choice of column scores for testing independence in ordered 2 x k contingency tables the numerous statistical methods for testing no association between a binary response rows and k ordered categories columns group naturally into two classes those that require preassigned numerical column scores and those that do not an example of the former would be a logistic regression analysis and of the latter would be a wilcoxon rank sum test in this paper we demonstrate that the perceived advantage of not preassigning scores is illusory we do this by presenting an example from our consulting experience in which the midrank scores used by the rank tests that do not require preassigned scores are clearly inappropriate our recommendations are to assign reasonable column scores whenever possible and to consider equally spaced scores when the choice is not apparent midranks as scores should always be examined for their appropriateness before a rank test is applied

bounds on the rate of 2 d bit stuffing encoders a method for bounding the rate of bit stuffing encoders for 2 d constraints is presented instead of considering the original encoder we consider a related one which is quasi stationary we use the quasi stationary property in order to formulate linear requirements that must hold on the probabilities of the constrained arrays that are generated by the encoder these requirements are used as part of a linear program the minimum and maximum of the linear program bound the rate of the encoder from below and from above respectively a lower bound on the rate of an encoder is also a lower bound on the capacity of the corresponding constraint for some constraints our results lead to tighter lower bounds than what was previously known

approximating the solution of nonlinear hammerstein fuzzy integral equations abstract in this paper we prove the convergence of the method of successive approximations used to approximate the solution of nonlinear hammerstein fuzzy integral equations and define the notion of numerical stability of the algorithm with respect to the choice of the first iteration this numerical stability is proved and finally some numerical experiments confirm the theoretical results and illustrate the accuracy of the method

the positive capacity region of two dimensional run length constrained channels a binary sequence satisfies a one dimensional d k constraint if every run of zeros with possible exception of the first and the last runs has length at least d and at most k a binary two dimensional array satisfies a d k constraint if each row and each column satisfies the one dimensional d k constraint few models have been proposed in the literature to handle two dimensional data the diamond model the square model the hexagonal model and the triangular model the constraints in the different directions might be asymmetric and hence many kind of constraints are defined depending on the number of directions in the model for example a two dimensional array in the diamond model satisfies a d1 k1 d2 k 2 constraint if it satisfies the one dimensional d1 k1 constraint horizontally and the one dimensional d 2 k2 constraint vertically in this correspondence the region in which the capacity is zero or positive in the various models is examined asymmetric constraints in the diamond model and symmetric constraints in the other models are considered in particular an almost complete solution for asymmetric constraints in the diamond model is provided

on the fuzzy initial value problem abstract the initial value problem x t f t x t x 0 x 0 with fuzzy initial value and with deterministic or fuzzy function f is considered two different approaches viz the extension principle and the use of extremal solutions of deterministic initial value problems are applied generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated

error estimation in the approximation of the solution of nonlinear fuzzy fredholm integral equations using a fixed point technique the sequence of successive approximations and a recent quadrature formula for fuzzy number valued functions it is constructed a numerical method for the solution of nonlinear fuzzy fredholm integral equations moreover the error estimate of the method and a criterion to stop the corresponding algorithm are given

entropy bounds for constrained two dimensional random fields the maximum entropy and thereby the capacity of two dimensional 2 d fields given by certain constraints on configurations is considered upper and lower bounds are derived a new class of 2 d processes yielding good lower bounds is introduced asymptotically the process achieves capacity for constraints with limited long range effects the processes are general and may also be applied to e g data compression of digital images results are given for the binary hard square model which is a 2 d run length limited model and some other 2 d models with simple constraints

on tests that are uniformly more powerful than the wilcoxon mann whitney test summary the wilcoxon mann whitney rank sum test u test is often used in the nonparametric analysis of controlled clinical trials we suggest a test that is uniformly more powerful than the u test in the following sense the test will reject ho whenever the u test rejects ho and there are situations when the new test will reject ho but the u test is not able to reject ho the new test is not a mathematical curiosity because it is constructed to achieve high power against those alternatives in which the better one of two given treatments has not only higher expectation but also less variability than the other treatment and furthermore the reduction in the p values given by the new test as compared to the u test is often decisive the principle is rather general and can be applied to practically all nonparametric tests a table of significance levels is given for continuous distributions and an algorithm is presented that allows the determination of the exact distribution in the general case 1 nonrandomized test procedures and the u test treatises on statistical theory usually consider only randomized test procedures the application of these procedures requires the use of a random device in order to obtain the final decision and is therefore not accepted by some researchers especially in the field of controlled clinical trials in the interest of reproducible results these researchers apply nonrandomized variants of the classical test procedures we are proposing an alternative method to randomization on the boundary for test statistics with discrete null distributions this method results in test procedures that are in general uniformly more powerful than the known nonrandomized tests the basic idea is very simple and will be illustrated in the case of the common u test that has been proposed by wilcoxon 1945 and mann and whitney 1947 the p value for the nonrandomized u test is determined as follows the observations x x1 x2 xn xnl i xn in the two treatment groups of size n1 resp n n1 are ranked under ho all n possible permutations of the rank vector are equally likely the probability p u under ho that the rank sum u u x in the first group takes a given value u is equal to the proportion of permutations with u as rank sum in the first group

improved exact inference about conditional association in three way contingency tables abstract we propose modified exact inferential methods for contingency tables ordinary exact inference is conservative because of the discreteness for estimating a common odds ratio in several 2 2 tables two modifications of the ordinary exact confidence interval maintain at least a fixed confidence level but tend to be much narrower one approach inverts results of a test with a modified p value utilizing the test statistic and table probabilities the second approach inverts one two sided test rather than two one sided tests this approach is much less conservative when the true odds ratio is relatively small or large we also generalize results of cohen and sackrowitz and relate modified p values to construction of exact unbiased and admissible tests for an ordinal alternative to conditional independence

two dimensional constrained coding based on tiling a new variable rate coding technique is presented for two dimensional 2 d constraints for certain constraints such as the 0 2 runlength limited rll and 3 rll constraints the technique is shown to improve on previously published lower bounds on the capacity of the constraint

modelling ordered categorical data recent advances and future challenges this article summarizes recent advances in the modelling of ordered categorical ordinal response variables we begin by reviewing some models for ordinal data introduced in the literature in the past 25 years we then survey recent extensions of these models and related methodology for special types of applications such as for repeated measurement and other forms of clustering we also survey other aspects of ordinal modelling such as small sample analyses power and sample size considerations and availability of software throughout we suggest problem areas for future research and we highlight challenges for statisticians who deal with ordinal data

an analytical method for solving linear fredholm fuzzy integral equations of the second kind in this paper we use the parametric form of a fuzzy number and convert a linear fuzzy fredholm integral equation to two linear systems of integral equations of the second kind in the crisp case for fuzzy fredholm integral equations with kernels the sign of which is difficult to determine a new parametric form of the fuzzy fredholm integral equation is introduced we use the homotopy analysis method to find the approximate solution of the system and hence obtain an approximation for fuzzy solutions of the linear fuzzy fredholm integral equation of the second kind the proposed method is illustrated by solving some examples using the ham it is possible to find the exact solution or the approximate solution of the problem in the form of a series

the supremum and infimum of the set of fuzzy numbers and its application abstract in this paper we prove that the bounded set of fuzzy numbers must exist supremum and infimum and give the concrete representation of supremum and infimum as an application we obtain that the continuous fuzzy valued function on a closed interval exists supremum and infimum and give the precise representation we also show that the bounded fuzzy valued function on a closed interval can define the lower and upper sums and the lower and upper integrals of riemann and riemann stieltjes by the usual way

fuzzy integral equations abstract we introduce a definition of the integral of a fuzzy valued function that is only slightly different from the usual one yet that is more intuitive and that can be applied to a larger class of functions we show that our definition is equivalent to the extension principle for functions of this class we then apply integration theory of fuzzy valued functions to integral equations we define a measure of the fuzziness of a fuzzy valued function and show that using this measure we can view the integration of fuzzy functions as a fuzzy linear transformation from a certain subspace of fuzzy valued functions to the space of fuzzy numbers we then show how this can be used to study fredholm integral equations of the second kind

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

bounded variation absolute continuity and absolute integrability for fuzzy number valued functions in this paper the concepts of bounded variation and absolute continuity for the fuzzy number valued functions are presented and discussed by means of the representation of the absolute value for fuzzy numbers the relations among bounded variation absolute continuity kaleva s integral and fuzzy henstock integral are characterized especially the representation of absolute continuous fuzzy valued functions is given by henstock kaleva integral of absolutely henstock kaleva integrable fuzzy number valued functions

improving tests for superior treatment in contingency tables abstract when comparing two treatments on the basis of ordinal data a natural alternative is stochastic order as it implies that more favorable outcomes receive greater probability results of eaton can be used to provide a complete class of tests we find that many tests in current use are not in this class and hence are inadmissible more important we present methods of improving such tests often the increase in power is substantial although the size remains the same

bit stuffing algorithms and analysis for run length constrained channels in two and three dimensions a rigorous derivation is given of the coding rate of a variable to variable length bit stuffing coder for a two dimensional 1 spl infin constrained channel the coder studied is nearly a fixed to fixed length algorithm then an analogous variable to variable length bit stuffing algorithm for the three dimensional 1 spl infin constrained channel is presented and its coding rate is analyzed using the two dimensional method the three dimensional coding rate is demonstrated to be at least 0 502 which is proven to be within 4 of the capacity

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

two computational methods for solving linear fredholm fuzzy integral equation of the second kind in this paper we present two methods which will exploit hybrid legendre and block pulse functions and legendre wavelets as bases for l 2 0 1 x 0 1 to find an approximate solution for a system of linear fredholm integral equation of the second kind with two variables we will apply our methods for an applied example linear fredholm fuzzy integral equation of the second kind m friedman ma ming a kandel numerical solutions of fuzzy differential and integral equations fuzzy sets and systems 106 1999 35 48

two dimensional low pass filtering codes we describe a framework for designing encoders that transform arbitrary data sequences into two dimensional arrays satisfying certain constraints in particular constraints that guarantee arrays with limited high spatial frequency content we also exhibit specific codes that produce such arrays such codes are useful for holographic recording systems

numerical solution of linear fredholm fuzzy integral equations of the second kind by adomian method using parametric form of fuzzy numbers we convert a linear fuzzy fredholm integral equation of the second kind to a linear system of integral equations of the second kind in crisp case we use adomian method and find the approximate solution of this system and hence obtain an approximation for fuzzy solution of the linear fuzzy fredholm integral equation of the second kind we apply the method to some examples

towards fuzzy differential calculus part 2 integration on fuzzy intervals abstract the problem of integrating mappings over fuzzy domains has already received some attention in the literature however there is not a unique approach but several definitions have been proposed each carrying its own meaning this paper tries to emphasize the relationships and the differences between the various points of view

convex hull test for ordered categorical data when testing for stochastic order in ordered 2 x j contingency tables it is common to select the cutoff required to declare significance so as to ensure that the size of the test is exactly alpha conditionally on the margins it is valid however to use the margins to select not only the cutoff but also the form of the test linear rank tests which are locally most powerful and frequently used in practice suffer from the drawback that they may have power as low as zero to detect some alternatives of interest when the margins satisfy certain conditions the smirnov and convex hull tests are shown through exact conditional power calculations and simulations to avoid this drawback the convex hull test is also admissible and palindromic invariant and minimizes the required significance level to have limiting power of one as the alternative moves away from the null in any direction

numerical solutions of fuzzy differential and integral equations using the embedding method numerical procedures for solving fuzzy differential equations fdes and fuzzy integral equations fies with arbitrary kernels have been investigated sufficient conditions for convergence of the proposed algorithms are given and their applicability is illustrated with examples this work and its conclusions may narrow the gap between the theoretical research on fdes and fies and the practical applications already existing in the design of various fuzzy dynamical systems

quadrature rules for integrals of fuzzy number valued functions in this paper we introduce some quadrature rules for the henstock integral of fuzzy number valued mappings by giving error bounds for mappings of bounded variation and of lipschitz type we also consider generalizations of classical quadrature rules such as midpoint type trapezoidal and three point type quadrature finally we study fine quadrature rules and we present some numerical applications

on the construction of bounded delay encodable codes for constrained systems we present a new technique to construct sliding block modulation codes with a small decoding window our method which involves both state splitting and look ahead encoding crucially depends on a new local construction method for bounded delay codes we apply our method to construct several new codes all with a smaller decoding window than previously known codes for the same constraints at the same rate

on the existence and uniqueness of solutions of fuzzy volterra fredholm integral equations abstract in this paper we study the problem of existence and uniqueness of solutions of volterra fredholm integral equation for the fuzzy set valued mappings

a log linear model for ordinal data to characterize differential change among treatments we propose a family of log linear models for ordinal data that contain parameters reflecting change patterns to compare treatments relative to change from baseline under the most general model rates of change can depend not only upon the direction of change but also upon the level of the baseline classification we describe methods for selection of a parsimonious model and for tests of hypotheses concerning treatment differences interpretation of treatment differences in the follow up response profiles within baseline strata employs the concept of stochastic ordering data from two clinical trials illustrate the proposed procedure