on lie gradings ii abstract to obtain fine gradings of gl n c it is necessary to study maximal abelian subgroups mad subgroups of diagonizable automorphisms in a ut gl n c we described two finite sets h n and k n such that each mad subgroup consisting only from inner automorphisms is conjugated to some element of h n and each mad subgroup containing at least one outer automorphism is conjugated to some element of k n using these results concerning k n we then easily find all mad subgroups in a ut o n c for n 8 and in a ut sp 2n c

saddlepoint approximations for estimating equations summary the saddlepoint method is used to approximate to the distribution of an estimator defined by an estimating equation two different approaches are available one of which is shown to be equivalent to the technique of field hampel 1982 two recent formulae for the tail probability due to lugannani rice 1980 and to robinson 1982 respectively which are uniformly accurate over the whole range of the estimator are compared numerically with the exact results and those computed by field hampel they are found to be of comparable accuracy while avoiding the use of numerical integration the most accurate is that of lugannani rice hampel 1974 introduced a new technique for approximating to the probability density of an estimator defined by an estimating equation it is an example of what he called small sample asymptotics where high accuracy is achieved for quite small sample sizes n even down to single figures in the original version it gave an approximation to the logarithmic derivative of the density function which was integrated numerically to get the density the distribution function was obtained by a second numerical integration which could then be used to renormalize both the density and the distribution function field hampel 1982 develop the technique in detail and compare its performance with that of other approximation methods as hampel had pointed out his approach is closely related to the saddlepoint method of daniels 1954 which was applied to sample means and ratios of means following the private communication referred to by field hampel 1982 p 31 it was realized that the first numerical integration was unnecessary and that hampel s approach could be shortened to give a direct approximation to the density which is in fact a saddlepoint approximation the purpose of the present paper is to extend the use of the saddlepoint method to estimating equations there are two distinct ways of doing this which lead to different approximations of similar accuracy one appears to be more convenient for approxi mating to tail probabilities by numerical integration and was used to compute the saddlepoint approximations quoted in field hampel s tables 1 and 2 the other gives the density directly in a form equivalent to their equation 4 3 which is then integrated

generalized tricircular projections on minimal norm ideals in b h abstract a nonzero projection p on a complex banach space x is said to be a generalized tricircular projection if there exist distinct modulus one complex numbers and not equal to 1 and nonzero projections q and r on x such that p q r i and p q r is an isometry we determine the structure of generalized tricircular projections on minimal norm ideals in b h different from the hilbert schmidt class

the weyl group of the fine grading of sl n c associated with tensor product of generalized pauli matrices we consider the fine grading of sl n c induced by tensor product of generalized pauli matrices in the paper regard pgl n c as the inner automorphism group of sl n c based on the classification of maximal abelian subgroups of pgl n c consisting of diagonalizable automorphisms by havlicek et al we prove that any such subgroup k of pgl n c is a symplectic abelian group and its weyl group which describes the symmetry of the fine grading induced by the action of k is just the isometry group of the symplectic abelian group k for a finite symplectic abelian group it is also proved that its isometry group is always generated by the transvections contained in it

a lie grading which is not a semigroup grading this note is devoted to the construction of a graded lie algebra whose grading is not given by a semigroup thus providing a counterexample to an assertion by patera and zassenhaus

gradings on semisimple algebras abstract the classification of gradings by abelian groups on finite direct sums of simple finite dimensional nonassociative algebras over an algebraically closed field is reduced by means of the use of loop algebras to the corresponding problem for simple algebras this requires a good definition of free products of group gradings

lie algebras with a set grading abstract consider g a lie algebra graded by an arbitrary set i set grading we show that g decomposes as the sum of well described graded ideals plus maybe a certain linear subspace under mild conditions the simplicity of g is characterized and it is shown that the above decomposition is actually the direct sum of the family of its minimal graded ideals each one being a simple set graded lie algebra

fine gradings on simple classical lie algebras abstract the fine abelian group gradings on the simple classical lie algebras including d 4 over algebraically closed fields of characteristic 0 are determined up to equivalence this is achieved by assigning certain invariant to such gradings that involve central graded division algebras and suitable sesquilinear forms on free modules over them

non semigroup gradings of associative algebras abstract it is known that there are lie algebras with non semigroup gradings i e such that the binary operation on the grading set is not associative we provide a similar example in the class of associative algebras

improvements of jackknife confidence limit methods summary edgeworth expansion is applied to studentized parameter estimates when the standard error has been computed by a jackknife method adjustments to the usual jackknife confidence limit formulae are obtained this approach is contrasted with a bootstrap approach in numerical illustrations for estimation of a ratio jackknife methods are nonparametric methods for estimating the bias and standard error of an estimate t approximate confidence limits for the estimand can be found by using a large sample normal approximation for t somewhat curiously little is known about possible improvements to the normal approximation in this context although improvements based on edgeworth expansions are familiar in other problems in this paper we show that edgeworth expansion methods can be applied in the jackknife context numerical results for a particular application raise the possibility that the resulting improvements of jackknife methods can be matched by a suitable use of bootstrap methods efron 1982

solution of the contractive projection problem in this paper we show that the class of j algebras a class of concrete jordan triple systems is stable under the action of norm one projections this result constitutes a general solution to a problem considered by the authors and others specifically our main result is theorem 2 let p be an arbitrary contractive projection defined on a j algebra m then p m is a jordan triple system in the triple product a b c a b c p f ab c t cb a for a b c e p m and p m has a faithful representation as a j algebra

on δ derivations of lie algebras and superalgebras we study derivations a construction simultaneously generalizing derivations and centroid first we compute derivations of current lie algebras and of modular zassenhaus algebra this enables us to provide examples of lie algebras having 12 derivations which are divisors of zero thus answering negatively a question of filippov second we note that derivations allow in some circumstances to construct examples of non semigroup gradings of lie algebras in addition to the recent ones discovered by elduque third we note that utilizing the construction of the grassmann envelope allows to obtain results about super derivations of lie superalgebras from the corresponding results about lie algebras in this way we prove that prime lie superalgebras do not possess nontrivial super derivations generalizing the recent result of kaygorodov

conditional bootstrap methods in the mean shift model summary bootstrap methods are not inherently conditional but they can be made so by appropri ate stratification of the simulated samples which bootstrap produces we show how stratification can work in a bootstrap analysis of mean shift in nile river flow data the results are compared with both parametric and semiparametric likelihood analyses the paper ends with some general remarks on conditional bootstraps bootstrap methods efron 1982 are simulation methods for assessing frequency properties of statistical analyses the simulation model being that which is fitted to the actual data often the simulation involves monte carlo rather than theoretical evaluation unfortunately the bootstrap as usually described is inherently unconditional rubin s 1981 discussion bears on this point however if the bootstrap simulation results can be properly stratified then it is possible to estimate relevant conditional probabilities stratification can sometimes be effected in the model itself as when fixing the values of sample sizes or explanatory variables otherwise one must run the unconditional bootstrap and stratify its output in the present paper we illustrate the latter situation in the analysis of a mean shift model section 2 describes the application and its analysis by parametric methods sections 3 and 4 produce alternative bootstrap analyses the first of which uses the stratification of an unconditional monte carlo simulation the second bootstrap analysis involves estimation of a likelihood function the final section contains some brief general remarks about conditional bootstraps

more non semigroup lie gradings this note is devoted to the construction of two very easy examples of respective dimensions 4 and 6 of graded lie algebras whose grading is not given by a semigroup the latter one being a semisimple algebra it is shown that 4 is the minimal possible dimension for such graded lie algebras

saddlepoint approximations in resampling methods summary saddlepoint approximations are shown to be easy to use and accurate in a variety of simple bootstrap and randomization applications examples include mean estimation ratio estimation two sample comparisons and autoregressive estimation

gradings on symmetric composition algebras abstract the group gradings on the symmetric composition algebras over arbitrary fields are classified applications of this result to gradings on the exceptional simple lie algebras are considered too

projections as convex combinations of surjective isometries on c ω we study properties of operators that are in the convex hull of a finite set of surjective isometries on the banach space of complex valued continuous maps defined on a compact and connected topological space we characterize those projections that are in the convex combination of two surjective isometries and we show that they are generalized bi circular projections

generalized bi circular projections on certain hardy spaces abstract we characterize generalized bi circular projections on the hardy space of the torus and show that they can be represented as the average of the identity and an isometric reflection we also show that the average of two distinct isometries on the hardy space of the disk is a projection if and only if it is a generalized bi circular projection

fast and accurate approximate double bootstrap confidence intervals we propose two new methods for the construction of approximate iterated bootstrap confidence intervals our methods follow a saddlepoint approach but employ approximate rather than exact solutions to the nonlinear saddlepoint equations that arise in applying that technique the main advantages of the new techniques are that they yield confidence intervals with high coverage accuracy in small to moderately sized samples and that they achieve this accuracy with very substantial reductions in computation time compared to other methods

prepivoting to reduce level error of confidence sets summary approximate confidence sets for a parameter 0 may be obtained by referring a function of 0 and of the sample to an estimated quantile of that function s sampling distribution we call this function the root of the confidence set either asymptotic theory or bootstrap methods can be used to estimate the desired quantile when the root is not a pivot in the sense of classical statistics the actual level of the approximate confidence set may differ substantially from the intended level prepivoting is the transformation of a confidence set root by its estimated bootstrap cumulative distribution function prepivoting can be iterated bootstrap confidence sets generated from a root prepivoted one or more times have smaller error in level than do confidence sets based on the original root the first prepivoting is nearly equivalent to studentizing when that operation is appropriate further iterations of prepivoting make higher order corrections automatically

bootstrapping survival times in stochastic systems by using saddlepoint approximations summary the single bootstrap is implemented by using a saddlepoint approximation to determine estimates for the survival and hazard functions of first passage times in complicated semi markov processes the double bootstrap is also implemented by resampling saddlepoint inversions and provides bca confidence bands for these functions confidence intervals for the mean and variance of first passage times are easily computed a new characterization of the asymptotic hazard rate for survival times is presented and leads to an indirect method for constructing its bootstrap confidence interval

bootstrap confidence bands for sojourn distributions in multistate semi markov models with right censoring transient semi markov processes have traditionally been used to describe the transitions of a patient through the various states of a multistate survival model a survival distribution in this context is a sojourn through the states until passage to a fatal absorbing state or certain endpoint states using complete sojourn data this paper shows how such survival distributions and associated hazard functions can be estimated nonparametrically and also how nonparametric bootstrap pointwise confidence bands can be constructed for them when patients are subject to independent right censoring from each state during the sojourn limitations to the estimability of such survival distributions that result from random censoring with bounded support are clarified the methods are applicable to any sort of sojourn through any finite state process of arbitrary complexity involving feedback into previously occupied states copyright 2012 oxford university press

g invariant norms and bicircular projections abstract it is shown that for many finite dimensional normed vector spaces v over c a linear projection p v v will have nice structure if p i p is an isometry for some complex unit not equal to one from these results one can readily determine the structure of bicircular projections i e those linear projections p such that p i p is an isometry for every complex unit the key ingredient in the proofs is the knowledge of the isometry group of the given norm the proof techniques also apply to real vector spaces in such cases characterizations are given to linear projections p such that p i p 2p i is an isometry

small sample asymptotic distributions of m estimators of location summary asymptotic formulae for the distribution of m estimators i e maximum likelihood type estimators of location including the arithmetic mean are derived which numerical studies show to give relative errors for densities and tail areas of the order of magnitude of 1 down to sample sizes 3 and 4 even in the extreme tails the paper is the continuation of earlier work by the second author and is also closely related to daniels s work on the saddlepoint approximation the method consists in expanding the derivative of the logarithm of the unstandardized density of the estimator in powers of 1 n at each point using recentring by means of conjugate distributions this method yields a unified point of view for the comparison of other asymptotic methods namely saddlepoint method edgeworth expansion and large deviations approach which are also compared numerically

generalized bi circular projections abstract recall that a projection p on a complex banach space x is a generalized bi circular projection if p i p is a surjective isometry for some such that 1 and 1 it is easy to see that every hermitian projection is generalized bi circular a generalized bi circular projection is said to be nontrivial if it is not hermitian botelho and jamison showed that a projection p on c 0 1 is a nontrivial generalized bi circular projection if and only if p i p is a surjective isometry in this article we prove that if p is a projection such that p i p is a surjective isometry for some then either p is hermitian or is an n th unit root of unity we also show that for any n th unit root of unity there are a complex banach space x and a nontrivial generalized bi circular projection p on x such that p i p is an isometry

an empirical study of the ratio estimator and estimators of its variance abstract this paper reports results from an empirical study of the ratio estimator for a finite population total from each of six real populations 1 000 simple random samples 1 000 restricted random samples and three nonrandom samples of size 32 are drawn performance of the ratio estimator and of five estimators of its variance is compared with theoretical results generated using a prediction superpopulation models and b probability sampling distributions the results presented graphically show that theory based on prediction models can reveal relationships that are essential in making inferences but that are concealed in probability sampling analyses