asymptotic equivalence of nonparametric regression and white noise the principal result is that under conditions to any nonparametric regression problem there corresponds an asymptotically equivalent sequence of white noise with drift problems and conversely this asymptotic equivalence is in a global and uniform sense any normalized risk function attainable in one problem is asymptotically attainable in the other with the difference in normalized risks converging to zero uniformly over the entire parameter space the results are constructive a recipe is provided for producing these asymptotically equivalent procedures some implications and generalizations of the principal result are also discussed

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

right l groups geometric garside groups and solutions of the quantum yang baxter equation abstract right lattice ordered groups are introduced and studied as a general framework for garside groups and related groups without a garside element every right l group g has a two sided partially ordered subgroup n g which generalizes the quasi centre of an artin tits group the group n g splits into copies of z if g is noetherian the positive cone of a right l group is described as a structure that is known from algebraic logic a pair of left and right self similar hoops for noetherian right l groups g modularity of the lattice structure is characterized in terms of an operation on the set x g of atoms it is proved that modular garside groups are equivalent to finite projective spaces with a non degenerate labelling a concept of duality for x g is introduced and applied in the distributive case this gives a one to one correspondence between noetherian right l groups with duality and non degenerate unitary set theoretic solutions of the quantum yang baxter equation the description of garside groups via garside germs is extended to right l groups which yields a one sided enhancement and a new proof of dvurecenskij s non commutative extension of mundici s correspondence between abelian l groups and mv algebras

interpretation of af c algebras in łukasiewicz sentential calculus on donne un critere pour la non simplicite d une af c algebre 7b u en termes de proprietes de la theorie de la recursivite du groupe dimension k 0 7b u

simultaneous confidence regions for multivariate bioequivalence demonstrating bioequivalence of several pharmacokinetic pk parameters such as auc and cmax that are calculated from the same biological sample measurements is in fact a multivariate problem even though this is neglected by most practitioners and regulatory bodies who typically settle for separate univariate analyses we believe however that a truly multivariate evaluation of all pk measures simultaneously is clearly more adequate in this paper we review methods to construct joint confidence regions around multivariate normal means and investigate their usefulness in simultaneous bioequivalence problems via simulation some of them work well for idealised scenarios but break down when faced with real data challenges such as unknown variance and correlation among the pk parameters we study the shapes of the confidence regions resulting from different methods discuss how marginal simultaneous confidence intervals for the individual pk measures can be derived and illustrate the application to data from a trial on ticlopidine hydrochloride an r package is available

a note on methods of restoring consistency to the bootstrap summary we consider the property of consistency and its relevance for determining the performance of the bootstrap we analyse various parametric bootstrap approximations to the distributions of the hodges and stein estimators whose behaviour is typical of that of super efficient estimators employed in wavelet regression kernel density estimation and nonparametric curve fitting our results reveal not only some of the difficulties in selecting good modifications to the intuitive bootstrap but also that inconsistent bootstrap approximations may perform better than consistent versions even in large samples

superefficiency in nonparametric function estimation fixed parameter asymptotic statements are often used in the context of nonparametric curve estimation problems e g nonparametric density or regression estimation in this context several forms of superefficiency can occur in contrast to what can happen in regular parametric problems here every parameter point e g unknown density or regression function can be a point of superefficiency we begin with an example which shows how fixed parameter asymptotic statements have often appeared in the study of adaptive kernel estimators and how superefficiency can occur in this context we then carry out a more systematic study of such fixed parameter statements it is shown in four general settings how the degree of superefficiency attainable depends on the structural assumptions in each case

decomposition of garside groups and self similar l algebras abstract picantin s iterated crossed product representation of garside monoids is extended and reproved for a wide class of not necessarily noetherian partially ordered groups with a right invariant lattice structure it is shown that the tree like structure of such an iterated crossed product is equivalent to a partial cycle set closely related to a class of set theoretic solutions of the quantum yang baxter equation the decomposition of finite square free solutions is related to the crossed product representation of the corresponding structure group

improved confidence sets for the coefficients of a linear model with spherically symmetric errors on obtient des resultats de domination semblables a ceux de hwang et casella 1984 pour differentes distributions a symetrie spherique

small confidence sets for the mean of a spherically symmetric distribution summary suppose that x has a k variate spherically symmetric distribution with mean vector 0 and identity covariance matrix wepresent two spherical confidence sets for 0 both centred at a positive part stein estimator t x in the first we obtain the radius by approximating the upper a point of the sampling disiribution of iits x 9112 by the first two non zero terms of its taylor series about the origin we can analyse some of the properties of this confidence set and see that it performs well in terms of coverage probability volume and conditional behaviour in the second method we find the radius by using a parametric bootstrap procedure here even greater improvement in terms of volume over the usual confidence set is possible at the expense of having a less explicit radius function a real data example is provided and extensions to the unknown covariance matrix and elliptically symmetric cases are discussed

a covering theory for non involutive set theoretic solutions to the yang baxter equation abstract an algebraic theory of non involutive set theoretic solutions to the yang baxter equation by means of a non commutative version of braces is established the structure group and the permutation group of a solution is put into a context of discrete lie theory this leads to a natural concept of covering which shares the good properties of coverings in geometry and topology in the special case of quandle coverings in knot theory known anomalies are removed

a decomposition theorem for square free unitary solutions of the quantum yang baxter equation abstract it is known that every skew polynomial ring with generating set x and binomial relations in the sense of gateva ivanova trans amer math soc 343 1994 203 is an artin schelter regular domain of global dimension x moreover every such ring gives rise to a non degenerate unitary set theoretical solution r x 2 x 2 of the quantum yang baxter equation which fixes the diagonal of x 2 gateva ivanova s conjecture talk at the international algebra conference miskolc hungary 1996 states that conversely every such solution r comes from a skew polynomial ring with binomial relations an equivalent conjecture duke math j 100 1999 169 says that the underlying set x is r decomposable we prove these conjectures and construct an indecomposable solution r with x which shows that an extension to infinite x is false

minimum volume confidence regions for a multivariate normal mean vector summary since stein s original proposal in 1962 a series of papers have constructed confidence regions of smaller volume than the standard spheres for the mean vector of a multivariate normal distribution a general approach to this problem is developed here and used to calculate a lower bound on the attainable volume bayes and fiducial methods are involved in the calculation scheffe type problems are used to show that low volume by itself does not guarantee favourable inferential properties

good exact confidence sets for a multivariate normal mean a class of confidence sets with constant coverage probability for the mean of a p variate normal distribution is proposed through a pseudo empirical bayes construction when the dimension is greater than 2 by combining analytical results with some exact numerical calculations the proposed sets are proved to have a uniformly smaller volume than the usual confidence region sufficient conditions for the connectedness of the proposed confidence sets are also derived in addition our confidence sets could be used to construct tests for point null hypotheses the resultant tests have convex acceptance regions and hence are admissible by birnbaum tabular results of the comparison between the proposed region and other confidence sets are also given

lattice ordered effect algebras and l algebras abstract it is known that many logical algebras can be considered as l algebras in this paper we first construct an effect algebra e 0 1 that cannot be converted into an l algebra e 0 1 with negation such that x 0 is the orthosupplement then we prove that every lattice ordered effect algebra gives rise to an l algebra with the same orthosupplement finally we characterize le l algebras as lattice ordered effect algebras as an application we extend those results obtained in some recent references for orthomodular lattices and mv algebras

cluster seeking james stein estimators this paper considers the problem of estimating a high dimensional vector of parameters theta in mathbb r n from a noisy observation the noise vector is independent identically distributed gaussian with known variance for a squared error loss function the james stein js estimator is known to dominate the simple maximum likelihood ml estimator when the dimension n exceeds two the js estimator shrinks the observed vector toward the origin and the risk reduction over the ml estimator is greatest for theta that lie close to the origin js estimators can be generalized to shrink the data toward any target subspace such estimators also dominate the ml estimator but the risk reduction is significant only when theta lies close to the subspace this leads to the question in the absence of prior information about theta how do we design estimators that give significant risk reduction over the ml estimator for a wide range of theta in this paper we propose shrinkage estimators that attempt to infer the structure of theta from the observed data in order to construct a good attracting subspace in particular the components of the observed vector are separated into clusters and the elements in each cluster shrunk toward a common attractor the number of clusters and the attractor for each cluster are determined from the observed vector we provide concentration results for the squared error loss and convergence results for the risk of the proposed estimators the results show that the estimators give significant risk reduction over the ml estimator for a wide range of theta particularly for large n simulation results are provided to support the theoretical claims

discrete optimization for interpretable study populations and randomization inference in an observational study of severe sepsis mortality abstractmotivated by an observational study of the effect of hospital ward versus intensive care unit admission on severe sepsis mortality we develop methods to address two common problems in observational studies 1 when there is a lack of covariate overlap between the treated and control groups how to define an interpretable study population wherein inference can be conducted without extrapolating with respect to important variables and 2 how to use randomization inference to form confidence intervals for the average treatment effect with binary outcomes our solution to problem 1 incorporates existing suggestions in the literature while yielding a study population that is easily understood in terms of the covariates themselves and can be solved using an efficient branch and bound algorithm we address problem 2 by solving a linear integer program to use the worst case variance of the average treatment effect among values for unobserved potential outcomes that are compatible with the null hy

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions pointwise error global error measured in l p norms pointwise and global error in estimation of derivatives and for a wide range of smoothness classes including standard holder and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves finally the theory underlying the method is interesting as it exploits a correspondence between statistical questions and questions of optimal recovery and information based complexity

l algebras self similarity and l groups abstract every set x with a binary operation satisfying x y x z y x y z corresponds to a solution of the quantum yang baxter equation if the left multiplication is bijective w rump a decomposition theorem for square free unitary solutions of the quantum yang baxter equation adv math 193 2005 40 55 the same equation becomes a true statement of propositional logic if the binary operation is interpreted as implication in the present paper l algebras essentially defined by the mentioned equation are introduced and studied for example hilbert algebras locales left hoops pseudo mv algebras and l group cones are l algebras the main result states that every l algebra admits a self similar closure in a further step a structure group g x is associated to any l algebra x one more equation implies that the structure group g x is lattice ordered as an application we characterize the l algebras with a natural embedding into the negative cone of an l group in particular this implies mundici s equivalence between mv algebras and unital abelian l groups and dvurecenskij s non commutative generalization

shrinkage estimators skorokhod s problem and stochastic integration by parts for a broad class of error distributions that includes the spherically symmetric ones we give a short proof that the usual estimator of the mean in a d dimensional shift model is inadmissible under quadratic loss when d 3 our proof involves representing the error distribution as that of a stopped brownian motion and using elementary stochastic analysis to obtain a generalization of an integration by parts lemma due to stein in the gaussian case

empirical bayes confidence sets for the mean of a multivariate normal distribution abstract through the use of an empirical bayes argument a confidence set for the mean of a multivariate normal distribution is derived the set is a recentered sphere is easy to compute and has uniformly smaller volume than the usual confidence set an exact formula for the coverage probability is derived and numerical evidence is presented which shows that the empirical bayes set uniformly dominates the usual set in coverage probability

empirical bayes confidence intervals shrinking both means and variances summary we construct empirical bayes intervals for a large number p of means the existing intervals in the literature assume that variances are either equal or unequal but known when the variances are unequal and unknown the suggestion is typically to replace them by unbiased estimators however when p is large there would be advantage in borrowing strength from each other we derive double shrinkage intervals for means on the basis of our empirical bayes estimators that shrink both the means and the variances analytical and simulation studies and application to a real data set show that compared with the t intervals our intervals have higher coverage probabilities while yielding shorter lengths on average the double shrinkage intervals are on average shorter than the intervals from shrinking the means alone and are always no longer than the intervals from shrinking the variances alone also the intervals are explicitly defined and can be computed immediately

one sided orthogonality orthomodular spaces quantum sets and a class of garside groups abstract the classification of orthomodular lattices by means of their structure group is extended to non symmetric orthogonality relations spaces with a non hermitian sesquilinear form quantum sets and a new class of garside groups are given as examples

admissibility of the usual confidence set for the mean of a univariate or bivariate normal population the unknown variance case summary in the gaussian linear regression model with unknown mean and variance we show that the standard confidence set for one or two regression coefficients is admissible in the sense of joshi this solves a long standing open problem in mathematical statistics and this has important implications on the performance of modern inference procedures post model selection or post shrinkage particularly in situations where the number of parameters is larger than the sample size as a technical contribution of independent interest we introduce a new class of conjugate priors for the gaussian location scale model