projected score methods for approximating conditional scores summary this paper extends the projected score methods of small mcleish 1989 it is shown that the conditional score function may be approximated with arbitrarily small stochastic error in terms of a natural basis for the space of centred likelihood ratios the utility of using this basis is established by identifying a u statistic representation theorem and a class of expectation identities for the basis elements making higher order asymptotics more tractable the results are applied to a canonical exponential family model where it is shown that the projected scores with estimated nuisance parameters can provide an accurate approximation to the conditional score function

rigid animals in three dimensions the number of rigid clusters c n is enumerated for the central force model on a site diluted fcc lattice the exponent vr governing the growth of the radius of gyration of the rigid animal is also calculated it is found that c n approximately n0 333n and vr 0 32 or 0 02

orthogonal locally ancillary estimating functions for matched pair studies and errors in covariates summary we propose an estimating function method for two related applications matched pair studies and studies with errors in covariates under a functional model where a mismeasured unknown scalar covariate is treated as a fixed nuisance parameter our method addresses the severe inferential problem that is posed by an abundance of nuisance parameters in these two applications we propose orthogonal locally ancillary estimating functions for these two applications that depend on merely the mean model and partial modelling of the variances of the observations and observed mismeasured covariate if applicable and we achieve first order bias correction of inferences under a small dispersion and large sample size asymptotic simulation results confirm that the estimator proposed is largely improved over that using a regular profile estimating function we apply the approach proposed to a length of hospital stay study with a mismeasured covariate

a simple and accurate method for approximate conditional inference applied to exponential family models summary a simple method for approximate conditional inference is described the methodology is applied to natural exponential family models where it is shown to provide accurate approximations to fully conditional estimates the approximation technique can be applied much more generally than in this particular class of models the technique depends only on the construction of certain projected scores derived from higher order likelihood derivatives and their covariances and so can be used in many problems where it is relevant to control for the estimation of nuisance parameters

simple relaxed conditional likelihood when the data are sparse but not exceedingly so we face a trade off between bias and precision that makes the usual choice between conducting either a fully unconditional inference or a fully conditional inference unduly restrictive we propose a method to relax the conditional inference that relies upon commonly available computer outputs in the rectangular array asymptotic setting the relaxed conditional maximum likelihood estimator has smaller bias than the unconditional estimator and smaller mean square error than the conditional estimator

complexity and varieties for infinite groups i this two part paper generalizes the usual notion of complexity and varieties for modules over the group algebra of a finite group to a large class of infinite groups the context is modules of typefp for groups in kropholler s classlhf one of the main results is that the category of such modules is generated in a suitable sense by modules induced from finite elementary abelian subgroups this implies that an element of complete cohomology of such a module is nilpotent if and only if its restriction to every finite elementary abelian subgroup is nilpotent it also implies that the complexity of a module of typefp is finite and that the variety is supported on some finite collection of finite elementary abelian subgroups an example is given which shows that the complexity does not determine the rate of growth of the number of generators in a projective resolution in the way it does for finite groups

rigid backbone a new geometry for percolation understood in both cases from the nodes linksblobs picture of percolation clusters in fact it is fair to say that in the vast majority of percolation problems encountered to date even continuum percolation 6 the basic geometric features of the percolation clusters such as the scaling of the number of singly connected bonds are identical and the differences arise only at the level of the dynamics for the class of models of interest here however geometric connection the existence of a bond does not imply complete elastic connection because the number of constraints associated with the existence of a bond is smaller than the number of degrees of freedom associated with the sites it connects the percolation threshold is higher8 than that of the corresponding standard percolation problem thorpe6 has suggested the name rigidity percolation for this class of models distinguishing it from the elasticity percolation universality class mentioned above in this paper we study the simplest rigidity percolation model namely the two dimensional central force model 8 ex

periodic complexes and group actions in this paper we show that the cohomology of a connected cw complex is periodic if and only if it is the base space of a spherical fibration with total space that is homotopically finite dimensional as applications we characterize those discrete groups that act freely and properly on in x sm we construct nonstandard free actions of rank two simple groups on finite complexes y sn x sm and we prove that a finite p group p acts freely on such a complex if and only if it does not contain a subgroup isomorphic to z p 3

conditioning to reduce the sensitivity of general estimating functions to nuisance parameters summary a conditional method is presented that renders an estimating function insensitive to nuisance parameters the approach is a generalisation of the conditional score method to a general estimating function context and does not require complete specification of the probability model we exploit the informal relationship between general estimating functions and score functions to derive simple generalisations of sufficient and partially ancillary statistics referred to as g sufficient and g ancillary statistics respectively these two types of statistic are defined in a manner that does not require complete knowledge of the probability model and thus are more suitable for use with estimating functions if we condition on a g sufficient statistic for the nuisance parameters the resulting conditional estimating function is insensitive to nuisance parameters and in particular achieves the plug in unbiasedness property furthermore if the conditioning argument is also g ancillary for the parameters of interest then the conditional estimating function possesses an attractive optimality property

maximum likelihood estimation of a survival function with a change point for truncated and interval censored data this paper considers estimation of a survival function when there exists a change point and the survival time of interest is defined as elapsed time between two related events furthermore there exists censoring on observations on the occurrences of both events and truncation on observations on the occurrence of the second event and thus the survival time of interest to obtain the maximum likelihood estimator of a survival function an em algorithm is developed when the survival function is completely unknown before the change point and known up to a vector of unknown parameters after the change point the idea is a generalization of that discussed in moeschberger and klein simulations and an example are used to evaluate and illustrate the algorithm

an algorithm for enumerating rigid clusters a more efficient algorithm for enumerating rigid clusters is presented using this algorithm the author has extended the series of site rigid animals enumerated by prunet and blanc 1986 by five more terms and has also calculated for the first time the exponent governing the growth of the radius of gyration for rigid animals

the variety of an indecomposable module is connected let g be a finite group and let k be an algebraically closed field of characteristic p 0 the connections between the modular representation theory of finite groups and cohomology theory is currently a topic of great interest in particular cohomology theory has been used to attach to each finitely generated kgmodule m a homogeneous affine variety v m if we let 17 m be the corresponding projective variety then our main result is the following

projection as a method for increasing sensitivity and eliminating nuisance parameters l article analyse des methodes statistiques basees sur les fonctions d estimation qui sont des fonctions de parametres et de donnees a la fois destinees a permettre une inference sur un parametre inconnu une methode de projection peut etre utilisee pour la reduction de ces fonctions afin d accroitre la puissance d un test et l efficacite d un estimateur ponctuel ou bien de rendre une fonction d inference insensible a la valeur d un parametre de nuisance

on constrained quasi likelihood estimation for maximum likelihood or least squares parameter estimation subject to a constrained parameter the standard approach is to use the method of lagrange multipliers in this paper it is shown that the same formal procedure applies very generally for constrained quasi likelihood estimation even though there is ordinarily no objective function to maximize or minimize this allows for optimal estimation in a wide range of problems where maximum likelihood cannot be used such as for semiparametric models the method is very similar to projecting the free estimator or projecting the equations which it solves

cluster renormalisation study of site lattice animals in two and three dimensions the author studies site lattice animals using the cluster renormalisation cr approach previously applied to bond animals the exponent v characterising the asymptotic behaviour of the mean square radius of gyration of site animals and the fractal dimension df 1 v are determined in two and three dimensions he finds v 0 649 or 0 009 and v approximately 0 51 in two and three dimensions respectively in addition he studies restricted valence site animals on a triangular lattice using a single parameter cr approach

gorenstein complexes and recollements from cotorsion pairs we describe a general correspondence between injective resp projective recollements of triangulated categories and injective resp projective cotorsion pairs this provides a model category description of these recollement situations our applications focus on displaying several recollements that glue together various full subcategories of k r the homotopy category of chain complexes of modules over a general ring r when r is left noetherian these recollements involve complexes built from the gorenstein injective modules when r is a left coherent ring for which all flat modules have finite projective dimension we obtain the duals these results extend to a general ring r by replacing the gorenstein modules with the gorenstein ac modules introduced in 9 we also see that in any abelian category with enough injectives the gorenstein injective objects enjoy a maximality property in that they contain every other class making up the right half of an injective cotorsion pair

specific heats of classical spin systems and inhomogeneous differential approximants derives accurate compact expressions for the high temperature specific heats of classical s infinity 3 vector spin systems on fcc bcc and sc lattices for pure heisenberg xy and ising like couplings respectively the analysis of the appropriate series expansions demonstrates the utility of inhomogeneous differential approximants and supports the estimate alpha h 0 21 or 0 04

generalizations of ancillarity completeness and sufficiency in an inference function space on introduit l e ancillarite et l e exhaustivite complete des extensions naturelles des definitions d ancillarite et d exhaustivite complete a l espace des fonctions d estimation ou d inference

relation between elastic and scalar transport exponent in percolation using simple variational considerations the author shows that the elastic critical exponent tau is related to the conductivity index t and to the correlation length exponent nu by tau or t 2 nu it is conjectured that this relation might be an equality

bond diluted rigid animals for central force model an exact enumeration of rigid clusters is analysed using pade approximants the number of bond diluted rigid clusters with n b bonds is found to grow like n theta b b 2 165 or 0 005 n b with theta b 0 988 a value different from that of site diluted rigid animals

a novel method of calculating amplitude ratios by series expansions the author introduces a new method of calculating critical amplitude ratios using series which is both simple and powerful this method which gives estimates for the amplitude ratios that are neither biased by the values of the critical points nor by the critical exponents is applied to several models it is shown that this method produces results where no reliable estimates from series expansion exist in particular one finds 0 025 or 0 001 for at b2 for the 3d ising model and 220 or 10 for c c for the two dimensional percolation model in agreement with and with more accuracy than values obtained by other methods

the stable category and invertible modules for infinite groups abstract we construct a well behaved stable category of modules for a large class of infinite groups we then consider its picard group which is the group of invertible or endotrivial modules we show how this group can be calculated when the group acts on a tree with finite stabilisers

reducing sensitivity to nuisance parameters in semiparametric models a quasi score method summary this paper proposes a semiparametric extension of the projected score method of waterman lindsay 1996 for the elimination of nuisance parameters the procedure addresses cases where only the mean and the variance of the response variable are specified and where the mean function involves both parameters of interest and nuisance parameters important applications of the semiparametric model include quasilikelihood models for matched designs and for measurement error models carroll stefanski 1990 as a result of the optimality and information unbiasedness of the quasi score function a second order quasi score basis of estimating functions for the nuisance parameter is derived second order locally ancillary estimating functions small mcleish 1994 pp 81 4 are then obtained by solving a simple linear system that corresponds to a true projection for canonical exponential family distributions asymptotic arguments and simulation work show that the impact of nuisance parameters is considerably reduced when adopting the proposed approach

efficiency of projected score methods in rectangular array asymptotics summary the paper considers a rectangular array asymptotic embedding for multistratum data sets in which both the number of strata and the number of within stratum replications increase and at the same rate it is shown that under this embedding the maximum likelihood estimator is consistent but not efficient owing to a non zero mean in its asymptotic normal distribution by using a projection operator on the score function an adjusted maximum likelihood estimator can be obtained that is asymptotically unbiased and has a variance that attains the cram6r rao lower bound the adjusted maximum likelihood estimator can be viewed as an approximation to the conditional maximum likelihood estimator

a characterization of cohomological dimension for a big class of groups abstract it was shown by cornick and kropholler 1998 in 5 that if a group g is in h f then the finitistic dimension of z g findim z g is equal to the supremum of the projective dimensions of the z g modules m such that pd z h m for all finite subgroups h of g here we show that a locally h f group g is of type if and only if findim z g a group g is of type if for any z g module m pd z g m if and only if pd z h m for every finite subgroup h of g type was proposed in talelli 2007 18 as an algebraic characterization for those groups g that admit a finite dimensional model for e g we also deduce a number of corollaries for locally h f groups

rigid clusters enumeration animal counting techniques suggest that the fraction of rigid animals among all animals with s sites on a triangular lattice varied roughly as s1 2 0 46 or 0 01 s

locally ancillary quasi score models for errors in covariates we use the notion of locally ancillary estimating functions to develop a quasi score method for fitting regression models containing measurement error in the covariates suppose that interest is on the model e y u w for response y the observed data are y x w and x is a mismeasured surrogate for u we take a functional modeling approach treating the u as a fixed nuisance parameter beginning with quasi scores for the regression parameter and the unknown u we derive a bias corrected quasi score for the regression parameter that is second order locally ancillary for the nuisance u our method for this requires only the correct specification of the mean and variance functions for y and x in terms of u w and the regression parameter when an estimator for u is plugged into the corrected quasi score local approximations show that the bias is small we present simulations verifying this result and an example from child psychiatry both using log linear regression models

flat and stably flat modules abstract we study kropholler s generalisation of lazard s criterion for a module to be flat the context is complete cohomology and modules of type fp let g be a group in the class h 1 f of groups which act on a finite dimensional contractible cell complex with finite stabilisers and let r be a strongly g graded algebra we provide a characterisation of the stably flat r modules under the assumption that r1 is coherent of finite global dimension

finiteness conditions in the stable module category we study groups whose cohomology functors commute with filtered colimits in high dimensions we relate this condition to the existence of projective resolutions which exhibit some finiteness properties in high dimensions and to the existence of eilenberg mac lane spaces with finitely many n cells for all sufficiently large n to that end we determine the structure of completely finitary gorenstein projective modules over group rings the methods are inspired by representation theory and make use of the stable module category in which morphisms are defined through complete cohomology in order to carry out these methods we need to restrict ourselves to certain classes of hierarchically decomposable groups

diffusivity and radius of random animals percolation clusters and compact clusters defines a cluster diffusivity ds by introducing a random rearrangement between arbitrary cluster and perimeter sites such that the cluster remains connected monte carlo simulation is used on the square and simple cubic lattices to determine the dependence of ds and the radius of gyration rs on s the number of cluster sites three limiting cases are considered random animals p 0 percolation clusters p pc and compact clusters p pc the results for the exponent rho of rs are consistent with the best experimental and theoretical values the authors develop an elementary scaling theory for the observed power law dependence of ds in terms of different mechanisms dominant for small and large clusters a position space renormalisation group calculation in two dimensions yields corrections to the elementary theory for random animals and percolation clusters the predictions are consistent with the monte carlo results for ds

empirical likelihood in missing data problems missing data is a ubiquitous problem in medical and social sciences it is well known that inferences based only on the complete data may not only lose efficiency but may also lead to biased results if the data is not missing completely at random mcar the inverse probability weighting method proposed by horvitz and thompson 1952 is a popular alternative when the data is not mcar the horvitz thompson method however is sensitive to the inverse weights and may suffer from loss of efficiency in this paper we propose a unified empirical likelihood approach to missing data problems and explore the use of empirical likelihood to effectively combine unbiased estimating equations when the number of estimating equations is greater than the number of unknown parameters one important feature of this approach is the separation of the complete data unbiased estimating equations from the incomplete data unbiased estimating equations the proposed method can achieve semiparametric efficiency if the probabilit