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

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

on local and global measures of separation in fuzzy topological spaces this paper introduces as far as separation properties are concerned a new way of studying a fuzzy topological space we define separation functions which measure the degree to which points in a given space can be separated in a t 0 t 1 or t 2 like fashion both local and global measures are introduced and their interrelation their relation with previously defined properties their behaviour with regard to subspaces and products and their behaviour in the special classes of fuzzy neighborhood spaces and fuzzy uniform spaces are studied

reflection subgroups of coxeter systems let w r be a coxetcr system and t ij w wrw denote the corresponding set of reflections for u e w define n w tv tlf tw l w h w ere i denotes the length function of w r the principal result 3 3 of this paper is that if w is a reflection subgroup of w i e w w n t then e ti n z n w 1 is a set of coxeter generators for w in particular w is a coxeter group in 6 v deodhar gives a geometric proof of this last fact using properties of the root system of w it is easily seen that the coxeter gcncrators for w defined by deodhar coincide with those described above the idea of the proof here is that n may be regarded as a cocycle with n r r rer and coxeter systems are characterized among groups generated by involutions r by the existence of such a cocycle one then checks that w h n w n w w e w gives a suitable cocycle for w the same argument applies to a wider class of reflection systems g x characterized by the cxistcncc of a certain cocycle and so we give the proof for these the coxeter systems are precisely the reflection systems g x in which x consists of involutions the arrangement of this paper is as follows in section 2 we describe the main facts about reflection systems closely paralleling 1 chap iv sect 11 in section 3 we prove that reflection subgroups of a reflection system are themselves reflection systems in a canonical way 3 3 and obtain some additional facts about their canonical generators 3 4 3 11 in section 4 we specialize to the case of a coxeter system and give a geometric criterion 4 4 for a set of reflections to be the set of canonical generators of a reflection subgroup in terms of the inner products of the

some results on generalized residual entropy shannon s entropy plays an important role in the context of information theory since this entropy is not applicable to a system which has survived for some units of time the concept of residual entropy has been developed in the literature here we generalize the residual entropy by choosing a convex function f with f 1 0 in this paper some orderings and aging properties have been defined in terms of the generalized residual entropy function and their properties have been studied quite a few results available in the literature have been generalized and some distributions viz uniform exponential pareto power series finite range have been characterized through the generalized residual entropy

information properties of order statistics and spacings we explore properties of the entropy kullback leibler information and mutual information for order statistics the probability integral transformation plays a pivotal role in developing our results we provide bounds for the entropy of order statistics and some results that relate entropy ordering of order statistics to other well known orderings of random variables we show that the discrimination information between order statistics and data distribution the discrimination information among the order statistics and the mutual information between order statistics are all distribution free and are computable using the distributions of the order statistics of the samples from the uniform distribution we also discuss information properties of spacings for uniform and exponential samples and provide a large sample distribution free result on the entropy of spacings the results show interesting symmetries of information orderings among order statistics

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

the entropy of ordered sequences and order statistics the entropy of a sequence of random variables under order restrictions is examined a theorem that shows the amount of entropy reduction when the sequence is ordered is presented upper and lower bounds to the entropy reduction and conditions under which they are achieved are derived some interesting properties of the entropy of the individual order statistics are also presented it is shown that the difference between the average entropy of the individual order statistics and the entropy of a member of the original independent identically distributed iid population is a constant regardless of the original distribution finally the entropies of the individual order statistics are found to be symmetric about the median when the probability density function pdf of the original iid sequence is symmetric about its mean

results on residual renyi entropy of order statistics and record values this paper explores properties of the residual renyi entropy of some ordered random variables the residual renyi entropy of the kth order statistic from a continuous distribution function is represented in terms of the residual renyi entropy of the kth order statistic from uniform distribution the monotone behavior of the residual renyi entropy of order statistic under various conditions is discussed analogues results for the residual renyi entropy of record values are also given

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

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

cumulative tsallis entropy for maximum ranked set sampling with unequal samples abstract in this paper we consider the information content of maximum ranked set sampling procedure with unequal samples mrssu in terms of tsallis entropy which is a non additive generalization of shannon entropy we obtain several results of tsallis entropy including bounds monotonic properties stochastic orders and sharp bounds under some assumptions we also compare the uncertainty and information content of mrssu with its counterpart in the simple random sampling srs data finally we develop some characterization results in terms of cumulative tsallis entropy and residual tsallis entropy of mrssu and srs data

on the r o property in fuzzy topology abstract nine r 0 type axioms for fuzzy topological spaces including two previously introduced axioms 1 5 are studied a complete answer is given concerning their interrelations goodness and initiality also a complete answer is given with regard to all possible t 1 r 0 type and t 0 r 0 t 1 type properties where the t 0 and t 1 axioms considered in 7 are taken into account the special cases of fuzzy neighborhood spaces and fuzzy uniform spaces are studied

function spaces in fuzzy topology abstract function spaces in several categories of fuzzy topological spaces and fuzzy continuous maps are studied the approach is via the existence of splitting and conjoining structures and makes extensive use of the theory of monotopological categories

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

maximum likelihood estimation of the survival functions of stochastically ordered random variables abstract many times aspects of populations exist that must satisfy a stochastic ordering requirement nevertheless estimates may not bear out this stochastic ordering because of the inherent variability of the observations this article will consider the problem of finding maximum likelihood estimates of stochastically ordered survival functions for the cases a one survival function being fixed in advance and b estimating two survival functions when the data include censored observations a numerical example is discussed in detail to illustrate the solution to this problem

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

on the determination of fuzzy topological spaces and fuzzy neighbourhood spaces by their level topologies abstract in this paper we discuss the problem of the reconstruction of a fuzzy topological space or a fuzzy neighbourhood space from an a priori given family of level topologies necessary and sufficient conditions for the existence of a solution are given and it is proved that in the particular case of fuzzy neighbourhood spaces this solution is always unique

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

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

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

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

a fuzzy r 1 separation axiom abstract we introduce a new r 1 separation axiom for fuzzy topological spaces we identify from the literature which lower fuzzy separation axioms are compatible with our r 1 axiom we provide alternative descriptions for this r 1 axiom in the special setting of fuzzy neighborhood spaces this special case has standard higher fuzzy separation axioms in it we establish aspects of compatibility of our r 1 axiom with those higher separation axioms as well as with fuzzy compactness

the asymptotics of maximum likelihood and related estimators based on type ii censored data abstract some simple procedures are provided for establishing the asymptotic normality and uniform strong convergence of a class of functions that arise in the context of estimating parameters from a type ii censored sample these lead to an elementary yet rigorous treatment of the asymptotic properties of maximum likelihood estimators based on type ii censored data further applications include the treatment of asymptotics of some modified maximum likelihood mml estimators in particular conditions are provided for the consistency and limiting normality of the mml estimators of mehrotra and nanda and the asymptotic efficiencies of these estimators are evaluated

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

formulas for renyi information and related measures for univariate distributions in a recent paper song j stat plan infer 93 2001 51 considered renyi information of order and established its connection to the loglikelihood from this relation an intrinsic distribution measure was proposed and analytic expressions of this measure and renyi information were derived for some standard continuous distributions in this paper we derive analytical formulas for renyi and shannon entropies as well as for song s measure for 26 flexible families of univariate continuous distributions we believe that the results presented here will serve as an important reference for scientists and engineers in many areas

fisher information in order statistics abstract when we have n independently and identically distributed observations it is an interesting question how the fisher information is distributed among order statistics the recipe for the fisher information in order statistics is easy but the detailed calculation has been known to be complicated an indirect approach using a decomposition of the fisher information in order statistics simplifies the calculation some recurrence relations for the fisher information in order statistics are derived that facilitate the calculation the fisher information in the first r order statistics is an r multiple integral but it can be simplified to just a double integral by using the decomposition a recurrence relation further simplifies the double integral to a sum of single integrals an information plot is suggested from which we can read at once the fisher information in any set of consecutive order statistics for a parametric distribution

on the combinatorics of b b orbits on group compactifications abstract it is shown that there is an order isomorphism from the poset v of b b orbits on the wonderful compactification of a semi simple adjoint group g with weyl group w to an interval in reverse chevalley bruhat order on a non canonically associated coxeter group w in general neither finite nor affine moreover preserves the corresponding kazhdan lusztig polynomials springer s partly conjectural construction of kazhdan lusztig polynomials for the analogues of v for general coxeter groups w is completed by reducing it by a similar order isomorphism to known results involving a twisted chevalley bruhat order on w

on some entropy functionals derived from renyi information divergence we consider the maximum entropy problems associated with renyi q entropy subject to two kinds of constraints on expected values the constraints considered are a constraint on the standard expectation and a constraint on the generalized expectation as encountered in nonextensive statistics the optimum maximum entropy probability distributions which can exhibit a power law behaviour are derived and characterized the renyi entropy of the optimum distributions can be viewed as a function of the constraint this defines two families of entropy functionals in the space of possible expected values general properties of these functionals including nonnegativity minimum convexity are documented their relationships as well as numerical aspects are also discussed finally we work out some specific cases for the reference measure q x and recover in a limit case some well known entropies

on separation axioms in fuzzy topological spaces fuzzy neighborhood spaces and fuzzy uniform spaces submitted by l zadeh a certain number of separation axioms for fuzzy topological spaces are provided all of which are good extensions of the topological t t or t2 all valid implications between the different axioms are studied and counterexamples are given for the nonvalid ones in this paper we introduce and study a number of separation properties of fuzzy topological fuzzy neighborhood and fuzzy uniform spaces all of which are extensions of t t and t in topological spaces we started from the viewpoint that separation axioms at least t originated in the need to guarantee in some sense uniqueness of limits and we tried to generalize the topological t axiom in that direction it appears that a whole range of axioms from a fairly weak condition wt up to a rather strong one tj all give rise to acceptable extensions of the topological property moreover it seems likely that it will not be possible to single out one axiom in fuzzy topology as the best extension of topological t2 we showed in a preceding paper 181 on completions of fuzzy uniform spaces that tj as well as wt collapsing with wt in uniform spaces have a specific role to play in the same sense wt has already shown its usefulness in the study of hyperspaces 131 and t2 in its relation with compactness 191 as to our other axioms from t to wt they all seem to be natural generalizations of the corresponding properties in topological spaces and of the different t notions in fuzzy topological spaces they are moreover hereditary and t and ty excepted also productive as to the relations with other work in this field we first point out that our wt axiom is in fact the 1 hausforff separation property of hiihle 2 31 hutton and reilly in 4 and rodabaugh in 161 also studied separation current address is rijksuniversitair centrum 2020 antwerpen belgium

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

a level topologies criterion for lowen fuzzy uniformizability throughout x will be a nonempty set we consider a subset m of x as a crisp fuzzy set and denote this again by m we denote the strong cut of a fuzzy set e ix by and its weak cut by e i

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

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

ranked set sampling for replicated sampling designs summary 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

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

renyi information measure for a used item recently in the literature a measure of tail heaviness has been proposed based on renyi entropy this measure is very useful in the sense that it can be used to measure tail heaviness even for the distributions for which the kurtosis measure b 2 m 4 m 2 2 does not exist nadarajah and zografos nadarajah zografos information sciences 153 2003 119 138 have derived the expression for this measure for different univariate continuous distributions but this measure can only be used for the lifetime of a new item in case of used item this measure needs some modification in this paper we have modified the measure accordingly so that it can be used in the case of used item and also for the new item we have also derived expressions for this measure for sixteen different univariate distributions and ten other standard distributions derived from the general distributions used in reliability and survival analysis the most of the results obtained in the literature in this direction can be obtained as particular cases of our general results

ordering univariate distributions by entropy and variance this paper examines the role of variance and entropy in ordering distributions and random prospects there is no universal relation between entropy and variance orderings of distributions but we place their relationship in the context of a stronger ordering relation known as dispersion ordering further some conditions are identified under which variance and entropy order similarly when continuous variables are transformed we also analyze parametric changes which do not disturb the agreement between these rankings the results are conveniently tabulated in terms of distribution parameters

modified maximum likelihood method for the robust estimation of system parameters from very noisy data abstract when experiments are conducted there is always a chance of the occurrence of large measurement errors outliers common identification methods like generalized least squares maximum likelihood etc may not converge in these situations due to the presence of outliers here we present a method for the robust estimation of system parameters based on the censoring of data and employing the maximum likelihood estimation several simulated examples show that the modified maximum likelihood method works well in situations where other methods failed

a comparison of some ft 2 concepts abstract recently ghanim kerre and mashhour 2 have introduced and examined a fuzzy hausdorffness concept in this paper we compare it with some of the other existing concepts of fuzzy hausdorffness and note thereby that though it is a good extension of hausdorffness it is independent of several other fuzzy hausdorffness concepts currently in use

regression estimator in ranked set sampling summary situations in this paper we assume that the ranking is done on the basis of a concomitant variable x regression type rss estimators of the population mean of y will be proposed by utilizing this concomitant variable x in both the ranking process of the units and the estimation process when the population mean of x is known when x has unknown mean double sampling will be used to obtain an estimate for the population mean of x it is found that when x and y jointly follow a bivariate normal distribution our proposed rss regression estimator is more efficient than rss and simple random sampling srs naive estimators unless the correlation between x and y is low ipi 0 4 moreover it is always superior to the regression estimator under srs for all p when normality does not hold this approach could still perform reasonably well as long as the shape of the distribution of the concomitant variable x is only slightly departed from symmetry for heavily skewed distributions a remedial measure will be suggested an example of estimating the mean plutonium concentration in surface soil on the nevada test site nevada u s a will be considered

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

fuzzy neighborhood spaces abstract in this paper we introduce a subcategory of the category of fuzzy topological spaces and we prove basic properties of this subcategory the main idea behind these spaces is the notion of fuzzy neighborhoods which enables us to introduce a local theory of fuzzy topologies

fuzzy t neighbourhood spaces part 3 t separation axioms we address the problem of identifying a useful set of mutually compatible separation axioms for each category t fns of t neighbourhood spaces it should contain one of t complete regularity that characterizes t uniformizability as indeed we achieve here this seems to necessitate the parameterization of most separation axioms by the triangular norm t with the exception of the two axioms t0 and r0 which have categorical definitions valid in any topological category we introduce t separation axioms in terms of t and fuzzy closure operators only making them meaningful for the whole of fts we then characterize them within t fns in terms of t fuzzy closures of crisp fuzzy subsets and t neighbourhoods we study our t axioms and establish some entailments between them our t complete regularity commits us to investigate a t real line which is a t neighbourhood space induced by a certain probabilistic t metric due to hohle fuzzy sets and systems 1 1978 311

the entropy of consecutive order statistics calculation of the entropy of a set of consecutive order statistics is relatively more complicated than that of the entropy of the individual order statistic which has been studied by wong and chan 1990 we provide some fundamental relations occurring in the entropy of consecutive order statistics which are very useful for computations we first consider the decomposition of the entropy of order statistics and derive some recurrence relations in the first r order statistics we also establish a dual principle for the entropy of order statistics which yields a dual relation from a given relation in the entropy of order statistics

on uncertainty and information properties of ranked set samples ranked set sampling is a sampling design which has a wide range of applications in industrial statistics economics and environmental and ecological studies etc it is well known that ranked set samples provide more fisher information than simple random samples of the same size about the unknown parameters of the underlying distribution in parametric inferences in this paper we consider the uncertainty and information content of ranked set samples in both perfect and imperfect ranking scenarios in terms of shannon entropy renyi and kullback leibler kl information measures it is proved that under these information measures ranked set sampling design performs better than its simple random sampling counterpart of the same size the information content is also a monotone function of the set size in ranked set sampling moreover the effect of ranking error on the information content of the data is investigated

inferences under a stochastic ordering constraint the k sample case if x1 and x2 are random variables with distribution functions f1 and f2 then x1 is said to be stochastically larger than x2 if f1 f2 statistical inferences under stochastic ordering for the two sample case has a long and rich history in this article we consider the k sample case that is we have k populations with distribution functions f1 f2 fk k 2 and we assume that f1 f2 fk for k 2 the nonparametric maximum likelihood estimators of f1 and f2 under this order restriction have been known for a long time their asymptotic distributions have been derived only recently these results have very complicated forms and are hard to deal with when making statistical inferences we provide simple estimators when k 2 these are strongly uniformly consistent and their asymptotic distributions have simple forms if and are the empirical and our restricted estimators of fi then we show that asymptotically for all x and all u 0 with strict inequality in some cases this clearly show

fuzzy uniform spaces in this paper we introduce the notion of fuzzy uniform spaces in section 2 we give some basic results and show how a fuzzy topology is derived from a fuzzy uniformity in sections 3 and 4 we prove that the notions which are introduced are good extensions and we prove that the category of uniform spaces is nicely injected in the category of fuzzy uniform spaces finally in section 5 we prove some basic results

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

reflectors and coreflectors in the category of fuzzy topological spaces abstract using a certain number of very elementary functors explicit formulas are given for the bireflection or and bicoreflection of top topological spaces fns fuzzy neighborhood spaces and max maximal with respect to the level topologies fuzzy topological spaces in the category fts of fuzzy topological spaces a subcategory of fts minimal with respect to certain associated topologies and dual in a certain sense to max arises in a natural way and is proved to be nothing else than fns