Parametric ranked set sampling

Ranked set sampling was introduced by McIntyre (1952,Australian Journal of Agricultural Research,3, 385–390) as a cost-effective method of selecting data if observations are much more cheaply ranked than measured. He proposed its use for estimating the population mean when the distribution of the data was unknown. In this paper, we examine the advantage, if any, that this method of sampling has if the distribution is known, for a specific family of distributions. Specifically, we consider estimation of and for the family of random variables with cdf's of the formF(x–/). We find that the ranked set sample does provide more information about both and than a random sample of the same number of observations. We examine both maximum likelihood and best linear unbiased estimation of and , as well as methods for modifying the ranked set sampling procedure to provide even better estimation.This paper has been prepared with partial support from the United States Environmental Protection Agency under Cooperative Agreement Number CR821801-01-0. The contents have not been subjected to Agency review and therefore do not necessarily reflect the views or policies of the Agency and no official endorsement should be inferred.     

Parametric estimation for the simple linear regression model under moving extremes ranked set sampling design

Cost effective sampling design is a major concern in some experiments especially when the measurement of the characteristic of interest is costly or painful or time consuming.Ranked set sampling(RSS) was first proposed by McIntyre [1952. A method for unbiased selective sampling, using ranked sets. Australian Journal of Agricultural Research 3, 385-390]as an effective way to estimate the pasture mean. In the current paper, a modification of ranked set sampling called moving extremes ranked set sampling(MERSS) is considered for the best linear unbiased estimators(BLUEs) for the simple linear regression model. The BLUEs for this model under MERSS are derived. The BLUEs under MERSS are shown to be markedly more efficient for normal data when compared with the BLUEs under simple random sampling.     

Rounded data analysis based on multi-layer ranked set sampling

Observations of sampling are often subject to rounding, but are modeled as though they were unrounded. This paper examines the impact of rounding errors on parameter estimation with multi-layer ranked set sampling. It shows that the rounding errors seriously distort the behavior of covariance matrix estimate, and lead to inconsistent estimation. Taking this into account, we present a new approach to implement the estimation for this model, and further establish the strong consistency and asymptotic normality of the proposed estimators. Simulation experiments show that our estimates based on rounded multi-layer ranked set sampling are always more efficient than those based on rounded simple random sampling.     

Inference on a distribution function from ranked set samples

Consider independent observations \((X_i,R_i)\) with random or fixed ranks \(R_i\), while conditional on \(R_i\), the random variable \(X_i\) has the same distribution as the \(R_i\)-th order statistic within a random sample of size k from an unknown distribution function F. Such observation schemes are well known from ranked set sampling and judgment post-stratification. Within a general, not necessarily balanced setting we derive and compare the asymptotic distributions of three different estimators of the distribution function F: a stratified estimator, a nonparametric maximum-likelihood estimator and a moment-based estimator. Our functional central limit theorems generalize and refine previous asymptotic analyses. In addition, we discuss briefly pointwise and simultaneous confidence intervals for the distribution function with guaranteed coverage probability for finite sample sizes. The methods are illustrated with a real data example, and the potential impact of imperfect rankings is investigated in a small simulation experiment.

     

Finite population corrections for ranked set sampling

Ranked set sampling (RSS) for estimating a population mean is studied when sampling is without replacement from a completely general finite populationx=(x ₁,x ₂,...,x _N ). Explicit expressions are obtained for the variance of the RSS estimator and for its precision relative to that of simple random sampling without replacement. The critical term in these expressions involves a quantity =(x–)(x–) where is anN × N matrix whose entries are functions of the population sizeN and the set-sizem, but where does not depend on the population valuesx. A computer program is given to calculate for arbitraryN andm. When the population follows a linear (resp., quadratic) trend, then is a polynomial inN of degree 2m+2 (resp., 2m+4). The coefficients of these polynomials are evaluated to yield explicit expressions for the variance and the relative precision of for these populations. Unlike the case of sampling from an infinite population, here the relative precision depends upon the number of replications of the set sizem.Prepared with partial support from the Statistical Analysis and Computing Branch. Environmental Statistics and Information Division, Office of Policy, Planning, and Evaluation, United States Environmental Protection Agency, Washington, DC under a Cooperative Agreement Number CR-821531. The contents have not been subjected to Agency review and therefore do not necessarily reflect the views of the Agency and no official endorsement should be inferred.     

Sign tests using ranked set sampling with unequal set sizes

This paper considers sign test under ranked set sampling with unequal set sizes (RSSU), and proposes weighted sign tests associated with judgment ranks. The optimal weight vector is shown to be distribution-free, and RSSU is shown to be more efficient than ranked set sampling.     

Interval estimation of $$$$ in ranked set sampling

This article deals with constructing a confidence interval for the reliability parameter using ranked set sampling. Some asymptotic and resampling-based intervals are suggested, and compared with their simple random sampling counterparts using Monte Carlo simulations. Finally, the methods are applied on a real data set in the context of agriculture.     

Log-logistic parameters estimation using moving extremes ranked set sampling design

In statistical parameter estimation problems, how well the parameters are estimated largely depends on the sampling design used. In the current paper, a modification of ranked set sampling(RSS) called moving extremes RSS(MERSS) is considered for the estimation of the scale and shape parameters for the log-logistic distribution. Several traditional estimators and ad hoc estimators will be studied under MERSS. The estimators under MERSS are compared to the corresponding ones under SRS. The simulation results show that the estimators under MERSS are significantly more efficient than the ones under SRS.     

Resampling based inference for a distribution function using censored ranked set samples

This article deals with constructing confidence intervals/bands for a distribution function based on censored ranked set samples. Toward this end, a resampling plan is suggested and its validity is investigated. Monte Carlo simulations are used to compare performances of the bootstrap confidence intervals with their asymptotic analogs, and their modifications by jackknife. An environmental data set is finally analyzed.     

Estimation of a parameter of Morgenstern type bivariate exponential distribution by ranked set sampling

Ranked set sampling is applicable whenever ranking of a set of sampling units can be done easily by a judgement method or based on the measurement of an auxiliary variable on the units selected. In this work, we consider ranked set sampling, in which ranking of units are done based on measurements made on an easily and exactly measurable auxiliary variable X which is correlated with the study variable Y. We then estimate the mean of the study variate Y by the BLUE based on the measurements made on the units of the ranked set sampling regarding the study variable Y, when (X ,Y) follows a Morgenstern type bivariate exponential distribution. We then consider unbalanced multistage ranked set sampling and estimate the mean of the study variate Y by the BLUE based on the observations made on the units of multistage ranked set sample regarding the study variable Y. Efficiency comparison is also made on all estimators considered in this work.     

Group sequential comparison of two binomial proportions under ranked set sampling design

Ranked set sampling (RSS) is a technique for incorporating auxiliary (concomitant) information into estimation and testing procedures right at the design stage. In this paper, we propose group sequential testing procedures for comparing two treatments with binary outcomes under an RSS scheme with perfect ranking. We compare the power, the average sample sizes and type I errors of the proposed tests to those of the group sequential tests based on simple random sampling schemes. We illustrate the usefulness of the methodology by using data from a clinical trial on leukemia.     

Exact inference for a simple step-stress model from the exponential distribution under time constraint

In reliability and life-testing experiments, the researcher is often interested in the effects of extreme or varying stress factors such as temperature, voltage and load on the lifetimes of experimental units. Step-stress test, which is a special class of accelerated life-tests, allows the experimenter to increase the stress levels at fixed times during the experiment in order to obtain information on the parameters of the life distributions more quickly than under normal operating conditions. In this paper, we consider the simple step-stress model from the exponential distribution when there is time constraint on the duration of the experiment. We derive the maximum likelihood estimators (MLEs) of the parameters assuming a cumulative exposure model with lifetimes being exponentially distributed. The exact distributions of the MLEs of parameters are obtained through the use of conditional moment generating functions. We also derive confidence intervals for the parameters using these exact distributions, asymptotic distributions of the MLEs and the parametric bootstrap methods, and assess their performance through a Monte Carlo simulation study. Finally, we present two examples to illustrate all the methods of inference discussed here.     

Proportion estimation in ranked set sampling in the presence of tie information

Ranked set sampling (RSS) is a statistical technique that uses auxiliary ranking information of unmeasured sample units in an attempt to select a more representative sample that provides better estimation of population parameters than simple random sampling. However, the use of RSS can be hampered by the fact that a complete ranking of units in each set must be specified when implementing RSS. Recently, to allow ties declared as needed, Frey (Environ Ecol Stat 19(3):309–326, 2012) proposed a modification of RSS, which is to simply break ties at random so that a standard ranked set sample is obtained, and meanwhile record the tie structure for use in estimation. Under this RSS variation, several mean estimators were developed and their performance was compared via simulation, with focus on continuous outcome variables. We extend the work of Frey (2012) to binary outcomes and investigate three nonparametric and three likelihood-based proportion estimators (with/without utilizing tie information), among which four are directly extended from existing estimators and the other two are novel. Under different tie-generating mechanisms, we compare the performance of these estimators and draw conclusions based on both simulation and a data example about breast cancer prevalence. Suggestions are made about the choice of the proportion estimator in general.     

Empirical process approach to some two-sample problems based on ranked set samples

We study the asymptotic properties of both the horizontal and vertical shift functions based on independent ranked set samples drawn from continuous distributions. Several tests derived from these shift processes are developed. We show that by using balanced ranked set samples with bigger set sizes, one can decrease the width of the confidence band and hence increase the power of these tests. These theoretical findings are validated through small-scale simulation studies. An application of the proposed techniques to a cancer mortality data set is also provided. Part of the research was conducted while Kaushik Ghosh was visiting Statistical Research and Applications Branch of the National Cancer Institute on an Intergovernmental Personnel Assignment.     

Exact two-sample nonparametric test for quantile difference between two populations based on ranked set samples

Creation of a ranked set sample, by its nature, involves judgment ranking error within set units. This ranking error usually distorts statistical inference of the population characteristics. Tests may have inflated sizes, confidence intervals may have incorrect coverage probabilities, and the estimators may become biased. In this paper, we develop an exact two-sample nonparametric test for quantile shift between two populations based on ranked set samples. This test is based on two independent exact confidence intervals for the quantile of interest corresponding to the two populations and rejects the null hypothesis of equal quantiles if these intervals are disjoint. It is shown that a pair of 83 and 93% confidence intervals provide a 5 and 1% test for the equality of quantiles. The proposed test is calibrated for the effect of judgment ranking error so that the test has the correct size even under a wide range of judgment ranking errors. A small scale simulation study suggests that the test performs quite well for cycle sizes as small as 2.     

Inference for ordered parameters in multinomial distributions

This paper discusses inference for ordered parameters of multinomial distributions. We first show that the asymptotic distributions of their maximum likelihood estimators (MLEs) are not always normal and the bootstrap distribution estimators of the MLEs can be inconsistent. Then a class of weighted sum estimators (WSEs) of the ordered parameters is proposed. Properties of the WSEs are studied, including their asymptotic normality. Based on those results, large sample inferences for smooth functions of the ordered parameters can be made. Especially, the confidence intervals of the maximum cell probabilities are constructed. Simulation results indicate that this interval estimation performs much better than the bootstrap approaches in the literature. Finally, the above results for ordered parameters of multinomial distributions are extended to more general distribution models. This work was supported by National Natural Science Foundation of China (Grant No. 10371126)     

Molecules of a partially ordered set

It is shown that in a general partially ordered set (P, ≤) the notion of a molecule (i.e., a nonzero elementm ofP such that any two nonzero elements ofP which are ≤m have a nonzero lower bound) is in close analogy to the notion of an atom in a Boolean ring. Presented by R. P. Dilworth.     

A simple query language based on set algebra

A design process for a query language based on set algebra is described. Key principles used in the design are: make explicit assumptions about the end users background, delimit the scope of the language, and make it simple by omitting all features that have not been found necessary. The language closely mirrors concepts well known from algebra and set theory: it contains no join or relational division, and it has a high expressive power.     

Average height in a partially ordered set

The average height of an element x in a finite poset P is the expect below x in a random linear extension of P. We prove a number of theorems about average height, some intuitive and some not, using a recent result of L.A. Shepp.     

More examples on ordered set reconstruction

We show that there are nonisomorphic ordered sets P and Q that have the same maximal and minimal decks and a rank k such that there is a map B from the elements of rank k in P to the elements of rank k in Q such that P{x} is isomorphic to Q{B(x)} for all x of rank k in P. The examples are preceded by a criterion as to when two nonisomorphic ordered sets will have a rank k and a map B as above.