superinduction for pattern groups abstract it is well known that the representation theory of the finite group of unipotent upper triangular matrices u n over a finite field is a wild problem by instead considering approximately irreducible representations supercharacters one obtains a rich combinatorial theory analogous to that of the symmetric group where we replace partition combinatorics with set partitions this paper studies diaconis isaacs concept of superinduction in pattern groups while superinduction shares many desirable properties with usual induction it no longer takes characters to characters we begin by finding sufficient conditions guaranteeing that superinduction is in fact induction it turns out for two natural embeddings of u m in u n superinduction is induction we conclude with an explicit combinatorial algorithm for computing this induction analogous to the pieri formulas for the symmetric group

shrinkage priors for bayesian prediction we investigate shrinkage priors for constructing bayesian predictive distributions it is shown that there exist shrinkage predictive distributions asymptotically dominating bayesian predictive distributions based on the jeffreys prior or other vague priors if the model manifold satisfies some differential geometric conditions kullback leibler divergence from the true distribution to a predictive distribution is adopted as a loss function conformal transformations of model manifolds corresponding to vague priors are introduced we show several examples where shrinkage predictive distributions dominate bayesian predictive distributions based on vague priors

maximum entropy and the nearly black object summary maximum entropy me inversion is a non linear inversion technique for inverse problems where the object to be recovered is known to be positive it has been applied in areas ranging from radio astronomy to various forms of spectroscopy sometimes with dramatic success in some cases me has attained an order of magnitude finer resolution and or an order of magnitude smaller noise level than that obtainable by standard linear methods the dramatic successes all seem to occur in cases where the object to be recovered is nearly black essentially zero in the vast majority of samples we show that near blackness is required both for signal to noise enhancements and for superresolution however other methods in particular minimum 1 norm reconstruction may exploit near blackness to an even greater extent

single observation unbiased priors this paper studies a class of default priors which we call single observation unbiased priors soup a prior for a parameter is a soup if the corresponding posterior mean of the parameter based on a single observation is an unbiased estimator of the parameter we prove that under mild regularity conditions a default prior for a convolution parameter is noninformative in the sense of yielding a posterior inference invariant under amalgamation only if it is a soup therefore when amalgamation invariance is desirable as in our motivating example of performing imputation for census undercount soup is the only possible coherent noninformative prior for bayesian predictions that will be utilized under aggregation the use of soup also mutually calibrates bayesian and frequentist inferences for aggregates of convolution parameters across many small areas we describe approaches that identify soups in many common models in particular a constructive duality method that identifies soups in pairs of distribution families we introduce o completeness a necessary and sufficient condition for a prior distribution to be uniquely characterized by the corresponding posterior mean uniqueness of the soup is determined by the o completeness of the dual family o completeness of a natural exponential family is implied by its completeness hence the diaconis ylvisaker characterization of the conjugate prior for natural exponential families via linear posterior expectation is a direct consequence of the completeness of such families for most of the examples we have examined the inverse of the variance function is the soup for the mean parameter of the corresponding family suggesting that hartigan s results on asymptotic unbiasedness can be generalized to some families with discrete parameters we also discuss a possible extension of berger s result on the inadmissibility of unbiased estimators as the nonexistence of soup can be a first step in establishing such inadmissibility

supercharacter formulas for pattern groups c andre and n yan introduced the idea of a supercharacter theory to give a tractable substitute for character theory in wild groups such as the unipotent uppertriangular group u n f q in this theory superclasses are certain unions of conjugacy classes and supercharacters are a set of characters which are constant on superclasses this paper gives a character formula for a supercharacter evaluated at a superclass for pattern groups and more generally for algebra groups

a supercharacter analogue for normality diaconis and isaacs define in 8 diaconis and isaacs 2008 a supercharacter theory for algebra groups over a finite field by constructing certain unions of conjugacy classes called superclasses and certain reducible characters called supercharacters this work investigates the properties of algebra subgroups h g which are unions of some set of the superclasses of g we call such subgroups supernormal after giving a few useful equivalent formulations of this definition we show that products of supernormal subgroups are supernormal and that all normal pattern subgroups are supernormal we then classify the set of supernormal subgroups of un q the group of unipotent upper triangular matrices over the finite field fq and provide a formula for the number of such subgroups when q is prime following this we give supercharacter analogues for clifford s theorem and mackey s method of little groups specifically we show that a supercharacter restricted to a supernormal subgroup decomposes as a sum of supercharacters with the same degree and multiplicity we then describe how the supercharacters of an algebra group of the form un uh ua where ua is supernormal and a2 0 are parametrized by uh orbits of the supercharacters of ua and the supercharacters of the stabilizer subgroups of these orbits

geometry of an accelerated model with censored data from the geometrical point of view a statistical model can be considered as a manifold with parameter plays the role of a coordinate system in this paper the geometrical quantities of a statistical model equipped with k step stress accelerated life test and progressively type ii censoring are obtained where censoring is allowed not only at each stress change time but also at each failure time as an application of these quantities the asymptotic expansions of the bayesian prediction are investigated finally some computation and simulation results are presented to illustrate our main results

the maximum likelihood prior consider an estimate theta of a parameter theta based on repeated observations from a family of densities f theta evaluated by the kullback leibler loss function k theta theta int log f theta f theta f theta the maximum likelihood prior density if it exists is the density for which the corresponding bayes estimate is asymptotically negligibly different from the maximum likelihood estimate the bayes estimate corresponding to the maximum likelihood prior is identical to maximum likelihood for exponential families of densities in predicting the next observation the maximum likelihood prior produces a predictive distribution that is asymptotically at least as close in expected truncated kullback leibler distance to the true density as the density indexed by the maximum likelihood estimate it frequently happens in more than one dimension that maximum likelihood corresponds to no prior density and in that case the maximum likelihood estimate is asymptotically inadmissible and may be improved upon by using the estimate corresponding to a least favorable prior as in brown the asymptotic risk for an arbitrary estimate near maximum likelihood is given by an expression involving derivatives of the estimator and of the information matrix admissibility questions for these near ml estimates are determined by the existence of solutions to certain differential equations

an analog of nilpotence arising from supercharacter theory abstract the goal of this paper is to generalize several group theoretic concepts such as the center and commutator subgroup central series and ultimately nilpotence to a supercharacter theoretic setting and to use these concepts to show that there can be a strong connection between the structure of a group and the structure of its supercharacter theories we then use these concepts to show that the upper and lower annihilator series of j can be described in terms of certain central series for the algebra group g 1 j defined by s when s is the algebra group supercharacter theory defined by diaconis isaacs

simultaneous prediction of independent poisson observables simultaneous predictive distributions for independent poisson observables are investigated a class of improper prior distributions for poisson means is introduced the bayesian predictive distributions based on priors from the introduced class are shown to be admissible under the kullback leibler loss a bayesian predictive distribution based on a prior in this class dominates the bayesian predictive distribution based on the jeffreys prior

secondary bifurcations in the buckling problem abstract the secondary bifurcation points and secondary states of a thin plate buckling problem are numerically studied in this paper the continuation methods and local perturbation techniques are successfully implemented to trace the first few secondary branch solutions our computer graphic output shows that the secondary branch process is an unsymmetric wrinkling

asymptotically minimax bayes predictive densities given a random sample from a distribution with density function that depends on an unknown parameter 0 we are interested in accurately estimating the true parametric density function at a future observation from the same distribution the asymptotic risk of bayes predictive density estimates with kullback leibler loss function d f f f f log f f is used to examine various ways of choosing prior distributions the principal type of choice studied is minimax we seek asymptotically least favorable predictive densities for which the corresponding asymptotic risk is minimax a result resembling stein s paradox for estimating normal means by maximum likelihood holds for the uniform prior in the multivariate location family case when the dimensionality of the model is at least three the jeffreys prior is minimax though inadmissible the jeffreys prior is both admissible and minimax for one and two dimensional location problems

homotopy algorithm for symmetric eigenvalue problems the homotopy method can be used to solve eigenvalue eigenvector problems the purpose of this paper is to report the numerical experience of the homotopy method of computing eigenpairs for real symmetric tridiagonal matrices together with a couple of new theoretical results in practice it is rerely of any interest to compute all the eigenvalues the homotopy method having the order preserving property can provide any specific eigenvalue without calculating any other eigenvalues besides this advantage we note that the homotopy algorithm is to a large degree a parallel algorithm numerical experimentation shows that the homotopy method can be very efficient especially for graded matrices

supercharacters and superclasses for algebra groups we study certain sums of irreducible characters and compatible unions of conjugacy classes in finite algebra groups these groups generalize the unimodular upper triangular groups over a finite field and the supercharacter theory we develop extends results of carlos andre and ning yan that were originally proved in the upper triangular case this theory sometimes allows explicit computations in situations where it would be impractical to work with the full character table we discuss connections with the kirillov orbit method and with gelfand pairs and we give conditions for a supercharacter or a superclass to be an ordinary irreducible character or conjugacy class respectively we also show that products of supercharacters are positive integer combinations of supercharacters

computational methods for bifurcation problems with symmetries with special attention to steady state and hopf bifurcation points we show how group theoretical methods can be employed to utilize the symmetry of a bifurcation problem in numerical computations we extend the approach by werner 1988 by presenting methods for the detection of bifurcation points and the computation of multiple hopf points the essential numerical point is the utilization of certain reduced instead of full systems involving appropriate subgroups of the underlying symmetry group the group theoretical tool is an a priori knowledge of the interaction of certain subgroups 0 and of at possibly multiple steady state or hopf bifurcation points minimal 0 breaking bifurcation we introduce a bifurcation graph which shows graphically this a priori information its edges represent possible symmetry breaking bifurcations our analysis follows the lines of golubitsky stewart and schaeffer 1988 but it is aimed to numerical applications we have chosen a 4 box brusselator model in order to explain our notions and ideas and to discuss the numerical procedure

geometry of exponential family with competing risks and censored data abstract employing the differential geometrical methods in statistics suggested by amari 1985 and amari et al 1987 considering the exponential family with censored data and competing risks as a manifold of a statistical model the geometry of the manifold is investigated based on two information sources as an application of the geometry the asymptotic expansions of the bootstrap prediction bayesian prediction and their risk evaluations are investigated the results show that these expansions are related to the coefficients of connections and metric tensors and the predictive density function is the estimative density function in the asymptotic sense finally taking rayleigh distribution and prostatic cancer data as examples some computation and simulation results are presented to illustrate our main results

universal coding information prediction and estimation a connection between universal codes and the problems of prediction and statistical estimation is established a known lower bound for the mean length of universal codes is sharpened and generalized and optimum universal codes constructed the bound is defined to give the information in strings relative to the considered class of processes the earlier derived minimum description length criterion for estimation of parameters including their number is given a fundamental information theoretic justification by showing that its estimators achieve the information in the strings it is also shown that one cannot do prediction in gaussian autoregressive moving average arma processes below a bound which is determined by the information in the data

admissible predictive density estimation let xj np v xi and y j np v yi be independent p dimensional multivariate normal vectors with common unknown mean based on observing x x we consider the problem of estimating the true predictive density p yj of y under expected kullback leibler loss our focus here is the characterization of admissible procedures for this problem we show that the class of all generalized bayes rules is a complete class and that the easily interpretable conditions of brown and hwang 1982 are sucient for a formal bayes rule to be admissible 1 introduction let xj np v xi and yj np v yi be independent p dimensional multivariate normal vectors with a common unknown mean 2 r p we assume that vx 0 and vy 0 are known we let p xj and p yj denote the conditional densities of x and y suppressing the dependence onvx and vy throughout based on observing only x x we consider the problem of estimating the density p yj of y the natural action spacea0 consists of all proper densities on r p i e

spl alpha parallel prior and its properties it is known that the jeffreys prior plays an important role in statistical inference in this paper we generalize the jeffreys prior from the point of view of information geometry and introduce a one parameter family of prior distributions which we named the spl alpha parallel priors the spl alpha parallel prior is defined as the parallel volume element with respect to the spl alpha connection and coincides with the jeffreys prior when spl alpha 0 further we analyze asymptotic behavior of the various estimators such as the projected bayes estimator the estimator obtained by projecting the bayes predictive density onto the original class of distributions and the minimum description length mdl estimator when the spl alpha parallel prior is used the difference of these estimators from maximum likelihood estimator mle due to the spl alpha prior is shown to be regulated by an invariant vector field of the statistical model although the jeffreys prior always exists the existence of spl alpha parallel prior with spl alpha spl ne 0 is not always guaranteed hence we consider conditions for the existence of the spl alpha parallel prior elucidating the conjugate symmetry in a statistical model

information geometry of generalized bayesian prediction using alpha divergences as loss functions in this paper the methods of information geometry are employed to investigate a generalized bayes rule for prediction taking alpha divergences as the loss functions optimality and asymptotic properties of the generalized bayesian predictive densities are considered we show that the bayesian predictive densities minimize a generalized bayes risk we also find that the asymptotic expansions of the densities are related to the coefficients of the alpha connections of a statistical manifold in addition we discuss the difference between two risk functions of the generalized bayesian predictions based on different priors finally using the non informative priors i e jeffreys and reference priors uniform prior and conjugate prior two examples are presented to illustrate the main results

on asymptotic properties of predictive distributions summary second order asymptotic theory of predictive distributions is investigated it is shown that estimative distributions with asymptotically efficient estimators can be improved to predictive distributions that do not belong to the model the model is assumed to be a multidimensional curved exponential family predictive distributions can be constructed by shifting estimative distributions in a direction orthogonal to the model the average kullback leibler divergence from the true distribution to a predictive distribution is represented as the sum of two components one of them depends on the choice of the estimative distribution and the other depends on the shift orthogonal to the model the optimal shift orthogonal to the model is obtained a quantity that can be regarded as the mixture mean curvature of the model in the space of all probability distributions is introduced the difference in average kullback leibler divergence between an estimative distribution and the optimal predictive distribution obtained by shifting it can be regarded as the mixture mean curvature of the model an asymptotic expression for bayesian predictive distributions is also obtained

exact minimax estimation of the predictive density in sparse gaussian models we consider estimating the predictive density under kullback leibler loss in an l0 sparse gaussian sequence model explicit expressions of the first order minimax risk along with its exact constant asymptotically least favorable priors and optimal predictive density estimates are derived compared to the sparse recovery results involving point estimation of the normal mean new decision theoretic phenomena are seen suboptimal performance of the class of plug in density estimates reflects the predictive nature of the problem and optimal strategies need diversification of the future risk we find that minimax optimal strategies lie outside the gaussian family but can be constructed with threshold predictive density estimates novel minimax techniques involving simultaneous calibration of the sparsity adjustment and the risk diversification mechanisms are used to design optimal predictive density estimates

improved minimax predictive densities under kullback leibler loss let x n p u x i and y n p v y i be independentp dimensional multivariate normal vectors with common unknown mean based on only observing x x we consider the problem of obtaining a predictive density p y x for y that is close to p y as measured by expected kullback leibler loss a natural procedure for this problem is the formal bayes predictive density p u y x under the uniform prior u 1 which is best invariant and minimax we show that any bayes predictive density will be minimax if it is obtained by a prior yielding a marginal that is superharmonic or whose square root is superharmonic this yields wide classes of minimax procedures that dominate p u y x including bayes predictive densities under superharmonic priors fundamental similarities and differences with the parallel theory of estimating a multivariate normal mean under quadratic loss are described

a bound on the financial value of information it is shown that each bit of information at most doubles the resulting wealth in the general stock market setup this information bound on the growth of wealth is actually attained for certain probability distributions on the market investigated by j kelly 1956 the bound is shown to be a special case of the result that the increase in exponential growth of wealth achieved with true knowledge of the stock market distribution f over that achieved with incorrect knowledge g is bounded above by the entropy of f relative to g

generic movement of eigenvalues for equivariant self adjoint matrices abstract in the numerical treatment of bifurcation problems one of the main tasks is to control the spectrum of matrices in a parametrized family if the original problem possesses symmetry then the matrices are additionally equivariant previously the generic eigenvalue behavior in a one parameter family of equivariant matrices has been studied for the case of general matrices golubitsky 1988 and for infinitesimally symplectic matrices dellnitz 1992 however in applications the situation frequently occurs that the matrices of the family are self adjoint we classify the generic eigenvalue behavior in such a family of equivariant matrices by the type of underlying symmetry

a shrinkage predictive distribution for multivariate normal observables summary we investigate shrinkage methods for constructing predictive distributions we consider the multivariate normal model with a known covariance matrix and show that there exists a shrinkage predictive distribution dominating the bayesian predictive distribution based on the vague prior when the dimension is not less than three kullback leibler divergence from the true distribution to a predictive distribution is adopted as a loss function somiie key w ords invariance james stein estimator kullback leibler divergence stein s prior vague prior

iterative character constructions for algebra groups abstract we construct a family of orthogonal characters of an algebra group which decompose the supercharacters defined by diaconis and isaacs 2008 6 like supercharacters these characters are given by nonnegative integer linear combinations of kirillov functions and are induced from linear supercharacters of certain algebra subgroups we derive a formula for these characters and give a condition for their irreducibility generalizing a theorem of otto 2010 20 we also show that each such character has the same number of kirillov functions and irreducible characters as constituents in proving these results we observe as an application how a recent computation by evseev 2010 7 implies that every irreducible character of the unitriangular group ut n q of unipotent n n upper triangular matrices over a finite field with q elements is a kirillov function if and only if n 12 as a further application we discuss some more general conditions showing that kirillov functions are characters and describe some results related to counting the irreducible constituents of supercharacters

nonsubjective priors via predictive relative entropy regret we explore the construction of nonsubjective prior distributions in bayesian statistics via a posterior predictive relative entropy regret criterion we carry out a minimax analysis based on a derived asymptotic predictive loss function and show that this approach to prior construction has a number of attractive features the approach here differs from previous work that uses either prior or posterior relative entropy regret in that we consider predictive performance in relation to alternative nondegenerate prior distributions the theory is illustrated with an analysis of some specific examples

harmonic bayesian prediction under alpha divergence we investigate bayesian shrinkage methods for constructing predictive distributions we consider the multivariate normal model with a known covariance matrix and show that the bayesian predictive density with respect to stein s harmonic prior dominates the best invariant bayesian predictive density when the dimension is greater than or equal to 3 alpha divergence from the true distribution to a predictive distribution is adopted as a loss function

information theory and superefficiency the asymptotic risk of efficient estimators with kullback leibler loss in smoothly parametrized statistical models is k 2n where k is the parameter dimension and n is the sample size under fairly general conditions we given a simple information theoretic proof that the set of parameter values where any arbitrary estimator is superefficient is negligible the proof is based on a result of rissanen that codes have asymptotic redundancy not smaller than k 2 log n except in a set of measure 0