Estimating the Population Mean Using Extreme Ranked Set Sampling

Precision of the estimate of the population mean using ranked set sample (RSS) relative to using simple random sample (SRS), with the same number of quantified units, depends upon the population and success in ranking. In practice, even ranking a sample of moderate size and observing the ith ranked unit (other than the extremes) is a difficult task. Therefore, in this paper we introduce a variety of extreme ranked set sample (ERSSs) to estimate the population mean. ERSSs is more practical than the ordinary ranked set sampling, since in case of even sample size we need to identify successfully only the first and/or the last ordered unit or in case of odd sample size the median unit. We show that ERSSs gives an unbiased estimate of the population mean in case of symmetric populations and it is more efficient than SRS, using the same number of quantified units. Example using real data is given. Also, parametric examples are given.     

Non‐Parametric Selected Ranked Set Sampling

A nonparametric selected ranked set sampling is suggested. The estimator of population mean based on the new approach is compared with that using the simple random sampling (SRS), the ranked set sampling (RSS) and the median ranked set sampling (MRSS) methods. The estimator of population mean using the new approach is found to be more efficient than its counter‐parts for almost all the cases considered.     

Random median sampling to enhance the precision of population estimates

We describe the characteristics of a sampling procedure called random median sampling that was proposed to enhance the precision of population estimates. In performing random median sampling, we first select a sampling item at random from the sampling area. We roughly compare the abundance of individuals in the selected item with that of the adjacent two items in order to identify the item that has median abundance, i.e., the item that has the second largest abundance among the three items. We count the number of individuals of the item having the median abundance. This procedure is repeated n times in the sampling area (i = 1, 2, ..., n). Let m _i be the ith median abundance. The estimates of the mean abundance per sampling item and the variance of estimates are given by Σm _i /n and Σ(m _i –Σm _i /n)²/n(n – 1), respectively. This method is a local application of the median ranked set sampling that was proposed by Muttlak (J Appl Stat Sci 6:245–255, 1997). Random median sampling is effective when the correlation coefficient between adjacent items is small. If the correlation coefficient is close to zero, random median sampling reduces the variance of estimates to 45 or 32% of that in simple random sampling when the distribution follows a normal distribution or a Laplace distribution, respectively. The sample size required to achieve a given precision of estimate decreases accordingly. The effectiveness of random median sampling, however, is small if the correlation coefficient is large. The condition in which random median sampling is superior to simple random sampling is also discussed.     

Ranked set sampling with unequal samples

A ranked set sampling procedure with unequal samples (RSSU) is proposed and used to estimate the population mean. This estimator is then compared with the estimators based on the ranked set sampling (RSS) and median ranked set sampling (MRSS) procedures. It is shown that the relative precisions of the estimator based on RSSU are higher than those of the estimators based on RSS and MRSS. An example of estimating the mean diameter at breast height of longleaf-pine trees on the Wade Tract in Thomas County, Georgia, is presented.     

Median Ranked Set Sampling with Size Biased Probability of Selection

Median ranked set sampling may be combined with size biased probability of selection. A two-phase sample is assumed. In the first phase, units are selected with probability proportional to their size. In the second phase, units are selected using median ranked set sampling to increase the efficiency of the estimators relative to simple random sampling. There is also an increase in the efficiency relative to ranked set sampling (for some probability distribution functions). There will be a loss in efficiency depending on the amount of errors in ranking the units, the median ranked set sampling can be used to reduce the errors in ranking the units selected from the population. Estimators of the population mean and the population size are considered. The median ranked set sampling with probability proportion to size and with errors in ranking is considered and compared with ranked set sampling with errors in ranking. Computer simulation results for some probability distributions are also given.     

New Ranked Set Sampling for One‐Parameter Family of Distributions

Bhoj (1997c) proposed a new ranked set sampling (NRSS) procedure for a specific two‐parameter family of distributions when the sample size is even. This NRSS procedure can be applied to one‐parameter family of distributions when the sample size is even. However, this procedure cannot be used if the sample size is odd. Therefore, in this paper, we propose a modified version of the NRSS procedure which can be used for one‐parameter distributions when the sample size is odd. Simple estimator for the parameter based on proposed NRSS is derived. The relative precisions of this estimator are higher than those of other estimators which are based on other ranked set sampling procedures and the best linear unbiased estimator using all order statistics.     

Ranked Set Sampling and the Line Intercept Method: A More Efficient Procedure

Ranked set sampling is a method which may be used to increase the efficiency of the estimator of the mean of a population. Ranked set sampling with size biased probability of selection (i.e., the items are selected with probability proportion to its size) is combined with the line intercept method to increase the efficency of estimating cover, density and total amount of some variable of interest (e.g. biomass). A two-stage sampling plan is suggested with line intercept sampling in the first stage. Simple random sampling and ranked set sampling are compared in the second stage to show that the unbiased estimators of density, cover and total amount of some variable of interest based on ranked set sampling have smaller variances than the usual unbiased estimator based on simple random sampling. Efficiency is increased by reducing the number of items which are measured on a transect or by increasing the number of independent transects utilized in a study area. An application procedure is given for estimation of coverage, density and number of stems of mountain mahogany (Cercocarpus montanus) in a study area east of Laramie, Wyoming.     

Estimation of Ratio Using Rank Set Sampling

Ranked set sampling (RSS) as suggested by McIntyre (1952) and developed by Takahasi and Wakimoto (1968) is used to estimate the ratio. It is proved that by using RSS method the efficiency of the estimator relative to the simple random sampling (SRS) method has increased. Computer simulated results are given. An example using real data is presented to illustrate the computations.     

Sampling Hubbell's neutral theory of biodiversity

In the context of neutral theories of community ecology, a novel genealogy‐based framework has recently furnished an analytic extension of Ewens’ sampling multivariate abundance distribution, which also applies to a random sample from a local community. Here, instead of taking a multivariate approach, we further develop the sampling theory of Hubbell's neutral spatially implicit theory and derive simple abundance distributions for a random sample both from a local community and a metacommunity. Our result is given in terms of the average number of species with a given abundance in any randomly extracted sample. Contrary to what has been widely assumed, a random sample from a metacommunity is not fully described by the Fisher log‐series, but by a new distribution. This new sample distribution matches the log‐series expectation at high biodiversity values (θ > 1) but clearly departs from it for species‐poor metacommunities (θ < 1). Our theoretical framework should be helpful in the better assessment of diversity and testing of the neutral theory by using abundance data.     

Ranked set sampling for replicated sampling designs

In practical ecological sampling studies, a certain design (such as plot sampling or line-intercept sampling) is usually replicated more than once. For each replication, the Horvitz-Thompson estimation of the objective parameter is considered. Finally, an overall estimator is achieved by averaging the single Horvitz-Thompson estimators. Because the design replications are drawn independently and under the same conditions, the overall estimator is simply the sample mean of the Horvitz-Thompson estimators under simple random sampling. This procedure may be wisely improved by using ranked set sampling. Hence, we propose the replicated protocol under ranked set sampling, which gives rise to a more accurate estimation than the replicated protocol under simple random sampling.     

Parameters Estimation in a Simple Linear Regression Using Rank Set Sampling

Ranked set sampling (RSS) as suggested by MCINTYRE (1952) and TAKAHASI and WAKIMOTO (1968) may be used to estimate the parameters of the simple regression line. The objective is to use the RSS method to increase the efficiency of the estimators relative to the simple random sampling (SRS) method. Estimators of the slope and intercept are considered. Computer simulated results are given, and an example using real data presented to illustrate the computations.     

Estimation of Parameters for One-Way Layout with Rank Set Sampling

Ranked set sampling (RSS) as suggested by McIntyre (1952) and independently by Takahasi and Wakimoto (1968) may be used to estimate the parameters of the one-way layout. The objective is to use the RSS method to increase the efficiency of the estimators relative to the simple random (SRS) method. Estimators of the populations (treatments) effect are considered. Computer simulated results are given, and an example using real data presented to illustrate the computations.     

祁连山高寒草原主要植物种群格局对冬季放牧的响应

以肃南裕固族自治县鹿场冬季牧场为研究对象,采用偏离系数及t检验方法结合Ripley''s K函数及蒙特卡罗随机模拟对紫花针茅种群与醉马草种群进行格局分析,进一步采用计盒维数与信息维数对紫花针茅种群与醉马草种群进行分形分析,揭示在放牧系统中优势植物的种群变化。格局分析结果表明：牧场植物的种群格局出现5种变化规律。紫花针茅种群的偏离系数在放牧率3.64 AUM/hm²时^#,出现最小值。醉马草种群的偏离系数在放牧率4.16AUM/hm²时,出现最大值。在0-1 m的研究尺度内,紫花针茅种群与醉马草种群的Ripley''s K函数结果均在蒙特卡洛随机模拟区间内,呈随机分布格局。分形分析结果表明：伴随放牧率增大,紫花针茅种群的计盒维数区间为[1.596,1.962],醉马草种群的计盒维数区间为[1.831,1.945];紫花针茅种群的信息维数区间为[1.590,1.899],醉马草种群的信息维数区间为[1.633,1.913]。在放牧率4.34 AUM/hm²时,紫花针茅种群与醉马草种群空间占据差值达到最大,信息维数相对较低,种群分布相对均匀,格局变化相对缓慢,经济效益与生态效益相平衡,最适合管理牧场。     

Estimation of a Joint Distribution Function with Multivariate Interval‐Censored Data when the Nonparametric MLE is not Unique

A nonparametric estimator of a joint distribution function F₀ of a d‐dimensional random vector with interval‐censored (IC) data is the generalized maximum likelihood estimator (GMLE), where d ≥ 2. The GMLE of F₀ with univariate IC data is uniquely defined at each follow‐up time. However, this is no longer true in general with multivariate IC data as demonstrated by a data set from an eye study. How to estimate the survival function and the covariance matrix of the estimator in such a case is a new practical issue in analyzing IC data. We propose a procedure in such a situation and apply it to the data set from the eye study. Our method always results in a GMLE with a nonsingular sample information matrix. We also give a theoretical justification for such a procedure. Extension of our procedure to Cox's regression model is also mentioned.     

Pair Rank Set Sampling

Ranked set sampling (RSS) as suggested by McIntyre (1952) may be modified to introduced a new sampling method called pair rank set sampling (PRSS), which might be used in some area of application instead of the RSS to increase the efficiency of the estimators relative to the simple random sampling (SRS) method. Estimators of the population mean are considered. An example using real data is presented to illustrate computations.     

Some properties of the point processes generated by integral density modulation of renewal processes

Integral density modulation of point processes is defined, and the properties of the modulated point processes are described. When a homogeneous renewal process is modulated by a step random signal, the mathematical expressions are derived of the probability density functions of the sums of r-successive inter point intervals, the intensity functions and the first order correlation coefficient of intervals. These quantities are calculated and illustrated for several parameter values. Modulated point sequences are generated by computer simulation method. The interval histograms and the serial correlation coefficients of counts and of intervals of the sequences are obtained. The results are compared with the theoretical results on the point processes modulated by the step random signal.Abbreviations IDM Integral density modulation - SRS Step random signal - PP Point process - PS Point sequence - PDF Probability density function - CPDF Conditional probability density function - JCPDF Joint conditional probability density function - SED Special Earlangian distribution - SCC Serial correlation coefficients - SIC Serial intensity coefficients     

Improved nonparametric estimation of the optimal diagnostic cut‐off point associated with the Youden index under different sampling schemes

A diagnostic cut‐off point of a biomarker measurement is needed for classifying a random subject to be either diseased or healthy. However, the cut‐off point is usually unknown and needs to be estimated by some optimization criteria. One important criterion is the Youden index, which has been widely adopted in practice. The Youden index, which is defined as the maximum of (sensitivity + specificity ?1), directly measures the largest total diagnostic accuracy a biomarker can achieve. Therefore, it is desirable to estimate the optimal cut‐off point associated with the Youden index. Sometimes, taking the actual measurements of a biomarker is very difficult and expensive, while ranking them without the actual measurement can be relatively easy. In such cases, ranked set sampling can give more precise estimation than simple random sampling, as ranked set samples are more likely to span the full range of the population. In this study, kernel density estimation is utilized to numerically solve for an estimate of the optimal cut‐off point. The asymptotic distributions of the kernel estimators based on two sampling schemes are derived analytically and we prove that the estimators based on ranked set sampling are relatively more efficient than that of simple random sampling and both estimators are asymptotically unbiased. Furthermore, the asymptotic confidence intervals are derived. Intensive simulations are carried out to compare the proposed method using ranked set sampling with simple random sampling, with the proposed method outperforming simple random sampling in all cases. A real data set is analyzed for illustrating the proposed method.     

Linear Rank Tests Derived from a Superpopulation Model

A superpopulation model generates the probabilities of a Bernouilli random variable. The ranks of the involved variables are considered as survey weights. The distribution f each linear rank statistic is derived under the null hypothesis for the two sample problem and for the case k2 when a simple random sampling or stratified sampling is used. The growth of a population of insects and the behavior of patients with imsomnia are studied using these procedures.     

On the KOLMOGOROV-SMIRNOV Test for the POISSON Distribution with Unknown Mean

The standard tables for the KOLMOGOROV -SMIRNOV test are valid only in the case of testing whether a set of observations is from a completely specified cumulative distribution, F₀(X), with all parameters known. If the parameters are unknown and must be estimated from the sample, then the tables are not valid. A table is given in this paper for use with the KOLMOGOROV -SMIRNOV statistic in the case of testing whether a set of observations is from the POISSON distribution with an unknown mean that must be estimated from the sample. The table is obtained from a Monte Carlo calculation.