Rank-ordered Set Sampling Research Articles

Utilizing Bayesian inference in accelerated testing models under constant stress via ordered ranked set sampling and hybrid censoring with practical validation

This research investigates the application of the ordered ranked set sampling (ORSSA) procedure in constant-stress partially accelerated life-testing (CSPALTE). The study adopts the assumption that the lifespan of a specific item under operational stress follows a half-logistic probability distribution. Through Bayesian estimation methods, it concentrates on estimating the parameters, utilizing both asymmetric loss function and symmetric loss function. Estimations are conducted using ORSSAs and simple random samples, incorporating hybrid censoring of type-I. Real-world data sets are utilized to offer practical context and validate the theoretical discoveries, providing concrete insights into the research findings. Furthermore, a rigorous simulation study, supported by precise numerical calculations, is meticulously conducted to gauge the Bayesian estimation performance across the two distinct sampling methodologies. This research ultimately sheds light on the efficacy of Bayesian estimation techniques under varying sampling strategies, contributing to the broader understanding of reliability analysis in CSPALTE scenarios.

Uniform k-Tuple Partially Rank-Ordered Set Sampling

Ranked Set Sampling (RSS), introduced by McIntyre, and other related methods, such as Partially Rank-Ordered Set Sampling (PROSS), have shown that inclusion of a ranking mechanism produces estimators with lower variance than their simple random sample (SRS)-based counterparts. Like RSS, PROSS takes only one measurement from each partially ranked-ordered set. We propose a sampling plan called Uniform k-Tuple Partially Rank-Ordered (UKPRSS) where a measurement is collected from each group of a partially rank-ordered set. This article demonstrates estimators from UKPRSS have lower variance than their SRS counterparts. In addition, there is a reduction in the number units needing to be screened when compared to PROSS. Estimation of the mean and distribution function are investigated theoretically.

Ordered partially ordered judgment subset sampling with applications to parametric inference

ABSTRACTIn 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.

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.

On distribution function estimation with partially rank-ordered set samples: estimating mercury level in fish using length frequency data

ABSTRACTWe study the non-parametric estimation of a continuous distribution function F based on the partially rank-ordered set (PROS) sampling design. A PROS sampling design first selects a random sample from the underlying population and uses judgement ranking to rank them into partially ordered sets, without measuring the variable of interest. The final measurements are then obtained from one of the partially ordered sets. Considering an imperfect PROS sampling procedure, we first develop the empirical distribution function (EDF) estimator of F and study its theoretical properties. Then, we consider the problem of estimating F, where the underlying distribution is assumed to be symmetric. We also find a unique admissible estimator of F within the class of nondecreasing step functions with jumps at observed values and show the inadmissibility of the EDF. In addition, we introduce a smooth estimator of F and discuss its theoretical properties. Finally, we expand on various numerical illustrations of our results via several simulation studies and a real data application and show the advantages of PROS estimates over their counterparts under the simple random and ranked set sampling designs.

Information content of partially rank-ordered set samples

Partially rank-ordered set (PROS) sampling is a generalization of ranked set sampling in which rankers are not required to fully rank the sampling units in each set, hence having more flexibility to perform the necessary judgemental ranking process. The PROS sampling has a wide range of applications in different fields ranging from environmental and ecological studies to medical research and it has been shown to be superior over ranked set sampling and simple random sampling for estimating the population mean. We study Fisher information content and uncertainty structure of the PROS samples and compare them with those of simple random sample (SRS) and ranked set sample (RSS) counterparts of the same size from the underlying population. We study uncertainty structure in terms of the Shannon entropy, Renyi entropy and Kullback–Leibler (KL) discrimination measures.

Efficiency comparisons for partially rank-ordered set sampling

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 study the efficiency of the PROSS sample mean under perfect rankings for various PROSS schemes. We obtain conditions under which one PROSS scheme is always more efficient than another, and we also obtain conditions under which how the efficiencies of two PROSS schemes compare depends on the particular distribution. We completely determine how PROSS schemes compare in the two-subset case, and we also prove a conjecture of Ozturk (Environ Ecol Stat 18:757–779, 2011) about how the efficiency of the PROSS sample mean compares to that of the RSS sample mean.

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

ABSTRACTIn 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.

An improved procedure for estimation of malignant breast cancer prevalence using partially rank ordered set samples with multiple concomitants.

Rank-based sampling designs are widely used in situations where measuring the variable of interest is costly but a small number of sampling units (set) can be easily ranked prior to taking the final measurements on them and this can be done at little cost. When the variable of interest is binary, a common approach for ranking the sampling units is to estimate the probabilities of success through a logistic regression model. However, this requires training samples for model fitting. Also, in this approach once a sampling unit has been measured, the extra rank information obtained in the ranking process is not used further in the estimation process. To address these issues, in this paper, we propose to use the partially rank-ordered set sampling design with multiple concomitants. In this approach, instead of fitting a logistic regression model, a soft ranking technique is employed to obtain a vector of weights for each measured unit that represents the probability or the degree of belief associated with its rank among a small set of sampling units. We construct an estimator which combines the rank information and the observed partially rank-ordered set measurements themselves. The proposed methodology is applied to a breast cancer study to estimate the proportion of patients with malignant (cancerous) breast tumours in a given population. Through extensive numerical studies, the performance of the estimator is evaluated under various concomitants with different ranking potentials (i.e. good, intermediate and bad) and tie structures among the ranks. We show that the precision of the partially rank-ordered set estimator is better than its counterparts under simple random sampling and ranked set sampling designs and, hence, the sample size required to achieve a desired precision is reduced.

Efficiency bounds for a generalization of ranked-set sampling

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.

A New Synthetic Exponentially Weighted Moving Average Control Chart for Monitoring Process Dispersion

Exponentially weighted moving average (EWMA) control charts have been widely recognized as a potentially powerful process monitoring tool of the statistical process control because of their excellent speed in detecting small to moderate shifts in the process parameters. Recently, new EWMA and synthetic control charts have been proposed based on the best linear unbiased estimator of the scale parameter using ordered ranked set sampling (ORSS) scheme, named EWMA-ORSS and synthetic-ORSS charts, respectively. In this paper, we extend the work and propose a new synthetic EWMA (SynEWMA) control chart for monitoring the process dispersion using ORSS, named SynEWMA-ORSS chart. The SynEWMA-ORSS chart is an integration of the EWMA-ORSS chart and the conforming run length chart. Extensive Monte Carlo simulations are used to estimate the run length performances of the proposed control chart. A comprehensive comparison of the run length performances of the proposed and the existing powerful control charts reveals that the SynEWMA-ORSS chart outperforms the synthetic-R, synthetic-S, synthetic-D, synthetic-ORSS, CUSUM-R, CUSUM-S, CUSUM-ln S2, EWMA-ln S2 and EWMA-ORSS charts when detecting small shifts in the process dispersion. A similar trend is observed when the proposed control chart is constructed under imperfect rankings. An application to a real data is also provided to demonstrate the implementation and application of the proposed control chart. Copyright © 2014 John Wiley & Sons, Ltd.

Ordered Double Ranked Set Samples and Applications to Inference

SYNOPTIC ABSTRACTBalakrishnan and Li (2005) introduced the use of ordered ranked set sampling (ORSS) and derived the best linear unbiased estimators (BLUEs) under ORSS (BLUEs-ORSS). In this study, we extend the work to ordered double ranked set sampling (ODRSS) scheme by using the idea of order statistics from independent and nonidentically distributed random variables. The BLUEs of the location and the scale parameters of a location-scale family of distributions are derived using ODRSS (BLUEs-ODRSS). It is shown that the BLUEs-ODRSS are uniformly better than the BLUEs-ORSS for the two-parameter exponential, normal, and generalized geometric distributions. Furthermore, we also study the properties of the distribution-free confidence intervals for quantiles and tolerance intervals based on ODRSS. We show that the confidence and tolerance intervals under the ODRSS scheme are more precise than their counterparts based on the ORSS scheme.

New Synthetic Control Charts for Monitoring Process Mean and Process Dispersion

A statistical quality control chart is widely recognized as a potentially powerful tool that is frequently used in many manufacturing and service industries to monitor the quality of the product or manufacturing processes. In this paper, we propose new synthetic control charts for monitoring the process mean and the process dispersion. The proposed synthetic charts are based on ranked set sampling (RSS), median RSS (MRSS), and ordered RSS (ORSS) schemes, named synthetic‐RSS, synthetic‐MRSS, and synthetic‐ORSS charts, respectively. Average run lengths are used to evaluate the performances of the control charts. It is found that the synthetic‐RSS and synthetic‐MRSS mean charts perform uniformly better than the Shewhart mean chart based on simple random sampling (Shewhart‐SRS), synthetic‐SRS, double sampling‐SRS, Shewhart‐RSS, and Shewhart‐MRSS mean charts. The proposed synthetic charts generally outperform the exponentially weighted moving average (EWMA) chart based on SRS in the detection of large mean shifts. We also compare the performance of the synthetic‐ORSS dispersion chart with the existing powerful dispersion charts. It turns out that the synthetic‐ORSS chart also performs uniformly better than the Shewhart‐R, Shewhart‐S, synthetic‐R, synthetic‐S, synthetic‐D, cumulative sum (CUSUM) ln S2, CUSUM‐R, CUSUM‐S, EWMA‐ln S2, and change point CUSUM charts for detecting increases in the process dispersion. A similar trend is observed when the proposed synthetic charts are constructed under imperfect RSS schemes. Illustrative examples are used to demonstrate the implementation of the proposed synthetic charts. Copyright © 2014 John Wiley & Sons, Ltd.

A New Maximum Exponentially Weighted Moving Average Control Chart for Monitoring Process Mean and Dispersion

Maximum exponentially weighted moving average (MaxEWMA) control charts have attracted substantial interest because of their ability to simultaneously detect increases and decreases in both the process mean and the process variability. In this paper, we propose new MaxEWMA control charts based on ordered double ranked set sampling (ODRSS) and ordered imperfect double ranked set sampling (OIDRSS) schemes, named MaxEWMA‐ODRSS and MaxEWMA‐OIDRSS control charts, respectively. The proposed MaxEWMA control charts are based on the best linear unbiased estimators obtained under ODRSS and OIDRSS schemes. Extensive Monte Carlo simulations are used to estimate the average run length and standard deviation of the run length of the proposed MaxEWMA control charts. The run length performances and the diagnostic abilities of the proposed MaxEWMA control charts are compared with that of their counterparts based on simple random sampling (SRS), ordered ranked set sampling (ORSS) and ordered imperfect ranked set sampling schemes (OIRSS) schemes, that is, MaxEWMA‐SRS, maximum generally weighted moving average (MaxGWMA‐SRS), MaxEWMA‐ORSS and MaxEWMA‐OIRSS control charts. It turns out that the proposed MaxEWMA‐ODRSS and MaxEWMA‐OIDRSS control charts perform uniformly better than the MaxEWMA‐SRS, MaxGWMA‐SRS, MaxEWMA‐ORSS and MaxEWMA‐OIRSS control charts in simultaneous detection of shifts in the process mean and variability. An application to real data is also provided to illustrate the implementations of the proposed and existing MaxEWMA control charts. Copyright © 2014 John Wiley & Sons, Ltd.

Nonparametric density estimation using partially rank-ordered set samples with application in estimating the distribution of wheat yield

We study nonparametric estimation of an unknown density function f based on the ranked-based observations obtained from a partially rank-ordered set (PROS) sampling design. PROS sampling design has many applications in environmental, ecological and medical studies where the exact measurement of the variable of interest is costly but a small number of sampling units can be ordered with respect to the variable of interest by any means other than actual measurements and this can be done at low cost. PROS observations involve independent order statistics which are not identically distributed and most of the commonly used nonparametric techniques are not directly applicable to them. We rst develop kernel density estimates of f based on an imperfect PROS sampling procedure and study its theoretical properties. Then, we consider the problem when the underlying distribution is assumed to be symmetric and introduce some plug-in kernel density estimators of f. We use an EM type algorithm to estimate misplacement probabilities associated with an imperfect PROS design. Finally, we expand on various numerical illustrations of our results via several simulation studies and a case study to estimate the distribution of wheat yield using the total acreage of land which is planted in wheat as an easily obtained auxiliary information. Our results show that the PROS density estimate performs better than its SRS and RSS counterparts.

Improved Exponentially Weighted Moving Average Control Charts for Monitoring Process Mean and Dispersion

Exponentially weighted moving average (EWMA) control charts are mostly used to monitor the manufacturing processes. In this paper, we propose some improved EWMA control charts for detecting the random shifts in the process mean and process dispersion. These EWMA control charts are based on the best linear unbiased estimators obtained under ordered ranked set sampling (ORSS) and ordered imperfect ranked set sampling (OIRSS), named EWMA‐ORSS and EWMA‐OIRSS charts, respectively. Monte Carlo simulations are used to estimate the average run length, median run length and standard deviation of run length of the proposed EWMA control charts. It is observed that the EWMA‐ORSS mean control chart is able to detect the random shifts in the process mean substantially quicker than the Shewhart‐cumulative sum and the Shewhart‐EWMA control charts based on the RSS scheme. Both EWMA‐ORSS and EWMA‐OIRSS location charts perform better than the classical EWMA, hybrid EWMA, Shewhart‐EWMA and fast initial response‐EWMA charts. The EWMA‐ORSS dispersion control chart performs better than the simple random sampling based CS‐EWMA and other EWMA control charts in efficient detection of the random shifts that occur in the process variability. An application to real data is also given to explain the implementation of the proposed EWMA control charts. Copyright © 2013 John Wiley & Sons, Ltd.

Prediction of order statistics and record values based on ordered ranked set sampling

In this paper, we consider two-sample prediction problems. First, based on ordered ranked set sampling (ORSS) introduced by Balakrishnan and Li [Ordered ranked set samples and applications to inference. Ann Inst Statist Math. 2006;58:757–777], we obtain prediction intervals for order statistics from a future sample and compare the results with the one based on the usual-order statistics. Next, we construct prediction intervals for record values from a future sequence based on ORSS and compare the results with the one based on an another independent record sequence developed recently by Ahmadi and Balakrishnan [Prediction of order statistics and record values from two independent sequences. Statistics. 2010;44:417–430].

Quantile inference based on partially rank-ordered set samples

This paper develops statistical inference for population quantiles based on a partially rank-ordered set (PROS) sample design. A PROS sample design is similar to a ranked set sample with some clear differences. This design first creates partially rank-ordered subsets by allowing ties whenever the units in a set cannot be ranked with high confidence. It then selects a unit for full measurement at random from one of these partially rank-ordered subsets. The paper develops a point estimator, confidence interval and hypothesis testing procedure for the population quantile of order p. Exact, as well as asymptotic, distribution of the test statistic is derived. It is shown that the null distribution of the test statistic is distribution-free, and statistical inference is reasonably robust against possible ranking errors in ranking process.

Two sample distribution-free inference based on partially rank-ordered set samples

This paper develops distribution-free inference for a location shift model based on a partially rank-ordered set (PROS) sample. In a PROS sample, a small set of experimental units is judgment ranked without measurement by allowing ties whenever the units cannot be ranked with high confidence. These tied units are replaced in partially ordered judgment subsets from which a unit is selected at random for a full measurement. Based on this sampling design, we construct an estimator, a test and a confidence interval for the location shift parameter. It is shown that the new sampling design is robust against any possible ranking error and has higher efficiency than competitor designs in the literature.

Ordered ranked set samples and applications to inference

Ranked set sampling (RSS) was first proposed by McIntyre [1952. A method for unbiased selective sampling, using ranked sets. Australian J. Agricultural Res. 3, 385–390] as an effective way to estimate the unknown population mean. Chuiv and Sinha [1998. On some aspects of ranked set sampling in parametric estimation. In: Balakrishnan, N., Rao, C.R. (Eds.), Handbook of Statistics, vol. 17. Elsevier, Amsterdam, pp. 337–377] and Chen et al. [2004. Ranked Set Sampling—Theory and Application. Lecture Notes in Statistics, vol. 176. Springer, New York] have provided excellent surveys of RSS and various inferential results based on RSS. In this paper, we use the idea of order statistics from independent and non-identically distributed (INID) random variables to propose ordered ranked set sampling (ORSS) and then develop optimal linear inference based on ORSS. We determine the best linear unbiased estimators based on ORSS (BLUE-ORSS) and show that they are more efficient than BLUE-RSS for the two-parameter exponential, normal and logistic distributions. Although this is not the case for the one-parameter exponential distribution, the relative efficiency of the BLUE-ORSS (to BLUE-RSS) is very close to 1. Furthermore, we compare both BLUE-ORSS and BLUE-RSS with the BLUE based on order statistics from a simple random sample (BLUE-OS). We show that BLUE-ORSS is uniformly better than BLUE-OS, while BLUE-RSS is not as efficient as BLUE-OS for small sample sizes ( n < 5 ).