Estimation of the mean of the exponential distribution using moving extremes ranked set sampling

Moving Extremes Ranked Set Sampling (MERSS) is a useful modification of Ranked Set Sampling (RSS). Unlike RSS, MERSS allows for an increase of set size without introducing too much ranking error. The method is considered parametrically under exponential distribution. Maximum likelihood estimator (MLE), and a modified MLE are considered and their properties are studied. The method is studied under both perfect and imperfect ranking (with error in ranking). It appears that these estimators can be real competitors to the MLE using the usual simple random sampling (SRS).     

Modified inference about the mean of the exponential distribution using moving extreme ranked set sampling

The maximum likelihood estimator (MLE) and the likelihood ratio test (LRT) will be considered for making inference about the scale parameter of the exponential distribution in case of moving extreme ranked set sampling (MERSS). The MLE and LRT can not be written in closed form. Therefore, a modification of the MLE using the technique suggested by Maharota and Nanda (Biometrika 61:601–606, 1974) will be considered and this modified estimator will be used to modify the LRT to get a test in closed form for testing a simple hypothesis against one sided alternatives. The same idea will be used to modify the most powerful test (MPT) for testing a simple hypothesis versus a simple hypothesis to get a test in closed form for testing a simple hypothesis against one sided alternatives. Then it appears that the modified estimator is a good competitor of the MLE and the modified tests are good competitors of the LRT using MERSS and simple random sampling (SRS).     

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.     

Modified Maximum Likelihood Estimator of Scale Parameter Using Moving Extremes Ranked Set Sampling

A modified maximum likelihood estimator (MMLE) of scale parameter is considered under moving extremes ranked set sampling (MERSS), and its properties are obtained. For some usual scale distributions, we obtain explicit form of the MMLE and prove the MMLE is an unbiased estimator under MERSS. The simulation results show that the MMLE using MERSS is always more efficient than the MLE using simple random sampling, when the same sample size is used. The simulation results also show that the loss of efficiency in using the MMLE instead of the MLE is very small for small sample.     

Entropy estimation and goodness-of-fit tests for the inverse Gaussian and Laplace distributions using paired ranked set sampling

ABSTRACT

In this paper, Vasicek [A test for normality based on sample entropy. J R Stat Soc Ser B. 1976;38:54–59] entropy estimator is modified using paired ranked set sampling (PRSS) method. Also, two goodness-of-fit tests using PRSS are suggested for the inverse Gaussian and Laplace distributions. The new suggested entropy estimator and goodness-of-fit tests using PRSS are compared with their counterparts using simple random sampling (SRS) via Monte Carlo simulations. The critical values of the suggested tests are obtained, and the powers of the tests based on several alternatives hypotheses using SRS and PRSS are calculated. It turns out that the proposed PRSS entropy estimator is more efficient than the SRS counterpart in terms of root mean square error. Also, the proposed PRSS goodness-of-fit tests have higher powers than their counterparts using SRS for all alternative considered in this study.     

Reducing sample size needed for cox-proportional hazards model analysis using more efficient sampling method

Abstract

In general, survival data are time-to-event data, such as time to death, time to appearance of a tumor, or time to recurrence of a disease. Models for survival data have frequently been based on the proportional hazards model, proposed by Cox. The Cox model has intensive application in the field of social, medical, behavioral and public health sciences. In this paper we propose a more efficient sampling method of recruiting subjects for survival analysis. We propose using a Moving Extreme Ranked Set Sampling (MERSS) scheme with ranking based on an easy-to-evaluate baseline auxiliary variable known to be associated with survival time. This paper demonstrates that this approach provides a more powerful testing procedure as well as a more efficient estimate of hazard ratio than that based on simple random sampling (SRS). Theoretical derivation and simulation studies are provided. The Iowa 65+ Rural study data are used to illustrate the methods developed in this paper.     

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.     

Unbiased variance estimation in a simple exponential population using ranked set samples

The problem considered in this paper is that of unbiased estimation of the variance of an exponential distribution using a ranked set sample (RSS). We propose some unbiased estimators each of which is better than the non-parametric minimum variance quadratic unbiased estimator based on a balanced ranked set sample as well as the uniformly minimum variance unbiased estimator based on a simple random sample (SRS) of the same size. Relative performances of the proposed estimators and a few other properties of the estimators including their robustness under imperfect ranking have also been studied.     

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.     

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.     

Ratio estimators based on a ranked set sample in a finite population setting

This paper considers the ratio estimator in a finite population setting in a ranked set sampling (RSS) design, where the sample is constructed either with or without replacement policies. It is shown that the proposed ratio estimator is slightly biased, but the amount of bias is smaller than the amount of bias of a simple random sample (SRS) ratio estimator. For the proposed ratio estimator, the paper provides explicit expressions for its mean square error and precision relative to the other competing estimators. It is shown that the new estimator has a substantial amount of improvement in efficiency with respect to SRS estimator. The proposed estimator is applied to two different finite population settings to estimate population mean.     

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.     

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.     

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.     

Estimation of system reliability for exponential distributions based on L ranked set sampling

Abstract

In the case where strength and stress both follow exponential distributions, this paper considers the maximum likelihood estimator (MLE) of the system reliability based on L ranked set sampling (LRSS). The proposed MLE is shown to have existence, uniqueness and asymptotic normality, and its asymptotic variance is obtained by the Fisher information matrix of LRSS. The values of asymptotic relative efficiencies show that the proposed MLE is always more efficient than the MLE using simple random sampling (SRS). However, the MLE using LRSS cannot be written in closed form. Therefore, the modified MLE is proposed using the technique replaced some terms in the maximum likelihood equations by their expectations. The newly modified MLE using LRSS is shown to be superior to the MLE using SRS. Finally, the proposed method is applied to a real data set on metastatic renal carcinoma study.     

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.     

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.     

Mixed ranked set sampling design

The main focus of agricultural, ecological and environmental studies is to develop well designed, cost-effective and efficient sampling designs. Ranked set sampling (RSS) is one method that leads to accomplish such objectives by incorporating expert knowledge to its advantage. In this paper, we propose an efficient sampling scheme, named mixed RSS (MxRSS), for estimation of the population mean and median. The MxRSS scheme is a suitable mixture of both simple random sampling (SRS) and RSS schemes. The MxRSS scheme provides an unbiased estimator of the population mean, and its variance is always less than the variance of sample mean based on SRS. For both symmetric and asymmetric populations, the mean and median estimators based on SRS, partial RSS (PRSS) and MxRSS schemes are compared. It turns out that the mean and median estimates under MxRSS scheme are more precise than those based on SRS scheme. Moreover, when estimating the mean of symmetric and some asymmetric populations, the mean estimates under MxRSS scheme are found to be more efficient than the mean estimates with PRSS scheme. An application to real data is also provided to illustrate the implementation of the proposed sampling scheme.     

Efficiency bounds for a generalization of ranked-set sampling

Abstract

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.