparametric estimators of the population mean. We present a simulation study to compare the performance of the suggested estimators. It turns out that RSS design can yield a substantial improvement in efficiency over the SRS design. Additionally, we apply the proposed methods to a real data set from an environmental study.     

Estimation of the means of the bivariate normal using moving extreme ranked set sampling with concomitant variable

The estimation of the means of the bivariate normal distribution, based on a sample obtained using a modification of the moving extreme ranked set sampling technique (MERSS) is considered. The modification involves using a concomitant random variable. Nonparametric-type methods as well as the maximum likelihood estimation are considered. The estimators obtained are compared to their counterparts based on simple random sampling (SRS). It appears that the suggested estimators are more efficient. Also, MERSS with concomitant variable is easier to use in practice than the usual ranked set sampling (RSS) with concomitant variable. The issue of robustness of the procedure is addressed. Real trees data set is used for illustration.     

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.     

Estimation of bivariate characteristics using ranked set sampling

The superiority of ranked set sampling (RSS) over simple random sampling (SRS) for estimating the mean of a population is well known. This paper introduces and investigates a bivariate version of RSS for estimating the means of two characteristics simultaneously. It turns out that this technique is always superior to SRS and the usual univariate RSS of the same size. The performance of this procedure for a specific distribution can be evaluated using simulation or numerical computation. For the bivariate normal distribution, the efficiency of the procedure with respect to that of SRS is evaluated exactly for set size m = 2 and 3. The paper shows that the proposed estimator is more efficient than the regression RSS estimators proposed by Yu & Lam (1997) and Chen (2001). Real data that consist of heights and diameters of 399 trees are used to illustrate the procedure. The procedure can be generalized to the case of multiple characteristics.     

An improved mean deviation exponentially weighted moving average control chart to monitor process dispersion under ranked set sampling

Quality-control charts are widely used to monitor and detect shifts in the process mean and dispersion. Abbasi and Miller [MDEWMA chart: an efficient and robust alternative to monitor process dispersion, J Stat Comput Simul 2013;83:247–268] suggested a robust mean deviation exponentially weighted moving average (MDEWMA) control chart for monitoring process dispersion under simple random sampling. In this study, an improved MDEWMA (IMDEWMA) control chart is proposed under ranked set sampling to monitor process dispersion. Detailed Monte Carlo simulations are performed from symmetric and asymmetric populations to investigate the performances of the proposed and existing control charts in terms of average run length (ARL), median run length and standard deviation of run length. An application to real-life data is also presented to illustrate the use of the IMDEWMA control chart. It is observed that the IMDEWMA control chart indicates a shift in process dispersion substantially quicker than the MDEWMA control chart, while maintaining comparable ARLs when the process is in control.     

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.