Estimation of the Half-Logistic Inverse Rayleigh Distribution Parameters via Ranked Set Sampling: Methods and Applications

The generalized Bilal (GB) distribution can be defined as the distribution of the median of three independent random variables drawn from the Weibull distribution. Its failure rate function can be monotonic (decreasing or increasing) or upside-down bathtub-shaped. In this study, we aim to reveal some important properties of the GB distribution that have not been considered before. The findings are both theoretical and practical. From the theoretical viewpoint, we present explicit expressions for both single and product moments of order statistics from the GB distribution. The L-moments are derived as well. From the practical viewpoint, the parameter estimations are accomplished using the maximum likelihood (ML) method, which is based on two different sampling schemes: simple random sampling (SRS) and ranked set sampling (RSS) schemes. Furthermore, the asymptotic confidence intervals for the SRS and RSS estimators are discussed. For the sake of comparison and illustration, a simulation study and a real data example are presented. Concluding remarks are given at the end.

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.

ABSTRACTRecently, 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.

Research on several forms of ranked set samples had been done by many researchers recently for estimating the population mean and other parameters. The results have ascertained that the ranked set samples are proven to be more efficient than the simple random samples in many situations. In this paper, modified bivariate ranked set sample (MBVRSS) was introduced to estimate a simple linear regression model when the regressors were assumed to be random. A primary data was collected on age and height in meter of 200 people from Ijebu-North Local Government Area of Ogun State. It is then shown that MBVRSS gives unbiased and highly efficient estimators of the regression model parameters, than those obtained by using bivariate simple random sampling (BVSRS), RSS or BVRSS using the same number of sample size. Key words: Modified bivariate ranked set sampling (MBVRSS), bivariate ranked-set sampling (BVRSS), bivariate simple random sampling (BVSRS) ranked set sampling (RSS), simple random sampling (SRS), simple linear regression.

Ranked Set Sampling and Ranked Set Sampling concomitant are more efficient than Simple Random Sampling. This can be determined by calculating the Relative Precision which is a ratio value from the variance of the mean from each sampling technique. From the research of Ranked Set Sampling, obtained , and so Ranked Set Sampling is more efficient than Simple Random Sampling. For the research of Ranked Set Sampling concomitant, obtained , and so Ranked Set Sampling concomitant is more efficient than Simple Random Sampling, and for simple linear regression analysis obtained , , , so simple linear regression model of Ranked Set Sampling is more efficient than simple linear regression model of Simple Random Sampling

This study is focused on ranked set sampling theory which is constructive and cost efficient. Estimation for population mean for different sample sizes based on new ranked set sampling (NRSS) scheme is explored. A simulation study has been conducted and a comparison has been made with simple random sampling (SRS), ranked set sampling (RSS), extreme ranked set sampling (ERSS) and quartile ranked set sampling (QRSS) for normal and non – normal distributions.

In statistical parameter estimation problems, how well the parameters are estimated largely depends on the sampling design used. In this article, a modified best linear unbiased estimator of the shape parameter β from log-logistic distribution is considered when scale parameter α is known and when α is unknown under simple random sampling (SRS) and ranked set sampling (RSS). In addition, a modified BLUE of β, when α is known using an RSS version based on the order statistic that maximizes the Fisher information for a fixed set size, will be considered. Theoretical properties of the suggested estimators are compared with its counterpart estimators under SRS. It is found that these estimators under RSS can be real competitors against those under SRS.

The maximum livelihood estimator (MLE) using a ranked set sample (RSS) usually has no closed expression because the maximum likelihood equation involves both hazard and inverse hazard functions, and may no longer be efficient when the judgment ranking is imperfect. In this paper, we consider a modified MLE (MMLE) using RSS for general parameters, which has the same expression as the MLE using a simple random sample (SRS), except that the SRS in the MLE is replaced by the RSS. The results show that, for the location parameter, the MMLE is always more efficient than the MLE using SRS, and for the scale parameter, the MMLE is at least as efficient as the MLE using SRS, when the same sample size is used. Under the perfect judgment ranking, numerical examples also show that the MMLE has good efficiency relative to the MLE based on RSS. When the judgment error is present, we conduct simulations to show that the MMLE is more robust than the MLE using RSS.

ABSTRACTIn this paper, ranked set sampling is proposed for constructing a non-parametric quality control chart based on sign statistic for monitoring the process center when the units are difficult and costly to measure. The characteristics of run length such as average, median and standard deviation are used as criteria to compare two sampling schemes: ranked set sampling and simple random sampling. Cauchy, double exponential and normal distributions are applied to investigate the performance of sign control chart based on studied sampling schemes. The proposed control chart is illustrated with a real data set. The results indicate that when the process starts to get out of control, the chart based on ranked set sampling reduces the run length characteristics values.

There are many situations where it is costly to measure the variable under the study, but this variable can be easily ranked. Ranked set sampling (RSS) utilizes the information of the ranks in addition to the measurements. It was proved that RSS is more efficient than simple random sampling (SRS) when estimating the population arithmetic mean for the same sample size. Because of its efficiency, RSS was employed to a wide range of applications and becoming an interesting method for many researchers. It was studied parametrically and non-parametrically and several estimators based on different statistical distributions were investigated. This research adopts Rician distribution and several measurements from this distribution were investigated based on RSS. The estimations include the arithmetic, geometric, harmonic, quadratic means as well as the median, variance, mean deviation, coefficient of variation, coefficient of skewness and kurtosis. Simulation study shows that the estimators based on RSS are more efficient than those based on SRS. However, the gain in the efficiency with variance and coefficient of kurtosis is not too much and even worse for small sample size.

Relative performance of bivariate ranked set sampling (BVRSS), with respect to ranked set sampling (RSS) and simple random sampling (SRS), for estimating the population mean using ratio and regression methods is investigated. Properties of the proposed estimators are derived. Simulation study is conducted to gain insight in the behaviour of the relative efficiency of BVRSS relative to other sampling scheme. Also, real data that consist of heights and diameters of 399 trees are used to illustrate the procedure. The result of the simulation study and the real data illustration indicate that using BVRSS for ratio and regression estimation is more efficient than using a SRS and an RSS in most of the cases.

Maximum likelihood estimation of dependence parameter using ranked set sampling

The article introduces a novel Bayesian AEWMA Control Chart that integrates different loss functions (LFs) like the square error loss function and Linex loss function under an informative prior for posterior and posterior predictive distributions, implemented across diverse ranked set sampling (RSS) designs. The main objective is to detect small to moderate shifts in the process mean, with the average run length and standard deviation of run length serving as performance measures. The study employs a hard bake process in semiconductor production to demonstrate the effectiveness of the proposed chart, comparing it with existing control charts through Monte Carlo simulations. The results underscore the superiority of the proposed approach, particularly under RSS designs compared to simple random sampling (SRS), in identifying out-of-control signals. Overall, this study contributes a comprehensive method integrating various LFs and RSS schemes, offering a more precise and efficient approach for detecting shifts in the process mean. Real-world applications highlight the heightened sensitivity of the suggested chart in identifying out-of-control signals compared to existing Bayesian charts using SRS.

Survival data is a type of data that measures the time from a defined starting point until the occurrence of a particular event, such as time to death from small cell lung cancer after diagnosis, Length of time in remission for leukemia patients, Length of stay (i.e., time until discharge) in hospital after surgery. The accelerated failure time (AFT) models are popular linear models for analyzing survival data. It provides a linear relationship between the log of the failure time and covariates that affect the expected failure time by contracting or expanding the time scale. This paper examines the performance of the Rank Set Sampling (RSS) on the AFT models for Progressive Type-II censoring-survival data. The Ranked Set Sampling (RSS) is a sampling scheme that selects a sample based on a baseline auxiliary variable for assessing survival time. Simulation studies show that this approach provides a more robust testing procedure, and a more efficient hazard ratio estimate than simple random sampling (SRS). The lung cancer survival data are used to demonstrate the method.

The terrible operating constraints of many real-world events cause systems to malfunction regularly. The failure of systems to perform their intended duties when they reach their lowest, highest, or both extreme operating conditions is a phenomenon that researchers rarely focus on. The multi-stress strength reliability R=P(W<X<Z) is deemed in this study for a component whose strength X falls between two stresses, W, and Z, where X, W, and Z are independently inverted Kumaraswamy distributed. Both maximum likelihood and maximum product spacing procedures are employed to obtain the reliability estimator under simple random sampling (SRS) and ranked set sampling (RSS) methodologies. Four scenarios for reliability estimators are considered. The reliability estimator in the first and second cases can be determined by applying the same sample design (RSS/SRS) to the strength and stress distributions. When the sample data for W and Z originate from RSS while those for X are acquired from SRS, the third reliability estimator is calculated. The drawn data of the strength and stress random variables, which are obtained from SRS and RSS, respectively, are taken into consideration in the final scenario. The effectiveness of the suggested estimators is compared using a comprehensive computer simulation. Lastly, three real data sets have been used to determine reliability estimators.