on measuring the conformity of a parameter set to a trend with applications abstract consider the hypothesis h 1 theta 1 or theta 2 or or theta k regarding a collection theta 1 theta 2 theta k of unknown parameters it is clear that this trend is reflected in certain possible parameter sets more than in others a quantification of this notion of conformity to a trend which was suggested by barlow and brunk 1972 is studied applications of the resulting theory to several order restricted hypothesis tests are presented author

multidimensional hermite hadamard inequalities and the convex order abstract the problem of establishing inequalities of the hermite hadamard type for convex functions on n dimensional convex bodies translates into the problem of finding appropriate majorants of the involved random vector for the usual convex order we present two results of partial generality which unify and extend the most part of the multidimensional hermite hadamard inequalities existing in the literature at the same time that lead to new specific results the first one fairly applies to the most familiar kinds of polytopes the second one applies to symmetric random vectors taking values in a closed ball for a given but arbitrary norm on r n related questions such as estimates of approximation and extensions to signed measures also are briefly discussed

p functions quasi convex functions and hadamard type inequalities we establish some results concerning p functions from the standpoint of abstract convexity in particular we show that the set of all p functions on a segment is the least set closed under pointwise sum supremum and convergence and containing the class of all nonnegative quasi convex functions on that segment further generalizations are derived of a recent inequality of hadamard type for p functions

cantor spectrum for the almost mathieu operator in this paper we use results on reducibility localization and duality for the almost mathieu operator left h b phi x right n x n 1 x n 1 b cos left 2 pi n omega phi right x n on l 2 and its associated eigenvalue equation to deduce that for b 0 2 and diophantine the spectrum of the operator is a cantor subset of the real line this solves the so called ten martini problem for these values of b and moreover we prove that for b 0 small or large enough all spectral gaps predicted by the gap labelling theorem are open

directional tests for one sided alternatives in multivariate models consider one sided testing problems for a multivariate exponential family model through conditioning or other considerations the problem oftentimes reduces to testing a null hypothesis that the natural parameter is a zero vector against the alternative that the natural parameter lies in a closed convex cone b the problems include testing homogeneity of parameters testing independence in contingency tables testing stochastic ordering of distributions and many others a test methodology is developed that directionalizes the usual test procedures such as likelihood ratio chi square fisher and so on the methodology can be applied to families of tests where the family is indexed by a size parameter so as to enable nonrandomized testing by p values for discrete models a refined family of tests provides a refined grid for better testing by p values the tests have essential monotonicity properties that are required for admissibility and for desirable power properties two examples are given

metal insulator transition for the almost mathieu operator we prove that for diophantine and almost every the almost mathieu operator h n n 1 ni 1 cos 2 n n exhibits localization for 2 and purely absolutely continuous spectrum for 2

chi squared tests for and against an order restriction on multinomial parameters abstract chi squared tests are considered for testing homogeneity against an order restriction and for testing an order restriction against all alternatives on multinomial parameters closed form expressions are obtained for the minimum chi squared estimator and the neyman modified chi squared estimator the asymptotic equivalence between these chi squared tests and the likelihood ratio test is established the latter has been studied by robertson 1978 their asymptotic null distribution is the chi bar squared distribution these chi squared tests are applied to a numerical example in horse racing also included are the results of some simulation studies conducted to investigate the rate of convergence to the limiting null distributions and to compare the powers of these tests chi squared tests are widely used statistical tools for making inferences about a collection p 1 p 2 pk of multinomial parameters many such collections encountered in practice exhibit a trend for instance pi may represent

an analysis of means type test for variances from normal populations after a brief review of the literature a test for homogeneity of variances is presented the test which is the variances analog of the analysis of means anom can be presented in a graphical form this graphical presentation makes it easy for practitioners to assess both practical and statistical significance tables of critical values are presented for balanced designs a large sample version of the test which uses anom critical values is also presented as well as a version of the test for unbalanced designs for which approximate critical values can be calculated monte carlo results indicate that this test has power comparable to well known normal based tests for homogeneity of variances

comparison of several treatments with a control using multiple contrasts abstract a problem frequently encountered in the practice of statistics is the comparison of several treatments with a control or standard we consider an experimental situation where prior knowledge indicates that all of the treatments are at least as effective as the control and the problem is to determine if any are significantly better than the control a number of statistical procedures have been proposed for this situation of which the best known is dunnett s 1955 multiple comparison procedure dunnett s test rejects equality of the treatments and the control for a large value of the maximum contrast of the data vector with several vectors that are located symmetrically within the alternative region we study a large class of such tests which includes dunnett s test as a particular case one of these tests which is based on the maximum contrast of the data vector with several orthogonal vectors is very easy to implement and has an uncomplicated good and fairly uniform power function over t

on the admissibility and consistency of tests for homogeneity of variances 1 introduction and summary the problem of testing homogeneity of variances in a one way fixed analysis of variance model has been studied for the past 50 years if we let s2 denote the sample variance for the ith population i 1 2 p then the s2 s are independently distributed and each estimates its corresponding variance a under the assumption of normality of each of the p populations and the further assumption that each s2 is based on n 1 the same number of observations laue 1965 studied a two parameter family of tests t x q which reject the homogeneity hypothesis when r x rq d where

refinements of hadamard type inequalities for quasi convex functions with applications to trapezoidal formula and to special means in this paper some inequalities of hadamard s type for quasi convex functions are given some error estimates for the trapezoidal formula are obtained applications to some special means are considered

hadamard type inequality for quasiconvex functions in higher dimensions in this article we study a hadamard type inequality for nonnegative evenly quasiconvex functions the approach of our study is based on the notion of abstract convexity we also provide an explicit calculation to evaluate the asymptotically sharp constant associated with the inequality over a unit square in the two dimensional plane

projected tests for order restricted alternatives consider the model where x ij i 1 2 k j 1 2 n are independent random variables distributed according to a one parameter exponential family with natural parameter i we test h 0 1 k versus h 1 c h 0 where 1 k and c is a cone determined by a 0 where the rows of a are contrasts with two nonzero elements we offer a method of generating good tests for h 0 versus h 1 the method is to take a good test for h 0 versus h 2 not h 0 and apply the test to projected sample points where the projection is onto c good tests for h 0 versus h 1 are tests that are schur convex good tests for h 0 versus h 1 are tests which are monotone with respect to a cone order we demonstrate that if the test function for h 0 versus h 2 is a constant size schur convex test then the resulting projected test is monotone

the ten martini problem we prove the conjecture known as the ten martini problem after kac and simon that the spectrum of the almost mathieu operator is a cantor set for all nonzero values of the coupling and all irrational frequencies

anderson localization for the completely resonant phases abstract for the almost mathieu operator h u n u n 1 u n 1 2 cos 2 n u n avila and jitomirskaya conjecture that for every phase r r 2 z z h satisfies anderson localization if e 2 in the present paper we verify the conjecture for e 7