rank regression in ranked set samples the use of statistical methods based on ranked set sampling can lead to a substantial improvement over analog methods associated with simple random sampling schemes this article develops a rank based estimator and testing procedures for linear models for ranked set samples the estimator is defined as the minimizer of the rank dispersion function with wilcoxon scores it is shown that the estimator of the regression parameter is asymptotically normal and has higher pitman asymptotic efficiency than a simple random sample rank regression estimator three testing procedures are developed to test a general linear hypothesis dispersion wald and aligned rank tests it is shown that all of these test statistics converge to a chi squared distribution and the aligned rank test reduces to a simple random sample analog of kruskal wallis test for one way analysis of variance under the assumption of perfect judgment ranking optimal allocation of order statistics are constructed for set sizes smaller than 7 the

estimation of parameters of the generalized geometric distribution using ranked set sampling the estimates of the parameters of the generalized geometric distribution are obtained by using ranked set sampling procedure these estimates are compared with the ordered least squares estimates given by downton 1954 annals of mathematical statistics 25 303 316 it is shown that the relative precisions of our estimators are higher than those of the ordered least squares estimators furthermore the relative precision of our estimator for the population mean is higher than the usual estimator based on ranked set sampling

a formal classification of hopf bogdanov takens singularities of codimension three abstract this paper is concerned with hopf bogdanov takens singularities that is singularities of 4 dimensional vector fields whose 1 jet is linearly conjugate to y x v u u v they first occur in generic 3 parameter families a normal form for these singularities and their unfoldings is given up to an arbitrary order working with a second order truncation of the normal form for the singularity we provide a formal classification at the level of codimension three the proof relies on blowing up techniques and estimation of abelian integrals previous results in the literature about the classification of 3 dimensional nilpotent singularities of codimension four are essential for our purposes

nonparametric tests for perfect judgment rankings the ranked set sampling literature includes both inference procedures that rely on the assumption of perfect rankings and inference procedures that are robust to violations of this assumption procedures that assume perfect rankings tend to be more efficient when rankings are in fact perfect but they may be invalid when perfect rankings fail as a result users of ranked set sampling must decide between efficiency and robustness and there is at present little to guide their decision in this article we introduce three rank based goodness of fit tests that may be consulted in making these decisions our strategy in producing these tests is to think of the judgment order statistic classes as separate samples compute the ranks of the units from each sample within the combined sample and use these ranks to test whether the judgment rankings are perfect consideration of both power and ease of use leads us to recommend use of a test that rejects when the concordance between the vector of mean ranks and its

nonparametric two sample methods for ranked set sample data a new collection of procedures is developed for the analysis of two sample ranked set samples providing an alternative to the bohn wolfe procedure these procedures split the data based on the ranks in the ranked set sample and lead to tests for the centers of distributions confidence intervals and point estimators the advantages of the new tests are that they require essentially no assumptions about the mechanism by which rankings are imperfect that they maintain their level whether rankings are perfect or imperfect that they lead to generalizations of the bohn wolfe procedure that can be used to increase power in the case of perfect rankings and that they allow one to analyze both balanced and unbalanced ranked set samples a new class of imperfect ranking models is proposed and the performance of the procedure is investigated under these models when rankings are random a theorem is presented which characterizes efficient data splits because random rankings are equivalent to iid samples thi

general ranked set sampling with cost considerations nahhas wolfe and chen 2002 biometrics 58 964 971 considered optimal set size for ranked set sampling rss with fixed operational costs this framework can be very useful in practice to determine whether rss is beneficial and to obtain the optimal set size that minimizes the variance of the population estimator for a fixed total cost in this article we propose a scheme of general rss in which more than one observation can be taken from each ranked set this is shown to be more cost effective in some cases when the cost of ranking is not so small we demonstrate using the example in nahhas wolfe and chen 2002 biometrics 58 964 971 by taking two or more observations from one set even with the optimal set size from the rss design can be more beneficial

judgment post stratification in finite mixture modeling an example in estimating the prevalence of osteoporosis judgment post stratification is used to supplement observations taken from finite mixture models with additional easy to obtain rank information and incorporate it in the estimation of model parameters to do this sampled units are post stratified on ranks by randomly selecting comparison sets for each unit from the underlying population and assigning ranks to them using available auxiliary information or judgment ranking this results in a set of independent order statistics from the underlying model where the number of units in each rank class is random we consider cases where one or more rankers with different ranking abilities are used to provide judgment ranks the judgment ranks are then combined to produce a strength of agreement measure for each observation this strength measure is implemented in the maximum likelihood estimation of model parameters via a suitable expectation maximization algorithm simulation studies are conducted to evaluate the performance of the estimators with or without the extra rank information results are applied to bone mineral density data from the third national health and nutrition examination survey to estimate the prevalence of osteoporosis in adult women aged 50 and over

invariant manifolds of maps close to a product of rotations close to the rotation axis abstract we consider families of maps depending on a parameter e such that for e 0 the map becomes a product of linear rotations in r 2m n and for e 0 the map is weakly attracting in the product of the rotation planes and weakly repelling in some complementary subspace we prove that the unstable manifold converges to the complementary subspace in the cr topology the case r included we consider both the local and the global manifolds for that we prove some results on families of maps near a norm one linear map which are of independent interest

a score regression approach to assess calibration of continuous probabilistic predictions calibration the statistical consistency of forecast distributions and the observations is a central requirement for probabilistic predictions calibration of continuous forecasts is typically assessed using the probability integral transform histogram in this article we propose significance tests based on scoring rules to assess calibration of continuous predictive distributions for an ideal normal forecast we derive the first two moments of two commonly used scoring rules the logarithmic and the continuous ranked probability score this naturally leads to the construction of two unconditional tests for normal predictions more generally we propose a novel score regression approach where the individual scores are regressed on suitable functions of the predictive variance this conditional approach is applicable even for certain nonnormal predictions based on the dawid sebastiani score two case studies illustrate that the score regression approach has typically more power in detecting miscalibrated forecasts than the other approaches considered including a recently proposed technique based on conditional exceedance probability curves

efficient regression analysis with ranked set sampling this article is motivated by a lung cancer study where a regression model is involved and the response variable is too expensive to measure but the predictor variable can be measured easily with relatively negligible cost this situation occurs quite often in medical studies quantitative genetics and ecological and environmental studies in this article by using the idea of ranked set sampling rss we develop sampling strategies that can reduce cost and increase efficiency of the regression analysis for the above mentioned situation the developed method is applied retrospectively to a lung cancer study in the lung cancer study the interest is to investigate the association between smoking status and three biomarkers polyphenol dna adducts micronuclei and sister chromatic exchanges optimal sampling schemes with different optimality criteria such as a d and integrated mean square error imse optimality are considered in the application with set size 10 in rss the improvement of the optimal schemes over simple random sampling srs is great for instance by using the optimal scheme with imse optimality the imses of the estimated regression functions for the three biomarkers are reduced to about half of those incurred by using srs

the effect of imperfect judgment rankings on properties of procedures based on the ranked set samples analog of the mann whitney wilcoxon statistic abstract in this article we address the issue of imperfect judgment rankings in ranked set sampling and in particular their effect on the properties of test procedures based on the ranked set samples analog of the mann whitney wilcoxon statistic u rss we consider the impact of these imperfect rankings on the null distribution of the statistic and the resulting effect on the nominal level of associated hypothesis tests we propose a model for the probabilities of imperfect judgment rankings based on the concept of expected spacings and use this model to study the properties of tests based on u rss this investigation includes both small sample monte carlo power simulations and a detailed analysis of the asymptotic relative efficiency properties of the u rss procedure we also examine as an indication of the merits of using ranked set sampling the relative cost of measuring the value of a sample item as compared to obtaining a judgment ordering of a set of sample items

thin algebras of embedding dimension three let r r0 r1 r2 be a finitely generated commutative graded k algebra with hilbert series hr n 0dimk rn n suppose we require that r have a presentation as r k x1 xg 1 r where g and r and the degrees d1 dr of the homogeneous polynomials 1 r are specified in advance no one knows precisely which hubert series such an r may have a lower bound is the coefficient wise inequality hr 1 g i 1r 1 di absolute value symbols denoting the initial non negative segment of a power series can this lower bound always be attained this deceptively simple sounding question has been answered affirmatively only when g 2 r froberg report no 37 department of mathematics university of stockholm stockholm sweden 1982 a iarrobino memoir no 188 amer math soc providence r i 1977 when r g and when r g 1 and char k 0 a iarrobino compressed algebras trans amer math soc 285 1984 337 378 this paper considers the case g 3 settling it affirmatively whenever k is an infinite field

nonparametric two sample procedures for ranked set samples data abstract ranked set samples have been shown to lead to improved methods of estimation in parametric settings under specific distributional forms when actual measurement of the sample observations is difficult but ranking them is relatively easy the earliest work with ranked set data concentrated on estimating a population mean or variance more recently a ranked set sample estimator of a cumulative distribution function was developed and used to obtain a simultaneous confidence interval for the function in this article we take the next logical step and use this ranked set empirical distribution function to construct distribution free competitors to the standard mann whitney wilcoxon estimation and testing procedures the appropriate null distribution tables for the associated test are presented for the case of perfect ranking asymptotic relative efficiency comparisons between the simple random sample mann whitney wilcoxon procedures and their ranked set analogues are discussed and the results of a s

analysis of longitudinal data with unequally spaced observations and time dependent correlated errors a linear model for repeated measurements is proposed in which the correlation structure includes a transformation of the time scale this transformation can produce nonstationary covariance structures within subjects with stationarity as a special case restricted maximum likelihood methods for parameter estimation are discussed the method is applied to simulated data as well as speech recognition data from the iowa cochlear implant project the growth curve for this audiologic performance measure is shown together with estimates of the standard errors of predictions at given months

modified maximum likelihood estimators using ranked set sampling the closed form maximum likelihood estimators mles of population mean and variance under ranked set sampling rss do not exist since the likelihood equations involve nonlinear functions and have usually no explicit solutions we derive modified maximum likelihood mml estimators for the population mean and variance under rss and show that they are considerably more efficient than rss estimators furthermore we suggest two new estimators for the unknown parameters using two modified ranked set sampling methods and show that these methods make the variances of both mml and rss estimators smaller

judgement post stratification with imprecise rankings summary judgement post stratification which is based on ideas similar to those in ranked set sampling relies on the ability of a ranker to forecast the ranks of potential observations on a set of units in practice the authors sometimes find it difficult to assign these ranks this note shows how one can borrow techniques from the literature on finite population sampling to allow a probabilistic ranking of the units in a set thus facilitating use of these sampling plans and improving estimation the same techniques provide one approach to estimation using a judgement post stratified sample with multiple rankers the technique is illustrated on allometric data relating brain weight to body weight in different species of mammals and on a study of student performance in graduate school

an exact control versus treatment comparison test based on ranked set samples a control versus treatment multiple comparison test procedure is developed based on ranked set samples the test constructed based on k independent exact median confidence intervals corresponding to the k populations rejects the null hypothesis of equal medians if the median confidence intervals of the control group and any one of the other k 1 treatment groups are disjoint the use of the proposed test is illustrated with data from an agricultural experiment

concomitants of multivariate order statistics with application to judgment poststratification we generalize the definition of a concomitant of an order statistic in the multivariate case develop general expressions for its density and establish related properties we study the concomitant of a normal random vector in detail and discuss methods for calculating its moments furthermore we apply the theory to develop new estimators of the mean from a judgment poststratified sample where poststrata are formed by rank classes of auxiliary variables our estimators are shown to be more efficient than existing ones and robust against violations of the normality assumption they are also well suited to applications requiring cost efficiency

using ranked set sampling with cluster randomized designs for improved inference on treatment effects abstractthis article examines the use of ranked set sampling rss with cluster randomized designs crds for potential improvement in estimation and detection of treatment or intervention effects outcome data in cluster randomized studies typically have nested structures where hierarchical linear models hlms become a natural choice for data analysis however nearly all theoretical developments in rss to date are within the structure of one level models thus implementation of rss at one or more levels of an hlm will require development of new theory and methods under rss structured crds developed to incorporate rss at different levels a nonparametric estimator of the treatment effect is proposed and its theoretical properties are studied under a general hlm that has almost no distributional assumptions we formally quantify the magnitude of the improvement from using rss over srs simple random sampling investigate the relationship between design parameters and relative efficiency and establ

a new ranked set sample estimator of variance summary we develop an unbiased estimator of the variance of a population based on a ranked set sample we show that this new estimator is better than estimating the variance based on a simple random sample and more efficient than the estimator based on a ranked set sample proposed by stokes also a test to determine the effectiveness of the judgment ordering process is proposed

selection through an associated characteristic with applications to the random effects model abstract the problem of choosing the best k objects out of n is considered when instead of measurements yi i 1 n of primary interest only associated measurements xi are readily obtainable large measurements are regarded as desirable it is assumed that the n pairs xi yi are a random sample from a continuous population a general expression is developed for the probability that the s objects with the largest x values include the k objects k s with the largest y values when x and y are bivariate normal with correlation a table of is presented for n 15 this table gives immediately the smallest s for which p where p is preassigned several applications are treated especially one to the random effects model for n varieties with r replications it is shown how reasonable values of r may be found to provide a reduction in the number of varieties from n to s such that the chosen s contain the k best varieties with at least probability p a method conditional on the observe

distribution of the maximum of concomitants of selected order statistics for a random sample of size n from an absolutely continuous bivariate population x y let x i n denote the i th order statistic of the x sample values the y value associated with x i n is denoted by y lbrack i n rbrack and is called the concomitant of the i th order statistic for 1 leq k leq n let v k n max y lbrack n k 1 n rbrack ldots y lbrack n n rbrack in this paper we discuss the finite sample and the asymptotic distributions of v k n we investigate the limit distribution of v k n as n rightarrow infty when k is held fixed and when k lbrack np rbrack 0 p 1 in both cases we obtain simple sufficient conditions and determine the associated norming constants we apply our results to some interesting situations including the bivariate normal population and the simple linear regression model

unequal allocation models for ranked set sampling with skew distributions ranked set sampling performs better than simple random sampling when units corresponding to each rank are allocated equally the performance of ranked set sampling further improves when appropriate unequal allocation is implemented instead of equal allocation the optimal allocation based on neyman s approach requires the sample size corresponding to each rank order to be proportional to its standard deviation in most practical situations the unavailability of standard deviations of the rank orders makes the optimal allocation impractical an attempt is made in this paper to make a near optimal allocation by exploiting knowledge of other more easily available characteristics of the population like skewness kurtosis and coefficient of variation for positively skewed distributions it is generally observed that the variances of the rank order statistics increase as the rank orders increase thus it is appropriate to observe ranks corresponding to the right tail more frequently than others we consider two right tail allocation models that assign more quantifications to i the largest order statistic and ii the two largest order statistics the performance of both the models is better than the equal allocation model and the latter is an improvement over the former under appropriate choices of allocation factors we study the role played by skewness kurtosis and the coefficient of variation for obtaining the allocation factor s and also in devising rules of thumb it appears that the coefficient of variation is a better measure to determine the rule of thumb

characterization of a ranked set sample with application to estimating distribution functions abstract ranked set sampling has been shown to provide improved estimators of the mean and variance when actual measurement of the observations is difficult but ranking of the elements in a sample is relatively easy this result holds even if the ranking is imperfect in this article we provide a characterization of a ranked set sample that makes the source of this additional information intuitively clear it is applied to show that the empirical distribution function of a ranked set sample is unbiased and has greater precision than that from a random sample the null distribution of a kolmogorov smirnov statistic based on this empirical distribution function is derived for the case in which perfect ranking is possible it is seen to be stochastically smaller than the usual kolmogorov smirnov statistic based on a simple random sample resulting in a smaller simultaneous confidence interval for the cumulative distribution function

monomial ideals almost complete intersections and the weak lefschetz property many algebras are expected to have the weak lefschetz property though this is often very difficult to establish we illustrate the subtlety of the problem by studying monomial and some closely related ideals our results exemplify the intriguing dependence of the property on the characteristic of the ground field and on arithmetic properties of the exponent vectors of the monomials

invariant manifolds of dynamical systems close to a rotation transverse to the rotation axis abstract we consider one parameter families of vector fields depending on a parameter such that for 0 the system becomes a rotation of r 2 r n around 0 r n and such that for 0 the origin is a hyperbolic singular point of saddle type with say attraction in the rotation plane and expansion in the complementary space we look for a local subcenter invariant manifold extending the stable manifolds to 0 afterwards the analogous case for maps is considered in contrast with the previous case the arithmetic properties of the angle of rotation play an important role

judgement post stratification for designed experiments in many scientific studies information that is not easily translated into covariates is ignored in the analysis however this type of information may significantly improve inference in this research we apply the idea of judgment post stratification to utilize such information specifically we consider experiments that are conducted under a completely randomized design sets of experimental units are formed and the units in a set are ranked estimation is performed conditional on the sets and ranks we propose a new estimator for a treatment contrast we improve the new estimator by rao blackwellization asymptotic distribution theory and corresponding inferential procedures for both estimators are developed simulation studies quantify the superiority of the new estimators and show their desirable properties for small and moderate sample sizes the impact of the new techniques is illustrated with data from a clinical trial

evaluating agreement with a gold standard in method comparison studies we develop a statistical model for method comparison studies where a gold standard is present and propose a measure of agreement this measure can be interpreted as a population correlation coefficient in a constrained bivariate model an estimator of this coefficient is proposed and its statistical properties explored applications of the new methodology to data from the medical literature are presented

efficient leave one out cross validation of kernel fisher discriminant classifiers abstract mika et al in neural network for signal processing vol ix ieee press new york 1999 pp 41 48 apply the kernel trick to obtain a non linear variant of fisher s linear discriminant analysis method demonstrating state of the art performance on a range of benchmark data sets we show that leave one out cross validation of kernel fisher discriminant classifiers can be implemented with a computational complexity of only o l 3 operations rather than the o l 4 of a naive implementation where l is the number of training patterns leave one out cross validation then becomes an attractive means of model selection in large scale applications of kernel fisher discriminant analysis being significantly faster than conventional k fold cross validation procedures commonly used

the weak and strong lefschetz properties for artinian k algebras abstract let a i 0ai be a standard graded artinian k algebra where chark 0 then a has the weak lefschetz property if there is an element l of degree 1 such that the multiplication l a i a i 1 has maximal rank for every i and a has the strong lefschetz property if l d a i a i d has maximal rank for every i and d the main results obtained in this paper are the following 1 every height three complete intersection has the weak lefschetz property our method surprisingly uses rank two vector bundles on p 2 and the grauert mulich theorem 2 we give a complete characterization including a concrete construction of the hilbert functions that can occur for k algebras with the weak or strong lefschetz property and the characterization is the same one 3 we give a sharp bound on the graded betti numbers achieved by our construction of artinian k algebras with the weak or strong lefschetz property and fixed hilbert function this bound is again the same for both properties some hilbert functions in fact force the algebra to have the maximal betti numbers 4 every artinian ideal in k x y possesses the strong lefschetz property this is false in higher codimension

the lefschetz properties of monomial complete intersections in positive characteristic abstract stanley proved that in characteristic zero all artinian monomial complete intersections have the strong lefschetz property we provide a positive characteristic complement to stanley s result in the case of artinian monomial complete intersections generated by monomials all of the same degree and also for arbitrary artinian monomial complete intersections in characteristic two to establish these results we first prove an a priori lower bound on the characteristics that guarantee the lefschetz properties we then use a variety of techniques to complete the classifications

on the structure of monomial complete intersections in positive characteristic abstract in this paper we study the lefschetz properties of monomial complete intersections in positive characteristic we give a complete classification of the strong lefschetz property when the number of variables is at least three which proves a conjecture by cook ii we also extend earlier results on the weak lefschetz property by dropping the assumption on the residue field being infinite and by giving new sufficient criteria

a nonparametric mean estimator for judgment poststratified data summary maceachern stasny and wolfe 2004 biometrics60 207 215 introduced a data collection method called judgment poststratification jps based on ideas similar to those in ranked set sampling and proposed methods for mean estimation from jps samples in this article we propose an improvement to their methods which exploits the fact that the distributions of the judgment poststrata are often stochastically ordered so as to form a mean estimator using isotonized sample means of the poststrata this new estimator is strongly consistent with similar asymptotic properties to those in maceachern et al 2004 it is shown to be more efficient for small sample sizes which appears to be attractive in applications requiring cost efficiency further we extend our method to jps samples with imprecise ranking or multiple rankers the performance of the proposed estimators is examined on three data examples through simulation

nonparametric maximum likelihood estimation based on ranked set samples abstract a ranked set sample consists entirely of independently distributed order statistics and can occur naturally in many experimental settings including problems in reliability when each ranked set from which an order statistic is drawn is of the same size and when the statistic of each fixed order is sampled the same number of times the ranked set sample is said to be balanced stokes and sager have shown that the edf f n of a balanced ranked set sample from the cdf f is an unbiased estimator of f and is more precise than the edf of a simple random sample of the same size the nonparametric maximum likelihood estimator mle f of f is studied in this article its existence and uniqueness is demonstrated and a general numerical procedure is presented and is shown to converge to f if the ranked set sample is balanced it is shown that the em algorithm with f n as a seed converges to the unique solution f of the problem s self consistency equations the consistency of every iterate of the em a

on the degrees of relations on x1d1 xndn x1 xn dn 1 in positive characteristic abstract we give a formula for the smallest degree of a non koszul relation on x 1 d 1 x n d n x 1 x n d n 1 k x 1 x n under certain assumptions on d 1 d n 1 where k is a field of positive characteristic p as an application of our result we give a formula for the diagonal f threshold of a diagonal hypersurface another application is a characterization depending on the characteristic p of k of the values of d 1 d n 1 satisfying certain assumptions such that the ring k x 1 x n 1 x 1 d 1 x n 1 d n 1 has the weak lefschetz property

on the interpretation and use of r 2 in regression analysis the coefficient of determination or squared multiple correlation coefficient r2 is a common output from computer regression packages we argue first that this statistic can be interpreted as an estimator of a population parameter only when the regressors are random in such a model the variation of r2 is discussed and a simple approximate confidence interval for the population coefficient of determination is proposed its use is illustrated on data from computerized tomography investigation of pigs

on estimating distribution functions using nomination samples abstract a nomination sample consists of independently distributed maxima from subsamples of a population with the same underlying distribution nomination sampling occurs when only the item with largest value is chosen from each of n independent subsamples if the subsamples differ in size then these observed order statistics are not identically distributed so estimation schemes built on assumptions of independently and identically distributed iid data are most likely inappropriate but we can exploit the structure of the data from the nomination sample by conditioning on the observed order of the independent maxima and form a least squares estimator of the distribution function that minimizes risk with respect to squared error loss using an approach similar to one found in ferguson where the case for iid data is presented the result is a product estimator that is consistent and compares favorably with the nonparametric maximum likelihood estimator proposed by boyles and samaniego as indicated by

isotonized cdf estimation from judgment poststratification data with empty strata in applications that require cost efficiency sample sizes are typically small so that the problem of empty strata may often occur in judgment poststratification jps an important variant of balanced ranked set sampling in this article we consider estimation of population cumulative distribution functions cdf from jps samples with empty strata in the literature the standard and restricted cdf estimators stokes and sager 1988 journal of the american statistical association 83 374381 frey and ozturk 2011 annals of the institute of statistical mathematics to appear do not perform well when simply ignoring empty strata in this article we show that the original isotonized estimator ozturk 2007 journal of nonparametric statistics 19 131 144 can handle empty strata automatically through two methods minmax and maxmin however blindly using them can result in undesirable results in either tail of the cdf we thoroughly examine minmax and maxmin and find interesting results about their behaviors and performance in the presence of empty strata motivated by these results we propose modified isotonized estimators to improve estimation efficiency through simulation and empirical studies we show that our estimators work well in different regions of the cdf and also improve the overall performance of estimating the whole function