unramified cohomology and witt groups of anisotropic pfister quadrics the unramified witt group of an anisotropic conic over a field k with char k 0 2 defined by the form 1 a b is known to be a quotient of the witt group w k of k and isomorphic to w k 1 a b ab w k we compute the unramified cohomology group h43rk c where c is the three dimensional anisotropic quadric defined by the quadratic form 1 a b ab c over k we use these computations to study the unramified witt group of c let k be a field of characteristic not two and k k be a finitely generated field extension the n th unramified cohomology group of k over k with coefficients in z 2 cf ct is denoted hnnr k k or just hnnr k when the ground field being considered is clear it is a subgroup of the galois cohomology group hn k z 2 abbreviated to hn k and is defined as hnnr k k n h6nt spec ov u2 vev k here v k is the set of all rank one discrete valuations on k that are trivial on k for v e v k ov denotes the corresponding discrete valuation ring it is well known that the group hnt spec ov j2 injects into hn k if x k is a smooth projective geometrically integral variety and k x is its function field then by abuse of terminology we refer to the groups hnrn k x as the unramified cohomology groups of x for a field f let w f be the witt group of quadratic forms over f sc for k and k k as above we can similarly define the unramified witt group of k k denoted wtr k k as the subgroup of w k defined by wnr k k n w ov vev k here w ov is the witt group of the discrete valuation ring ov which is a subgroup of w k sc chapter vi if x k is smooth projective then we refer to wnr k x as the unramified witt group of x recall that a variety x is said to be k rational if x is k birational to the projective space then the natural maps hn k hn k x resp w k w k x induce isomorphisms htn k hnrr k x resp w k wnr k x thus the unramified cohomology groups of a k rational variety resp the unramified witt received by the editors november 7 1995 1991 mathematics subject classification primary iie70 secondary 13k05 12g05

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

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

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

ranked set sampling with unequal samples summary a ranked set sampling procedure with unequal samples rssu is proposed and used to estimate the population mean this estimator is then compared with the estimators based on the ranked set sampling rss and median ranked set sampling mrss procedures it is shown that the relative precisions of the estimator based on rssu are higher than those of the estimators based on rss and mrss an example of estimating the mean diameter at breast height of longleaf pine trees on the wade tract in thomas county georgia is presented

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

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

on the structure of pythagorean fields abstract in this paper we study in detail the relationship between fans and valuations first uncovered by brocker see 2 our techniques are somewhat different as our valuations are constructed by brute force from the properties of the fields involved this approach has the advantage in that the valuation theory it yields can be directly related to the structure theory for spaces of orderings as developed by murray marshall see 11 14 we also discuss several applications which include results in k theory

cohomology and graded witt group kernels for extensions of degree four in characteristic two abstract in earlier work the authors determined the graded witt kernel g w q e f ker g w q f g w q e when e f is a biquadratic extension in characteristic 2 by calculating the cohomological kernel h 2 e f ker h 2 f h 2 e in this paper this result is extended to the cases where e f 4 and e is either cyclic or has dihedral galois closure in addition the use of izhboldin s q groups is generalized to obtain six term exact sequences that describe the behavior of these graded rings whenever four term exact sequences and homotopies describe the arithmetic of the extension these tools are valid in characteristic p although the applications here are in characteristic 2

the graded witt ring and galois cohomology ii a primary problem in the theory of quadratic forms over a field f of characteristic different from two is to prove that the rings hq f and gwf are isomorphic here hqf h gal fq f z 2z where fq is the quadratic closure of f and gwf is the graded witt nrng associated to the fundamental ideal of even dimensional forms in the witt ring wf of f in this paper we assume we are given a field extension k of f such that wk is close to wf or hq k is close to hq f a method is developed to obtain information about these graded rings over f and its 2 extensions from information about the corresponding graded ring of k this relative theory extends and includes the previously developed absolute case where k fq applications are also given to show that hq f and gwf are isomorphic for a collection of fields anrsing naturally from the theory of abstract witt rings

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

rigid elements valuations and realization of witt rings to the theory over the residue class field of the valuation this theorem generalizes to any 2 henselian valuation cf kl sect 12 21 let v be the valuation let a e k f o 1 be such that v a is not divisible by two the key idea behind the proof is that the quadratic form 1 a only represents elements in u a such an element a is called

random dirichlet environment viewed from the particle in dimension d ge 3 we consider random walks in random dirichlet environment rwde which is a special type of random walks in random environment where the exit probabilities at each site are i i d dirichlet random variables on zd rwde are parameterized by a 2d tuple of positive reals called weights in this paper we characterize for d 3 the weights for which there exists an absolutely continuous invariant probability distribution for the process viewed from the particle we can deduce from this result and from ann inst henri poincare probab stat 47 2011 1 8 a complete description of the ballistic regime for d 3

limiting velocity of high dimensional random walk in random environment we show that random walk in uniformly elliptic i i d environment in dimension 5 has at most one non zero limiting velocity in particular this proves a law of large numbers in the distributionally symmetric case and establishes connections between different conjectures

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

harmonic functions h transform and large deviations for random walks in random environments in dimensions four and higher we consider large deviations for nearest neighbor random walk in a uniformly elliptic i i d environment on d there exist variational formulae for the quenched and averaged rate functions i q and i a obtained by rosenbluth and varadhan respectively i q and i a are not identically equal however when d 4 and the walk satisfies the so called t condition of sznitman they have been previously shown to be equal on an open set a eq for every a eq we prove the existence of a positive solution to a laplace like equation involving and the original transition kernel of the walk we then use this solution to define a new transition kernel via the h transform technique of doob this new kernel corresponds to the unique minimizer of varadhan s variational formula at it also corresponds to the unique minimizer of rosenbluth s variational formula provided that the latter is slightly modified