Ranked Set Research Articles

A new adaptive membership function with CUB uncertainty with application to cluster analysis of Likert-type data

Likert-type scales are commonly used in both academia and industry to capture human feelings since they are user-friendly, easy-to-develop and easy-to administer. This kind of scales generate ordinal variables made up of a set of rank ordered items. Since the distance between two consecutive items cannot be either defined or presumed equal, this kind of variable cannot be analysed by either statistical methods defined on a metric space or parametric tests. Therefore, Likert-type variables cannot be used as segmentation variables of a traditional cluster analysis unless pre-transformed. In such context, fuzzy numbers have been suggested as a way to recode Likert-type variables. Fuzzy numbers are defined by a membership function whose form is usually determined by an expert. In practice, researchers usually define one membership function for each Likert-type scale, not considering the peculiar characteristics of neither questions nor respondents. In this way, the individual uncertainty against each question is considered equal and constant. To overcome this limitation and to reduce the expert’s subjectivity, in this study an adaptive membership function based on CUB model is suggested to pre-transform Likert-type variables into fuzzy numbers before the adoption of a clustering algorithm. After a theoretical presentation of the method, an application using real data will be presented to demonstrate how the method works.

Developing a Ranking Problem Library (RPLIB) from a data-oriented perspective

We present an improved library for the ranking problem called RPLIB. RPLIB includes the following data and features. (1) Real and artificial datasets of both pairwise data (i.e., information about the ranking of pairs of items) and feature data (i.e., a vector of features about each item to be ranked). These datasets range in size (e.g., from small \(n=10\) item datasets to large datasets with hundred of items), application (e.g., from sports to economic data), and source (e.g. real versus artificially generated to have particular structures). (2) RPLIB contains code for the most common ranking algorithms such as the linear ordering optimization method and the Massey method. (3) RPLIB also has the ability for users to contribute their own data, code, and algorithms. Each RPLIB dataset has an associated. JSON model card of additional information such as the number and set of optimal rankings, the optimal objective value, and corresponding figures.

Goodness-of-Fit Tests for Weighted Generalized Quasi-Lindley Distribution Using SRS and RSS with Applications to Real Data

This paper deals with the problem of goodness-of-fit tests (GFTs) for the weighted generalized quasi-Lindley distribution (WGQLD) using ranked set sampling (RSS) and simple random sampling (SRS) techniques. The tests are based on the empirical distribution function and sample entropy. These tests include the Kullback–Leibler, Kolomogorov–Smirnov, Anderson–Darling, Cramér–von Mises, Zhang, Liao, and Shimokawa, and Watson tests. The critical values (CV) and power of each test are obtained based on a simulation study by using SRS and RSS methods considering various sample sizes and alternatives. A rain data set is used to investigate the effectiveness of the suggested GFTs. Based on the same number of measured units for the various alternatives taken into consideration in this study, it is discovered that the RSS tests are more effective than those of their rivals in SRS. Additionally, as the set size increases, the GFTs’ power increases.

Computing the effect of measurement errors on ranked set sampling estimators of the population mean

SummaryThe effect of measurement errors shows the seriousness of the estimators or the estimation procedures and cannot be avoided in sensitive cases because “a miss is as good as a mile.” In this article, we have proposed the very first ranked set estimators like unbiased, ratio, product, difference, ratio exponential, and product exponential estimators under both correlated and uncorrelated measurement errors. The expressions for bias and mean square error of the proposed estimators are obtained up to the first order of approximation. The proposed ranked set estimators have been compared through relative efficiencies over unbiased estimators and the effect of measurement errors has been given through percentage computation of measurement errors. The theoretical conditions are obtained under which the proposed estimator has performed better. The performance of the proposed ranked set estimators under measurement errors are shown by efficiency comparisons, Monte‐Carlo simulation study, and real data study.

Surjective Lp-isometries on rank 1 idempotents

We have studied surjective Lp-isometries on rank 1 idempotents acting on a Hilbert space H for p>1. We show that if φ is a surjective Lp-isometry on the set of rank 1 idempotents, then it leaves the set of rank 1 projections invariant. By applying the classical Wigner's theorem and some further calculations, we prove that either φ or φ⁎ is induced by a unitary or an anti-unitary.

Row–Column Sampling Design Using Auxiliary Ranking Variables

In this paper, we present a new class of row–column sampling design when a sampling region is presented by non-overlapping quadrats of an rp-by-p row–column grid. Quadrats are rank-ordered on a relevant auxiliary variable, first in the rp rows of the grid and then in the so formed p ranked columns. This ranking enhances the precision of the sampling using ideas from standard rank set sampling methods. The sampling design falls in the traditional sampling framework where the sample selection probabilities are independent of the variables of interest. Selection probabilities are governed by a spatial design and the auxiliary information to facilitate better sampling coverage over the sampling region. The paper constructs unbiased estimators for population mean, total and provides variance estimates for them. It also shows when the new sampling design performs better than its competitors. The new sampling design is applied to a forest and two agricultural field samples to estimate the population means. Supplementary materials accompanying this paper appear online.

Ranked hesitant fuzzy sets for multi-criteria multi-agent decisions

This paper introduces and investigates ranked hesitant fuzzy sets, a novel extension of hesitant fuzzy sets that is less demanding than both probabilistic and proportional hesitant fuzzy sets. This new extension incorporates hierarchical knowledge about the various evaluations submitted for each alternative. These evaluations are ranked (for example by their plausibility, acceptability, or credibility), but their position does not necessarily derive from supplementary numerical information (as in probabilistic and proportional hesitant fuzzy sets). In particular, strictly ranked hesitant fuzzy sets arise when no ties exist, i.e., when for any fixed alternative, each submitted evaluation is either strictly more plausible or strictly less plausible than any other submitted evaluation. A detailed comparison with similar models from the literature is performed. Then in order to produce a natural strategy for multi-criteria multi-agent decisions with ranked hesitant fuzzy sets, canonical representations, scores and aggregation operators are designed in the framework of ranked hesitant fuzzy sets. In order to help implementation of this model, Mathematica code is provided for the computation of both scores and aggregators. The decision-making technique that is prescribed is tested with a comparative analysis with four methodologies based on probabilistic hesitant fuzzy information. A conclusion of this numerical exercise is that this methodology is reliable, applicable and robust. All these evidences show that ranked hesitant fuzzy sets are an intuitive extension of the hesitant fuzzy set model designed by V. Torra, that can be implemented in practice with the aid of computationally assisted algorithms.

Reducing the time required to find the Kemeny ranking by exploiting a necessary condition for being a winner

The ranking aggregation problem is gaining attention in different application fields due to its connection with decision making. One of the most famous ranking aggregation methods can be traced back to Kemeny in 1959. Unfortunately, the problem of determining the result of the aggregation proposed by Kemeny, known as Kemeny ranking as it minimizes the number of pairwise discrepancies from a set of rankings given by voters, has been proved to be NP-hard, which unfortunately prevents practitioners from using this method in most real-life problems. In this work, we introduce two exact algorithms for determining the Kemeny ranking. The best of these algorithms guarantees a reasonable search time up to 14 alternatives, showing an important reduction of the execution time in comparison to other algorithms found in the literature. Moreover, a dataset of profiles of rankings is provided and a study of additional aspects of the votes that may have impact on the execution time required to determine the winning ranking is also detailed.

Empirical likelihood inference for area under the receiver operating characteristic curve using ranked set samples.

The area under a receiver operating characteristic curve (AUC) is a useful tool to assess the performance of continuous-scale diagnostic tests on binary classification. In this article, we propose an empirical likelihood (EL) method to construct confidence intervals for the AUC from data collected by ranked set sampling (RSS). The proposed EL-based method enables inferences without assumptions required in existing nonparametric methods and takes advantage of the sampling efficiency of RSS. We show that for both balanced and unbalanced RSS, the EL-based point estimate is the Mann-Whitney statistic, and confidence intervals can be obtained from a scaled chi-square distribution. Simulation studies and two case studies on diabetes and chronic kidney disease data suggest that using the proposed method and RSS enables more efficient inference on the AUC.

Genetic relatedness and morphology as drivers of interspecific dominance hierarchy in hummingbirds.

A dominance hierarchy is the set of ranks occupied by species within an assemblage. Species with a high position within the dominance hierarchy tend to dominate subordinate species in contests for access to resources. In hummingbirds, greater weight and wing disc loading have been associated with highest ranks within the dominance hierarchy. Nevertheless, the limit to which the difference between the weight of contending species represents a competitive advantage has not yet been determined. Here, we determined the dominance hierarchy of a hummingbird assemblage exploiting the most abundant floral resource (Palicourea padifolia, Rubiaceae) in a cloud forest of central Veracruz, Mexico. Specifically, we tested whether species weight and wing disc loading influence the dominance hierarchy. Additionally, we tested whether the flowers visited per foraging bout increases with species weight and dominance. We further tested whether weight, wing disc loading, and the genetic relatedness between contenders influenced the dominance relationships in species-pair interactions. Our results indicate that the hierarchy is positively influenced by weight. Hummingbirds visited similar number of flowers regardless their weight or their dominance. Nevertheless, the probability that the heaviest contender won contests was positively associated with the differences of weight and genetic relatedness between contenders. Contrarily, the probability that the contender with greatest wing disc loading won contests was positively associated with differences of weight and negatively associated with the relatedness between contenders. However, these models only explained between 22% and 34% of the variation, respectively. Our results demonstrate that the weight was the major contributor to high dominance values. However, future studies should include (1) the temporal variability of the weight and (2) experimental predictor variables such the burst power of the hummingbirds to evaluate its effects on the dynamics of dominance hierarchies in hummingbird assemblages. All the hummingbird species present in the studied assemblage have developed wide behavioral mechanisms that compensate their morphological differences, which allow them to coexist, even when they compete for the access to the same resource.

Generalized estimator for computation of population mean under neutrosophic ranked set technique: An application to solar energy data

Generalized estimator for computation of population mean under neutrosophic ranked set technique: An application to solar energy data

Extreme Points of Gram Spectrahedra of Binary Forms

The Gram spectrahedron mathrm {Gram}(f) of a form f with real coefficients is a compact affine-linear section of the cone of psd symmetric matrices. It parametrizes the sum of squares decompositions of f, modulo orthogonal equivalence. For f a sufficiently general positive binary form of arbitrary degree, we show that mathrm {Gram}(f) has extreme points of all ranks in the Pataki range. We also calculate the dimension of the set of rank r extreme points, for any r. Moreover, we determine the pairs of rank two extreme points for which the connecting line segment is an edge of mathrm {Gram}(f). The proof of the main result relies on a purely algebraic fact of independent interest: Whenever d,rge 1 are integers with left( {begin{array}{c}r+1 2end{array}}right) le 2d+1, there exists a length r sequence f_1,dots ,f_r of binary forms of degree d for which the left( {begin{array}{c}r+1 2end{array}}right) pairwise products f_if_j, ile j, are linearly independent.

Using accounting dataset for agricultural sustainability assessment through a multi‐criteria approach: an Italian case study

AbstractSustainability has become a key issue for agricultural systems. In this context, sustainability assessment implies considering different aspects of agricultural activities, including environmental, economic and social issues, and it is usually based on practice‐based indicators. However, proposed methods and applications are still rare due to the difficulty to build a strong general framework and to find appropriate data. The present work develops a performance indicator framework using farm accounting data from the Italian Agricultural Accounting Information Network and proposes a multi‐criteria approach based on the multiple reference point weak‐strong composite indicators (MRP‐WSCI) method to assess the degree of sustainability at the farm level. This method, widely applied to sustainability assessment, calculates the relative rankings including the weak and strong approaches to sustainability and facilitates an understanding of the different indices used, but it is still unused in the agricultural sector. The proposed evaluation framework shows that MRP‐WSCI can handle a holistic set of indicators and rank the farms according to their level of sustainability, making it suitable for the agricultural farm sustainability assessment. Our results show that none of the farms in the case study sample can be classified as “good,” according to some of the indices produced, that is, the strong–strong and strong–weak index, while just two of them are good for the weak–weak index. Although the size of the sample used is small, the methodological approach proposed can be straightforwardly extended to larger samples. This methodology can provide information regarding the sustainability of farms to regional and/or national agricultural offices, agricultural extension workers, and farmers themselves, helping in understanding issues in specific dimensions.

Estimating System Reliability Using Neoteric and Median RSS Data for Generalized Exponential Distribution

In this work, we show how to estimate stress strength (SS) reliability when the stress (Y) and strength (X) distributions are generalized exponentials with a common scale parameter. The SS reliability estimator is considered in view of neoteric ranked set sampling (NRSS) and median ranked set sampling (MRRS). We acquire an estimate of the reliability (R) when such samples of the stress and strength random variables are gathered using the same NRSS technique. Furthermore, the reliability estimator is derived when the stress distribution data are in the pattern of MRSS with just an odd/even set size and the strength distribution data are derived from NRSS and vice versa. The simulation results are used to evaluate and understand the adequacy of a variety of estimators for the suggested schemes. Based on our simulated results, we found that NRSS-based stress strength reliability estimates are more efficient than MRSS-based stress strength reliability estimates. The analysis of real-world data is used to implement the recommended estimators.

Modeling the Ranked Antenatal Care Visits Using Optimized Partial Least Square Regression.

The frequency and timing of antenatal care visits are observed to be the significant factors of infant and maternal morbidity and mortality. The present research is conducted to determine the risk factors of reduced antenatal care visits using an optimized partial least square regression model. A data set collected during 2017-2018 by Pakistan Demographic and Health Surveys is used for modeling purposes. The partial least square regression model coupled with rank correlation measures are introduced for improved performance to address ranked response. The proposed models included PLSρs, PLSτA, PLSτB, PLSτC, PLS D, PLSτGK, PLS G, and PLS U. Three filter-based factor selection methods are executed, and leave-one-out cross-validation by linear discriminant analysis is measured on predicted scores of all models. Finally, the Monte Carlo simulation method with 10 iterations of repeated sampling for optimization of validation performance is applied to select the optimum model. The standard and proposed models are executed over simulated and real data sets for efficiency comparison. The PLSρs is found to be the most appropriate proposed method to model the observed ranked data set of antenatal care visits based on validation performance. The optimal model selected 29 influential factors of inadequate use of antenatal care. The important factors of reduced antenatal care visits included women's educational status, wealth index, total children ever born, husband's education level, domestic violence, and history of cesarean section. The findings recommended that partial least square regression algorithms coupled with rank correlation coefficients provide more efficient estimates of ranked data in the presence of multicollinearity.

Estimation of distribution function using L ranked set sampling and robust extreme ranked set sampling with application to reliability

Estimation of distribution function using L ranked set sampling and robust extreme ranked set sampling with application to reliability

A Suggestion for Using Ranked Set Sampling in Household Expenditure and Income Survey

The ranked set sampling method uses the ranking information of the units to provide a more representative sample of the population to the survey designers. The sampling distribution is closer to the actual distribution of the population. In this article, to ensure the effectiveness of ranked set sampling in extensive surveys conducted to prepare official statistics, we intend to use this sampling method to improve the efficiency of key estimates of household expenditure and income survey of the Statistics Center of Iran. The results show that using ranked set sampling to design household expenditure and income surveys can improve the efficiency of key estimates of the study, provided that the ranking variable used has a high correlation with the main variables of the study. Obviously, in the absence of a suitable and available variable for ranking the units, the information of the sampling frame can be used to construct a ranking variable correlated with the key variables of the survey.

Optimum stratification for equal allocation using ranked set sampling as a method of selection

Present investigation deals with the problem of optimum stratification on the auxiliary variable for equal allocation when method of selection of units in individual strata is done by ranked set sampling (RSS). A rule of obtaining approximate optimum strata boundaries has been developed by minimizing the variance and solution of minimal equations thereof. A numerical analysis on the basis of percent relative efficiency has also been made, which indicated remarkable gain in efficiency.

Reliability Estimation in Multicomponent Stress-Strength for Generalized Inverted Exponential Distribution Based on Ranked Set Sampling

Stress-strength models are considered of great significance due to their applicability in varied fields. We address the estimation of the system reliability of a multicomponent stress-strength model, say Rs,k, of an s out of k system when the pair stress and strengths are drawn from a generalized inverted exponential distribution. The system is deemed as working if at least s out of k strengths be more than its stress. We obtain the reliability estimators when the data of strength and stress distributions are collected from three sampling schemes, specifically; simple random sampling, ranked set sampling, and median ranked set sampling. We obtain four estimators of Rs,k out from median ranked set sampling. The behavior of different estimates is examined via a simulation study based on mean squared errors and efficiencies. The simulation studies point out that the reliability estimates of Rs,k, from the ranked set sampling scheme are preferred than other estimates picked from the simple random sample and median ranked set sampling in a majority of the situations. The theoretical studies are explained with the aid of real data analysis.

A class of ratio-cum-product estimators of population mean under extreme ranked set sampling (ERSS) and simple random sampling (SRS)

Extreme Ranked Set Sampling (ERSS) is a survey technique which seeks to improve the likelihood that collected sample data provides a good representation of the population and minimizes costs associated with obtaining them. The main goal of a statistical survey is to reduce sampling errors either by devising suitable sampling scheme or by formulating efficient estimator of the population parameters. In an attempt to address the problem of weak or loss of efficiency usually suffered in estimation of population mean under Simple Random Sampling (SRS), a class of ratio-cum-product estimators for population mean of the study variable Y is proposed based on ERSS using information on a single accompanying variable. Members of the proposed class of estimators were obtained by assigning various values to the scalars that helps in designing the estimators. These members were then transformed to a form that can be easily expanded using Taylor’s series approximation, from where various properties such as biases, relative biases, Mean Square Errors (MSEs), and Optimal Mean Square Errors (OMSEs) were derived under large sample approximation. Empirical study was conducted using three natural population data sets in order to investigate the performances and efficiency of the proposed classes of estimators under ERSS over its corresponding counterpart’s estimator based on SRS and some existing ratio and product estimators. This empirical study was followed up with a computer simulation study using R-software. The results revealed that the advocated class of estimators in ERSS produced smaller biases and MSEs which is an indicator of appreciable gain in efficiency and superiority over its corresponding counterpart estimator and some existing ratio type estimators in sample survey for all cases considered in this work and are therefore adjudged to provide a better alternative whenever efficiency is required.