bayesian inferences on umbrella orderings in regression applications with categorical predictors interest often focuses on comparing the null hypothesis of homogeneity to an ordered alternative this article proposes a bayesian approach for addressing this problem in the setting of normal linear and probit regression models the regression coefficients are assigned a conditionally conjugate prior density consisting of mixtures of point masses at 0 and truncated normal densities with a possibly unknown changepoint parameter included to accommodate umbrella ordering two strategies of prior elicitation are considered 1 a bayesian bonferroni approach in which the probability of the global null hypothesis is specified and local hypotheses are considered independent and 2 an approach which treats these probabilities as random a single gibbs sampling chain can be used to obtain posterior probabilities for the different hypotheses and to estimate regression coefficients and predictive quantities either by model averaging or under the preferred hypothesis the methods are applied to data from a carcinogenesis study

order restricted score parameters in association models for contingency tables abstract the row effects and column effects models for two way contingency tables have parameters for the row and column categories pertaining to the association between the variables for classifications having ordered categories it is often reasonable to assume that the association parameters have a corresponding ordering this article proposes order restricted estimates of the association parameters in these models the maximum likelihood solution can be determined by the solution of a simple isotonic regression of some of the model sufficient statistics the models are primarily log linear in form and can be expressed in terms of odds ratios for 2 2 subtables consisting of adjacent rows and adjacent columns for the order restricted solution these local log odds ratios have uniform sign goodness of fit statistics for this solution are related to corresponding statistics for collapsed tables and to statistics for testing equality of sets of the parameters the row effects model discussed in this art

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

unfoldings in knot theory in in r the link at infinity of a hypersurface v cie r defined by a good polynomial map f ty ii2 was used as an extended example f was called good if it had only isolated singularities and it was claimed that the link at infinity then always has a milnor fibration this is incorrect although we have found it to be a common misconception to correct it the definition of good must be modified as follows

analysis on binary responses with ordered covariates and missing data summary we consider the situation of two ordered categorical variables and a binary outcome variable where one or both of the categorical variables may have missing values the goal is to estimate the probability of response of the outcome variable for each cell of the contingency table of categorical variables while incorporating the fact that the categorical variables are ordered the probability of response is assumed to change monotonically as each of the categorical variables changes level a probability model is used in which the response is binomial with parameters pij for each cell i j and the number of observations in each cell is multinomial estimation approaches that incorporate gibbs sampling with order restrictions on pij induced via a prior distribution two dimensional isotonic regression and multiple imputation to handle missing values are considered the methods are compared in a simulation study using a fully bayesian approach with a strong prior distribution to induce ordering can lead to large gains in efficiency but can also induce bias utilizing isotonic regression can lead to modest gains in efficiency while minimizing bias and guaranteeing that the order constraints are satisfied a hybrid of isotonic regression and gibbs sampling appears to work well across a variety of scenarios the methods are applied to a pancreatic cancer case control study with two biomarkers copyright q 2007 john wiley sons ltd

quadratic loss of order restricted estimators for treatment means with a control we consider an experiment which consists of k treatment groups and a control group let the sample means yo y1 yk be independent normal variates with expected values a ml uk and with variances u2 no o2 n1 02 nk let ub wl wk be positive weights and let uo t be the weighted least squares estimators subject to the constraints yl fi i 1 k we establish that for large k e 1 mo 2 e yo mo 2 when wi ni i 0 1 k under suitable conditions we show that e a ai 2 e yi mi 2 i 0 1 k

inconsistency of the maximum likelihood estimator of a distribution having increasing failure rate average abstract marshall and proschan showed that the mle for a life distribution with increasing failure rate is strongly consistent ann math statist 1965 36 69 77 in this note the author shows that the mle for a life distribution with increasing failure rate average is not consistent in fact maximum likelihood estimation in the ifra case yields estimators of the average failure rate and of the distribution function which in general coverage a s to values other than the true values in the decreasing failure rate average case the mle fails to exist

nonparametric maximum likelihood estimation of survival functions with a general stochastic ordering and its dual summary this paper discusses estimation of survival functions under an arbitrary partial stochas tic ordering of the underlying populations this technique is especially useful when data are not easily acquired from one or more of the populations but data from the other populations are available estimation of the ordered survival functions entails use of a fenchel duality theorem that results in an algorithm to identify and resolve violations of the underlying ordering the algorithm produces estimates which converge to the maximum likelihood estimates of the survival functions given the underlying ordering the method is illustrated by an example in life testing of components and in clinical trials estimation of a survival function p t is often of prime interest if the functional form of the survival function is unknown nonparametric methods such as the product limit estimator kaplan meier 1958 are needed johansen 1978 showed that the product limit estimator is a maximum likelihood estimator in the class of all distributions under a generalized maximum likelihood framework kiefer wolfowitz 1956 since the generalized maximum likelihood esti mates for continuous survival functions place probability only on those time points at which observations occur suppose however that one desires to estimate a number of survival functions when a partial stochastic ordering exists among them and under censoring for example in a clinical trial of two treatments for a rare debilitating disease let pi p2 and p3 be the survival functions respectively for the disease free group and for the disease groups under the two treatments the well documented survival function p1 can be used to generate better estimates for p2 and p3 by using the order restrictions pi t p2 t and pi t p3 t brunk et al 1966 developed the maximum likelihood estimators for two stochastically ordered distributions for uncensored independent random samples dykstra

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

the min max algorithm and isotonic regression there is growing interest in statistical inference under order restrictions a major demand in this subject is to have a fast direct method to solve the least squares problem of partially ordered isotonic regression the min max algorithm is such a method in which the user searches for the global minimum and the local maximum successively a comparison of algorithms for partially ordered isotonic regression is included as an application using this efficient algorithm it is feasible to approximate critical values of isotonic tests by simulation

stability of steady states and existence of travelling waves in a vector disease model in this paper a host vector model is considered for a disease without immunity in which the current density of infectious vectors is related to the number of infectious hosts at earlier times spatial spread in a region is modelled in the partial integro differential equation by a diffusion term for the general model we first study the stability of the steady states using the contracting convex sets technique when the spatial variable is one dimensional and the delay kernel assumes some special form we establish the existence of travelling wave solutions by using the linear chain trick and the geometric singular perturbation method

on the use of stochastic ordering to test for trend with clustered binary data we introduce the use of stochastic ordering for defining treatment related trend in clustered exchangeable binary data for both when cluster sizes are fixed and when they vary randomly in the latter case there is a well documented tendency for such data to be sparse a problem we address by making an assumption of interpretability or equivalently marginal compatibility our procedures are based on a representation of the joint distribution of binary exchangeable random variables by a saturated model and may hence be considered nonparametric the definition of trend by stochastic ordering is proposed to ensure a flexibility that allows for various forms of monotone increases in response to the cluster as a whole to be included in the evaluation of the trend we obtain maximum likelihood estimates of probability functions under stochastic ordering using mixture likelihood based algorithms since the data are sparse we avoid the use of asymptotic results and obtain p values of the likelihood ratio procedures by permutation resampling we demonstrate how the proposed framework can be used in risk assessment and provide comparisons with existing procedures

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

an iterative procedure for obtaining i projections onto the intersection of convex sets abstract a frequently occuring problem is to find that probability distribution pd lying within a convex set which minimizes the i divergence between it and given pd r this document proposes an iterative procedure which requires only that the set be convex and not necessarily linear which under general conditions will converge to the desired i projection

maximum likelihood estimation under order restrictions by the prior feedback method abstract algorithms for deriving isotonic regression estimators in order restricted linear models and more generally restricted maximum likelihood estimators are usually quite dependent on the particular problem considered we propose here an optimization method based on a sequence of formal bayes estimates whose variances converge to zero this method akin to simulated annealing can be applied universally that is as long as these bayes estimators can be derived by exact computation or markov chain monte carlo sampling approximation we then give an illustration of our method for two real life examples

transforming contingency tables abstract we present a general theorem on transforming contingency tables and several applications where the transformation technique has allowed us to take advantage of the iterative proportional fitting procedure and has resulted in simple and useful procedures a further advantage of this technique is that it is sometimes possible to recognize closed form estimates in the transformed problem while they would be overlooked in the original setting we shall view the estimation problem as one of minimizing the kullback leibler information distance between two probability mass functions and will roughly follow the notation of csiszar 1976 although we have adopted the information distance point of veiw the duality between maximum likelihood estimation and minimum information estimation see e g darroch and ratcliff 1972 implies that the results of this paper can just as well be interpreted from the maximum likelihood point of view

stochastic reduction of loss in estimating normal means by isotonic regression cf barlow bartholomew bremner and brunk 1972 page 19 the estimator fi has been compared to the unrestricted estimator y and several results point to the superiority of the isotonic regression estimator when the order restriction holds it has been shown cf robertson wright and dykstra 1988 that the total error of the estimator is reduced through isotonic regression on any order restriction in the sense that

ordinal dominance curve based inference for stochastically ordered distributions summary in a variety of applications researchers collect data that are sampled under ordered experimental conditions in such situations it is reasonable to assume that outcomes are ordered stochastically by the level of the treatment for example a toxicologist may want to assess the effect of a toxin on the reproductive system by studying the number of offspring produced by an animal as a function of the dose that it received a variety of scientific questions arise in such a context the most basic is whether a dose response relationship is present if so determining the minimal toxic dose is of great interest the strength of the ordering between groups i j may be assessed by the probabilities or monotone functions thereof large probabilities indicate a strong ordering we find that such and other questions are naturally addressed by studying the ordinal dominance curve and the area under the curve auc we develop new non parametric order restricted estimators for the aucs in k2 samples and investigate them in detail we show that the restricted estimators are strongly consistent and have lower mean squared error compared with the unrestricted estimators in the two sample case estimators are considerably improved in those situations where the data are highly variable and or sample sizes are small and where the auc is close to its boundary value note that these are precisely the conditions under which estimating the auc is difficult simulations indicate that in the multisample case estimators are improved under a much broader set of conditions with substantial reductions in mean squared error these estimators provide a basis for a family of distribution free tests for the usual stochastic order in k2 samples limiting distributions are derived an accurate and simple approximation to the tail area is given for k 2 simulation studies show that the new tests perform well compared with existing alternatives the methodology proposed is illustrated by applying it to an aquatic toxicology study where it is used to determine the minimal toxic dose more accurately

bayesian inference of survival probabilities under stochastic ordering constraints abstract in the statistical analysis of survival data arising from two populations it often happens that the analyst knows a priori that the life lengths in one population are stochastically shorter than those in the other nevertheless survival probability estimates if determined separately from the corresponding samples may not be consistent with this prior assumption because of inherent statistical variability in the observations this problem has been considered in a number of papers during the past decade by adopting a generalized maximum likelihood approach our approach is bayesian and in essence nonparametric the a priori assumption regarding stochastic ordering is formulated naturally in terms of a joint prior distribution defined for pairs of survival functions nonparametric specification of the model based on hazard rates and using a few hyperparameters allows for sufficient flexibility in practical applications the numerical computations are based on a coupled version of the m

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

estimation of multivariate distributions under stochastic ordering abstract let f and g be the cdf s of two p dimensional multivariate distributions such that f is stochastically larger than g a straightforward derivation is given of the generalized maximum likelihood estimators of f and g based on random samples from each population an algorithmic approach to computing these estimators is described and motivating numerical examples are discussed the special case when f and g correspond to multivariate ordinal contingency tables is also presented the relationship of these results to those of robertson and wright 1974 is considered

a design for phase i trials in completely or partially ordered groups we propose a design for dose finding for cytotoxic agents in completely or partially ordered groups of patients by completely ordered groups we mean that prior to the study there is clinical information that would indicate that for a given dose the groups can be ordered with respect to the probability of toxicity at that dose with partially ordered groups at a given dose only some of the groups can be ordered with respect to the probability of toxicity at that dose the method we propose includes elements of the parametric model used in the continual reassessment method combined with the hwang peddada order restricted estimation procedure we evaluate the operating characteristics of these designs in a family of dose toxicity curves representing complete and partial orders copyright 2017 john wiley sons ltd

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

asymptotic theory for nonparametric estimation of survival curves under order restrictions f t g f t all t the nonparametric maximum likelihood estimators in these problems were derived by brunk franck hansen and hogg and dykstra in the present paper we derive their large sample distributions we present two sets of proofs depending on whether or not the data are right censored when centered and scaled by n 1r2 the estimators con verge in distribution to limiting processes related to the concave majorant of brownian motion the limiting distributions are not known in closed form but can be simulated for the purpose of forming asymptotic point wise confidence limits

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

persistence of travelling fronts of kdv burgers kuramoto equation we study the travelling fronts of kdv burgers kuramoto equation from geometric singular perturbation point of view motivated by the analogue between travelling fronts and heteroclinic orbits of the corresponding ordinary differential equations we prove the persistence of these waves for sufficiently small dissipation the result of numerical investigation also establishes our analysis

inference for likelihood ratio ordering in the two sample problem abstract we obtain the maximum likelihood estimators of two multinomial probability vectors under the constraint that they are likelihood ratio ordered we extend this estimation approach to the case of two univariate distributions and show strong consistency of the estimators we also derive and study the asymptotic distribution of the likelihood ratio statistic for testing the equality of two discrete probability distributions against the alternative that one distribution is greater than the other in the likelihood ratio ordering sense finally we examine a data set pertaining to average daily insulin dose from the boston collaborative drug surveillance program and compare our testing procedure to testing procedures for other stochastic orderings

association models and canonical correlation in the analysis of cross classifications having ordered categories abstract the association models considered in goodman 1979a for the analysis of cross classifications having ordered categories are presented in a somewhat different form in the present article to facilitate comparison of the results obtained using these models with those obtained using the earlier canonical correlation approach both the association models and the canonical correlation approach can provide meaningful scores for the row and column categories and these scores can be used to partition into relevant components the usual chi squared statistic for testing the null hypothesis of statistical independence between the row classification and column classification however while the usual procedure for testing the statistical significance of chi squared components is invalid when these components are based on the canonical correlations a corresponding procedure is valid when these components are obtained with the association models the components of association obtained with the association mo

quasi isometric classification of graph manifold groups we show that the fundamental groups of any two closed irreducible nongeometric graph manifolds are quasi isometric we also classify the quasi isometry types of fundamental groups of graph manifolds with boundary in terms of certain finite twocolored graphs a corollary is the quasi isometric classification of artin groups whose presentation graphs are trees in particular any two right angled artin groups whose presentation graphs are trees of diameter greater than 2 are quasi isometric further this quasi isometry class does not include any other right angled artin groups a finitely generated group can be considered geometrically when endowed with a word metric up to quasi isometric equivalence such metrics are unique in this article only finitely generated groups are considered given a collection of groups g gromov 11 proposed the fundamental questions of identifying which groups are quasi isometric to those in g rigidity and which groups in g are quasi isometric to each other classification in this article we focus on the classification question for graph manifold groups and right angled artin groups a compact 3 manifold m is called geometric if m m admits a geometric structure in the sense of thurston i e a complete locally homogeneous riemannian metric of finite volume thurston s geometrization conjecture see 29 17 21 23 22 provides that every irreducible 3 manifold of zero euler characteristic i e with boundary consisting only of tori and klein bottles admits a decomposition along tori and klein bottles into geometric pieces the minimal such decomposition being called the geometric decomposition there is a considerable literature on quasi isometric rigidity and classification of 3 manifold groups the rigidity results can be briefly summarized in the following form

a likelihood ratio test against stochastic ordering in several populations abstract the likelihood ratio test is often used to test hypotheses involving a stochastic ordering distribution theory for the likelihood ratio test has been developed only for two stochastically ordered distributions for testing equality of distributions against a stochastic ordering in several populations this paper derives the null asymptotic distribution of the likelihood ratio test statistic which is characterized by minimization problems and has no closed form a monte carlo simulation is conducted to study the limiting distribution because the limiting distribution depends on the specific values of the unknown distributions under the null hypothesis asymptotic and bootstrap approaches are proposed to overcome practical difficulties and implement tests based on the likelihood principle asymptotic validities for these tests are established and simulations are carried out to check their performances for finite sample sizes the tests are applied to an example involving data for survival time f

pointwise nonparametric maximum likelihood estimator of stochastically ordered survivor functions in this paper we consider estimation of survivor functions from groups of observations with right censored data when the groups are subject to a stochastic ordering constraint many methods and algorithms have been proposed to estimate distribution functions under such restrictions but none have completely satisfactory properties when the observations are censored we propose a pointwise constrained nonparametric maximum likelihood estimator which is defined at each time t by the estimates of the survivor functions subject to constraints applied at time t only we also propose an efficient method to obtain the estimator the estimator of each constrained survivor function is shown to be nonincreasing in t and its consistency and asymptotic distribution are established a simulation study suggests better small and large sample properties than for alternative estimators an example using prostate cancer data illustrates the method

simultaneous estimation of intrarater and interrater agreement for multiple raters under order restrictions for a binary trait it is valuable in many studies to assess both intrarater and interrater agreement most measures of intrarater agreement do not adjust for unequal estimates of prevalence between the separate rating occasions for a given rater and measures of interrater agreement typically ignore data from the second set of assessments when raters make duplicate assessments in the event when both measures are assessed there are instances where interrater agreement is larger than at least one of the corresponding intrarater agreements this implies that a rater agrees less with him herself and more with another rater in the situation of multiple raters making duplicate assessments on all subjects the authors propose properties for an agreement measure based on the odds ratio for a dichotomous trait i estimate a single prevalence across two reading occasions for each rater ii estimate pairwise interrater agreement from all available data iii bound the pairwise interrater agreement above by the corresponding intrarater agreements estimation of odds ratios under these properties is done by maximizing the multinomial likelihood with constraints using generalized log linear models in combination with a generalization of the lemke dykstra iterative incremental algorithm an example from a mammography examination reliability study is used to demonstrate the new method

the homflypt skein algebra of the torus and the elliptic hall algebra we give a generators and relations presentation of the homflypt skein algebra h of the torus t2 and we give an explicit description of the module corresponding to the solid torus using this presentation we show that h is isomorphic to the 1 specialization of the elliptic hall algebra of burban and schiffmann as an application for an iterated cable k of the unknot we use the elliptic hall algebra to construct a 3 variable polynomial that specializes to the colored homflypt polynomial of k we show that this polynomial also specializes to one constructed by cherednik and danilenko using the gln double affine hecke algebra this proves one of the connection conjectures in their recent work

estimation of the rate distortion function motivated by questions in lossy data compression and by theoretical considerations the problem of estimating the rate distortion function of an unknown not necessarily discrete valued source from empirical data is examined the focus is the behavior of the so called ldquoplug inrdquo estimator which is simply the rate distortion function of the empirical distribution of the observed data sufficient conditions are given for its consistency and examples are provided demonstrating that in certain cases it fails to converge to the true rate distortion function the analysis of its performance is complicated by the fact that the rate distortion function is not continuous in the source distribution the underlying mathematical problem is closely related to the classical problem of establishing the consistency of maximum likelihood estimators mles general consistency results are given for the plug in estimator applied to a broad class of sources including all stationary and ergodic ones a more general class of estimation problems is also considered arising in the context of lossy data compression when the allowed class of coding distributions is restricted analogous results are developed for the plug in estimator in that case finally consistency theorems are formulated for modified e g penalized versions of the plug in and for estimating the optimal reproduction distribution

bayesian analysis of stochastically ordered distributions of categorical variables abstract this article considers a finite set of discrete distributions all having the same finite support the problem of interest is to assess the strength of evidence produced by sampled data for a hypothesis of a specified stochastic ordering among the underlying distributions and to estimate these distributions subject to the ordering we present a bayesian approach that is an alternative to using the posterior probability of the hypothesis and the bayes factor in favor of the hypothesis we develop computational methods for the implementation of bayesian analyses we analyze examples to illustrate inferential and computational developments the methodology used for testing a hypothesis is seen to apply to a wide class of problems in bayesian inference and has some distinct advantages

on estimation for monotone dose response curves abstract estimating monotone dose response curves with independent normal errors has been studied in detail in the literature a simple procedure using a generalization of the studentized maximum modulus technique is proposed for constructing confidence bands for monotone regression functions this procedure is shown to compare well with its predecessors the center of this smooth and efficient confidence band is then used to estimate a smooth monotone regression function its pointwise mean squared errors compare favorably with the maximum likelihood estimate a weighted average of the isotonic regression and the least squares line and the rank transformed regression a numerical example of a binding inhibition assay is included to illustrate the proposed interval and point estimation procedures

designs for single or multiple agent phase i trials phase i trials of cytotoxic agents in oncology are usually dose finding studies that involve a single cytotoxic agent many statistical methods have been proposed for these trials all of which are based on the assumption of a monotonic dose toxicity curve for single agent trials this is a valid assumption in many trials however investigators are interested in finding the maximally tolerated dose based on escalating multiple cytotoxic agents when there are multiple agents monotonicity of the dose toxicity curve is not clearly defined in this article we present a design for phase i trials in which the toxicity probabilities follow a partial order meaning that there are pairs of treatments for which the ordering of the toxicity probabilities is not known at the start of the trial we compare the new design to existing methods for simple orders and investigate the properties of the design for two partial orders

nonparametric estimation of a distribution subject to a stochastic precedence constraint for any two random variables x and y with distributions f and g defined on 0 x is said to stochastically precede y if p x y 1 2 for independent x and y stochastic precedence denoted by x spy is equivalent to e g x 1 2 the applicability of stochastic precedence in various statistical contexts including reliability modeling tests for distributional equality versus various alternatives and the relative performance of comparable tolerance bounds is discussed the problem of estimating the underlying distribution s of experimental data under the assumption that they obey a stochastic precedence sp constraint is treated in detail two estimation approaches one based on data shrinkage and the other involving data translation are used to construct estimators that conform to the sp constraint and each is shown to lead to a root n consistent estimator of the underlying distribution the asymptotic behavior of each of the estimators is fully characterized conditions are given under which

an iterative procedure for general probability measures to obtain i projections onto intersections of convex sets the iterative proportional fitting procedure ipfp was introduced formally by deming and stephan in 1940 for bivariate densities this procedure has been investigated by kullback and ruschendorf it is well known that the ipfp is a sequence of successive i projections onto sets of probability measures with fixed marginals however when finding the i projection onto the intersection of arbitrary closed convex sets e g marginal stochastic orders a sequence of successive i projections onto these sets may not lead to the actual solution addressing this situation we present a new iterative i projection algorithm under reasonable assumptions and using tools from fenchel duality convergence of this algorithm to the true solution is shown the cases of infinite dimensional ipfp and marginal stochastic orders are worked out in this context

duality of i projections and maximum likelihood estimation for log linear models under cone constraints abstract important order restrictions and general log linear models for multinomial experiments can often be expressed by requiring that the vector composed of the logs of the probabilities fall within a closed convex cone here we exhibit a duality relationship by way of a fenchel duality theorem between these types of problems and projections in i divergence geometry we show that many cone constrained maximum likelihood estimation problems are exactly equivalent to an i projection onto a translation of the negative polar cone this duality relationship permits the concise characterization of cone restricted maximum likelihood estimates and the use of iterative algorithms proposed by csiszar 1975 and dykstra 1985b which are analogous to the iterative proportional fitting procedure computational aspects of maximum likelihood estimates are discussed and a relationship with least squares projections onto isotonic cones is presented we discuss examples in detail to exhibit the usefulness and power

an exact distribution free analysis for accelerated life testing at several levels of a single stress consider the following experiment at stress levels x1 xk n1 nk items are put on test the failure times of those items failing before times t 1 t k are recorded and the remaining failure times are censored let p t x denote the probability that an item stressed at level x fails on or before time t we consider models of the form p t x g t x where g is a real valued function specified up to an unknown parameter c and is an unspecified continuous distribution function under these assumptions a kolmogorov smirnov type test is discussed as a method for estimating c as well as p t o xo for specified t o and xo the method is exact and does not require failures at unaccelerated stress levels if n 1 n k the method extends to experiments that at each stress level are terminated after a specified number of failures is observed

a unified approach to likelihood inference on stochastic orderings in a nonparametric context abstract for data in a two way contingency table with ordered margins we consider various hypotheses of stochastic orders among the conditional distributions considered by rows and show that each is equivalent to requiring that an invertible transformation of the vectors of conditional row probabilities satisfies an appropriate set of linear inequalities this leads to the construction of a general algorithm for maximum likelihood estimation under multinomial sampling and provides a simple framework for deriving the asymptotic distribution of log likelihood ratio tests the usual stochastic ordering and the so called uniform and likelihood ratio orderings are considered as special cases in particular for each of these three orderings we determine the transformation required to apply the estimation algorithm we then consider testing the hypothesis that the rows are identically distributed against the alternative that they are stochastically ordered as well as testing each stochastic order against an u

estimation of order restricted means from correlated data xftanghotmail com summary in many applications researchers are interested in estimating the mean of a multivariate normal random vector whose components are subject to order restrictions various authors have demonstrated that the likelihood based methodology may perform poorly under certain conditions for such problems the problem is much harder when the underlying covariance matrix is nondiagonal in this paper a simple iterative algorithm is introduced that can be used for estimating the mean of a multivariate normal population when the components are subject to any order restriction the proposed methodology is illustrated through an application to human reproductive hormone data

the alexander polynomial of a plane curve singularity and the ring of functions on it we prove two formulae that express the alexander polynomial delta sp c of several variables of a plane curve singularity c in terms of the ring mathscr o sb c of germs of analytic functions on the curve one of them expresses delta sp c in terms of dimensions of some factors corresponding to a multi indexed filtration on the ring mathscr o sb c the other one gives the coefficients of the alexander polynomial delta sp c as euler characteristics of some explicitly described spaces complements to arrangements of projective hyperplanes

bayesian inference on order constrained parameters in generalized linear models summary in biomedical studies there is often interest in assessing the association between one or more ordered categorical predictors and an outcome variable adjusting for covariates for a k level predictor one typically uses either a k 1 degree of freedom df test or a single df trend test which requires scores for the different levels of the predictor in the absence of knowledge of a parametric form for the response function one can incorporate monotonicity constraints to improve the efficiency of tests of association this article proposes a general bayesian approach for inference on order constrained parameters in generalized linear models instead of choosing a prior distribution with support on the constrained space which can result in major computational difficulties we propose to map draws from an unconstrained posterior density using an isotonic regression transformation this approach allows flat regions over which increases in the level of a predictor have no effect bayes factors for assessing ordered trends can be computed based on the output from a gibbs sampling algorithm results from a simulation study are presented and the approach is applied to data from a time to pregnancy study

a survey of strategies for modeling cross classifications having ordinal variables abstract a survey is given of the main strategies for modeling cross classifications that contain ordinal variables models are described for two way tables in which one or both classifications are ordered and for multidimensional tables in which at least one classification is ordered primary emphasis is given to construction interpretation and implications of loglinear and logit models

confidence interval estimation subject to order restrictions this article deals with the construction of confidence intervals when the components of the location parameter mu of the random variable mathbf x which is elliptically symmetrically distributed are subject to order restrictions several domination results are proved by studying the derivative of the coverage probability of the confidence intervals centered at the improved point estimators consequently we strengthen the previously known results regarding the simple ordering and obtain several new results for other general forms of order restrictions including the simple tree ordering the umbrella ordering the simple and the double loop ordering and some combination of these these domination results are obtained under the assumption that sigma is a diagonal matrix when sigma is nondiagonal some new intervals are introduced which dominate the standard intervals centered at the unrestricted maximum likelihood estimator for various types of order restrictions similar results are obtained for scale parameters as well contrary to the location problems in case of the scale parameters satisfying the simple ordering we find that the restricted maximum likelihood estimator of the largest parameter fails to universally dominate the unrestricted maximum likelihood estimator a similar negative result is noted for simple tree order restriction

the first integral method to study the burgers korteweg de vries equation in this paper applying the theory of commutative algebra we propose a new approach which we currently call the first integral method to study the burgers korteweg de vries equation

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

smoothing spline models for the analysis of nested and crossed samples of curves abstract we introduce a class of models for an additive decomposition of groups of curves stratified by crossed and nested factors generalizing smoothing splines to such samples by associating them with a corresponding mixed effects model the models are also useful for imputation of missing data and exploratory analysis of variance we prove that the best linear unbiased predictors blups from the extended mixed effects model correspond to solutions of a generalized penalized regression where smoothing parameters are directly related to variance components and we show that these solutions are natural cubic splines the model parameters are estimated using a highly efficient implementation of the em algorithm for restricted maximum likelihood reml estimation based on a preliminary eigenvector decomposition variability of computed estimates can be assessed with asymptotic techniques or with a novel hierarchical bootstrap resampling scheme for nested mixed effects models our methods are applied to me

tests for order restrictions in binary data in this article a general procedure is presented for testing for equality of k independent binary response probabilities against any given ordered alternative the proposed methodology is based on an estimation procedure developed in hwang and peddada 1994 annals of statistics 22 67 93 and can be used for a very broad class of order restrictions the procedure is illustrated through application to two data sets that correspond to three commonly encountered order restrictions simple tree order simple order and down turn order

statistical inference for uniform stochastic ordering in several populations tion for k population problems under uniform stochastic ordering restrictions we derive closed form estimates even with right censored data by a reparameterization which reduces the problem to a well known isotonic regression problem we also derive the asymptotic distribution of the likelihood ratio statistic for testing equality of the k populations against the uniform stochastic ordering restriction this asymptotic distribution is of the chi bar square type as discussed by robertson wright and dykstra these distributional results are obtained by appealing to elegant results from empirical process theory and showing that the proposed test is asymptotically distribution free recurrence formulas are derived for the weights of the chi bar square distribution for particular cases the theory developed in this paper is illustrated by an example involving data for survival times for carcinoma of the oropharynx

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

a likelihood ratio test for stochastic ordering abstract this article presents a method of extending the likelihood ratio principle to a nonparametric setting the problem dealt with is a test for stochastic ordering the resulting test is shown via simulation to have good power it is shown to be consistent and unbiased

critical angles in polyhedral convex cones numerical and statistical considerations this work concerns the numerical computation of critical angles in polyhedral convex cones the set of proper critical angles is evaluated explicitly by solving a series of generalized eigenvalue problems involving the generators of the cone the local maximal angles are identified by using a necessary condition for local maximality the expected numbers of critical angles in random polyhedral convex cones are estimated experimentally

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