on the eccentric distance sum of trees and unicyclic graphs abstract let g be a simple connected graph with the vertex set v g the eccentric distance sum of g is defined as d g v v g e v d g v where e v is the eccentricity of the vertex v and d g v u v g d u v is the sum of all distances from the vertex v in this paper we characterize the extremal unicyclic graphs among n vertex unicyclic graphs with given girth having the minimal and second minimal eccentric distance sum in addition we characterize the extremal trees with given diameter and minimal eccentric distance sum

local shrinkage rules levy processes and regularized regression we use l evy processes to generate joint prior distributions and therefore penalty functions for a location parameter b b1 bp as p grows large this generalizes the class of local global shrinkage rules based on scale mixtures of normals illuminates new connections among disparate methods and leads to new results for computing posterior means and modes under a wide class of priors we extend this framework to large scale regularized regression problems where p n and provide comparisons with other methodologies

weight functions and fundamental fields for the penny shaped and the half plane crack in three space abstract weight functions permit to represent stress intensity factors as weighted averages of the externally impressed boundary tractions and body forces in this paper the relevant weight functions of all three modes are represented in closed form with the aid of elementary transcendentals their derivation as displacements of fundamental fields is simple and without recourse to integral transforms of any kind

prediction via orthogonalized model mixing abstract we introduce an approach and algorithms for model mixing in large prediction problems with correlated predictors we focus on the choice of predictors in linear models and mix over possible subsets of candidate predictors our approach is based on expressing the space of models in terms of an orthogonalization of the design matrix advantages are both statistical and computational statistically orthogonalization often leads to a reduction in the number of competing models by eliminating correlations computationally large model spaces cannot be enumerated recent approaches are based on sampling models with high posterior probability via markov chains based on orthogonalization of the space of candidate predictors we can approximate the posterior probabilities of models by products of predictor specific terms this leads to an importance sampling function for sampling directly from the joint distribution over the model space without resorting to markov chains compared to the latter ortho

an upper prediction limit for the arithmetic mean of a lognormal random variable in this article we construct an upper prediction limit for the arithmetic mean of a future sample from a lognormal distribution first we present three approximate distributions for the total of the future sample and compare them to the exact distribution then we use the exact and approximate distributions to construct the desired prediction limit we study confidence levels for the different methods via simulation we illustrate our results with an example and show via simulation that the desired confidence level is attained for only one of these methods we also show that the usual approach to this problem based on logarithmic transformation leads to a prediction limit for the median which can dramatically underestimate the prediction limit for the mean

a split and merge bayesian variable selection approach for ultrahigh dimensional regression summary we propose a bayesian variable selection approach for ultrahigh dimensional linear regression based on the strategy of split and merge the approach proposed consists of two stages split the ultrahigh dimensional data set into a number of lower dimensional subsets and select relevant variables from each of the subsets and aggregate the variables selected from each subset and then select relevant variables from the aggregated data set since the approach proposed has an embarrassingly parallel structure it can be easily implemented in a parallel architecture and applied to big data problems with millions or more of explanatory variables under mild conditions we show that the approach proposed is consistent i e the true explanatory variables can be correctly identified by the approach as the sample size becomes large extensive comparisons of the approach proposed have been made with penalized likelihood approaches such as the lasso elastic net sure independence screening and iterative sure independence screening the numerical results show that the approach proposed generally outperforms penalized likelihood approaches the models selected by the approach tend to be more sparse and closer to the true model

the theory of higher order weight functions for linear elastic plane problems abstract we generalize bueckner s fundamental field concept and develop higher order weight functions for calculating power expansion coefficients of a regular elastic field in a two dimensional body in the absence of body forces problems of the first and third kind are investigated integral formulas for the expansion coefficients are given for interior points and crack tips in these formulas the integration is performed over the boundary of the body crack faces included the prescribed boundary data tractions and or displacements of the regular field appear in the integrand in weighted form the weights are derived from fundamental fields of universal character the significance of these expansion coefficients in fracture analysis is also discussed

decoupling shrinkage and selection in bayesian linear models a posterior summary perspective selecting a subset of variables for linear models remains an active area of research this article reviews many of the recent contributions to the bayesian model selection and shrinkage prior literature a posterior variable selection summary is proposed which distills a full posterior distribution over regression coefficients into a sequence of sparse linear predictors

the bayesian lasso the lasso estimate for linear regression parameters can be interpreted as a bayesian posterior mode estimate when the regression parameters have independent laplace i e double exponential priors gibbs sampling from this posterior is possible using an expanded hierarchy with conjugate normal priors for the regression parameters and independent exponential priors on their variances a connection with the inverse gaussian distribution provides tractable full conditional distributions the bayesian lasso provides interval estimates bayesian credible intervals that can guide variable selection moreover the structure of the hierarchical model provides both bayesian and likelihood methods for selecting the lasso parameter slight modifications lead to bayesian versions of other lasso related estimation methods including bridge regression and a robust variant

a representation of the posterior mean for a location model summary an exact representation of the posterior mean is developed for a location model and the class of priors that are normal scale mixtures the result makes use of the conditional distribution of the maximum likelihood estimator and masreliez s theorem directions for future development are indicated an exact representation for the posterior mean e 01y is given where y is a 1 x n vector of observations from a location model f x 0 and 0 has a prior density p 0 that is a normal scale mixture let l 0 denote the likelihood function and let y 0 a where 0 is the maximum likelihood estimator and a is the maximal ancillary the representation makes use of two results the conditional distribution of the maximum likelihood estimator p 00 a barndorff nielsen 1983 and a result of masreliez 1975 it is shown that under a normal prior e ojy can be represented as a linear transformation of the score function of p 01 a where p 0la p 10 a p 0 do the representation can be viewed as a generalization of masreliez s result that deals with the model x 0 0 n m r2 and represents the posterior mean in terms of the derivative of the logarithm of the marginal density for x the appendix gives a sufficient regularity condition and a rigorous proof of an interchange of derivative and integral sign required for masreliez s theorem a representation is also developed in the case where the prior density is a normal scale mixture the results have a quantitative robustness appeal for example under any model and a normal prior the sensitivity of e oy with respect to aberrant observations is

estimating the degree of activity of jumps in high frequency data we define a generalized index of jump activity propose estimators of that index for a discretely sampled process and derive the estimators properties these estimators are applicable despite the presence of brownian volatility in the process which makes it more challenging to infer the characteristics of the small infinite activity jumps when the method is applied to high frequency stock returns we find evidence of infinitely active jumps in the data and estimate their index of activity

asymptotics of three dimensional macrocrack microcrack interaction abstract the perturbation in the mode i stress intensity factor along the edge of a half plane crack due to the opening of one or more co planar penny shaped cracks close to the edge calculated recently by the authors int j solids structures 30 2117 2139 1993 using the finite element method is shown to have the same logarithmic asymptotic behaviour as that of its plane strain two dimensional counterpart the approximate method of solution used in that paper however does not predict this asymptotic behaviour

three dimensional interactions of a half plane crack with point forces dipoles and moments abstract closed form solutions in terms of elementary functions are derived for stress intensity factors along the edge of a half plane crack induced by point forces dipoles centers of expansion and moments locations and orientations of these stress sources are arbitrary the solid is assumed to be either transversely isotropic with the crack parallel to the plane of isotropy or isotropic

bayesian inference for logistic models using polya gamma latent variables we propose a new data augmentation strategy for fully bayesian inference in models with binomial likelihoods the approach appeals to a new class of polya gamma distributions which are constructed in detail a variety of examples are presented to show the versatility of the method including logistic regression negative binomial regression nonlinear mixed effect models and spatial models for count data in each case our data augmentation strategy leads to simple effective methods for posterior inference that 1 circumvent the need for analytic approximations numerical integration or metropolis hastings and 2 outperform other known data augmentation strategies both in ease of use and in computational efficiency all methods including an efficient sampler for the polya gamma distribution are implemented in the r package bayeslogit supplementary materials for this article are available online

three dimensional elastic crack tip interactions with shear transformation strains abstract three dimensional elastic interactions between a half plane crack and sources of internal stress such as unconstrained shear transformation strains are analysed using bueckner s recent weight function method of three dimensional crack analysis analytical expressions arc given for the stress intensity factors induced along the crack front by unconstrained shear transformation strains for several simple shapes of the transformation domain centred at the origin when the centre of the transformation domain is not coincident with the origin of coordinates analytical expressions are too cumbersome to be of much practical value in such instances numerical results are presented for transformation domains in the shape of a sphere and an oblate spheroid the influence of the orientation of the latter is also studied

a unified finite element method for determining weight functions in two and three dimensions abstract a variational principle is introduced for determining singular fields in finite bodies with unbounded strain energy a specialized form of this variational principle is employed to determine the bueckner rice weight functions in two and three dimensions the cracked body considered is finite linear elastic and homogeneous but arbitrary anisotropy is permitted it contains an arbitrary configuration of planar cracks with continuously turning crack fronts a finite element implementation of the variational principle is given and this leads to a unified approach in the numerical computations of weight functions for all three fracture modes the finite element method presented is simple and yet accurate and it can be incorporated into standard finite element programs numerical examples are given for two dimensional crack geometries in isotropic bodies to illustrate the versatility and accuracy of the method

ep gig priors and applications in bayesian sparse learning in this paper we propose a novel framework for the construction of sparsity inducing priors in particular we define such priors as a mixture of exponential power distributions with a generalized inverse gaussian density ep gig ep gig is a variant of generalized hyperbolic distributions and the special cases include gaussian scale mixtures and laplace scale mixtures furthermore laplace scale mixtures can subserve a bayesian framework for sparse learning with nonconvex penalization the densities of ep gig can be explicitly expressed moreover the corresponding posterior distribution also follows a generalized inverse gaussian distribution we exploit these properties to develop em algorithms for sparse empirical bayesian learning we also show that these algorithms bear an interesting resemblance to iteratively reweighted l2 or l1 methods finally we present two extensions for grouped variable selection and logistic regression

bayes and empirical bayes multiplicity adjustment in the variable selection problem this paper studies the multiplicity correction effect of standard bayesian variable selection priors in linear regression our first goal is to clarify when and how multiplicity correction happens automatically in bayesian analysis and to distinguish this correction from the bayesian ockham s razor effect our second goal is to contrast empirical bayes and fully bayesian approaches to variable selection through examples theoretical results and simulations considerable differences between the two approaches are found in particular we prove a theorem that characterizes a surprising aymptotic discrepancy between fully bayes and empirical bayes this discrepancy arises from a different source than the failure to account for hyperparameter uncertainty in the empirical bayes estimate indeed even at the extreme when the empirical bayes estimate converges asymptotically to the true variable inclusion probability the potential for a serious difference remains

a numerical approach for determining weight functions in fracture mechanics the paper deals with the development of a numerical method for determining weight functions in two dimensional problems after a short review of some recent numerical techniques an original approach is presented the method is based on finite element calculations with coarse meshes and on the knowledge of some values of the stress intensity factor for one reference loading condition the validity of the method is demonstrated for a theoretical case and its accuracy and suitability are discussed with reference to practical applications

a statistical view of some chemometrics regression tools chemometrics is a field of chemistry that studies the application of statistical methods to chemical data analysis in addition to borrowing many techniques from the statistics and engineering literatures chemometrics itself has given rise to several new data analytical methods this article examines two methods commonly used in chemometrics for predictive modeling partial least squares and principal components regression from a statistical perspective the goal is to try to understand their apparent successes and in what situations they can be expected to work well and to compare them with other statistical methods intended for those situations these methods include ordinary least squares variable subset selection and ridge regression

sparse partial least squares regression for simultaneous dimension reduction and variable selection partial least squares regression has been an alternative to ordinary least squares for handling multicollinearity in several areas of scientific research since the 1960s it has recently gained much attention in the analysis of high dimensional genomic data we show that known asymptotic consistency of the partial least squares estimator for a univariate response does not hold with the very large p and small n paradigm we derive a similar result for a multivariate response regression with partial least squares we then propose a sparse partial least squares formulation which aims simultaneously to achieve good predictive performance and variable selection by producing sparse linear combinations of the original predictors we provide an efficient implementation of sparse partial least squares regression and compare it with well known variable selection and dimension reduction approaches via simulation experiments we illustrate the practical utility of sparse partial least squares regression in a joint analysis of gene expression and genomewide binding data

fracture mechanics weight functions in three dimensions subtraction of fundamental fields a new procedure is presented for the determination of the fracture mechanics weight functions that are required for the evaluation of stress intensity factors in cracked solids the procedure can be used with a standard three dimensional boundary element code the weight functions are proportional to the displacements on the boundary of the solid when the only loading is a pair of self equilibrated point forces at the crack front in previous work the highly singular crack tip fields that this loading produces have been modelled by replacing the crack front by a cylindrical cavity with appropriate displacement boundary conditions on the cavity walls it is shown here that results are dependent on the cavity radius and that convergence of the results cannot be guaranteed an alternative procedure based on the substraction of fundamental fields sff is demonstrated herein the high order singularities are removed from the field before the reduced problem is solved numerically using a standard boundary element method since the reduced problem is equivalent to an unloaded crack in a solid subjected to boundary tractions the usual quarter point displacement elements and quarter point traction singular elements can be used to improve the accuracy weight functions so obtained are used to evaluate stress intensity factors as a function of position on the crack front for a straight fronted crack in a rectangular bar subjected to various loadings both edge and central cracks are considered and the validity of the technique is demonstrated by comparing the results with previously published values

connective steiner 3 eccentricity index and network similarity measure abstract for a set s v g in a network g the steiner distance dg s of s is the minimum size among all connected subnetworks whose vertex sets contain s the steiner k eccentricity k v of a vertex v of g is the maximum steiner distance among all k vertex set s which contains the vertex v i e e k v max d s s v g s k v s based on steiner k eccentricity the connective steiner k eccentricity index is introduced as a newly structural invariant some properties of the connective steiner 3 eccentricity index are investigated firstly we present an o n2 polynomial time algorithm to calculate the connective steiner 3 eccentricity index of trees secondly some optimal problems among some network classes are discussed as its application finally we consider the network similarity measure based on the connective steiner 3 eccentricity index by two different methods we study its advantages numerical results show that the measure based on the connective steiner 3 eccentricity index has more advantages than the ones based on other topological indices graph energy randic index the largest adjacent eigenvalue the largest laplacian eigenvalue

on the eccentric distance sum of graphs the eccentric distance sum is a novel topological index that offers a vast potential for structure activity property relationships for a graph g it is defined as d g v ve v d v where e v is the eccentricity of the vertex v and d v u v g d u v is the sum of all distances from the vertex v motivated by g yu l feng a ilic on the eccentric distance sum of trees and unicyclic graphs j math anal appl 375 2011 934 944 in this paper we characterize the extremal trees and graphs with maximal eccentric distance sum various lower and upper bounds for the eccentric distance sum in terms of other graph invariants including the wiener index the degree distance eccentric connectivity index independence number connectivity matching number chromatic number and clique number are established in addition we present explicit formulae for the values of eccentric distance sum for the cartesian product applied to some graphs of chemical interest like nanotubes and nanotori

default bayesian analysis with global local shrinkage priors we provide a framework for assessing the default nature of a prior distribution using the property of regular variation which we study for global local shrinkage priors in particular we show that the horseshoe priors originally designed to handle sparsity are regularly varying and thus are appropriate for default bayesian analysis to illustrate our methodology we discuss four problems of noninformative priors that have been shown to be highly informative for nonlinear functions in each case we show that global local horseshoe priors perform as required global local shrinkage priors can separate a low dimensional signal from high dimensional noise even for nonlinear functions

a note on posterior moments for a normal mean with double exponential prior summary general results on the behaviour of the posterior mean and variance of the mean of a normal distribution with known variance using a double exponential prior are derived and complement the conjectures and particular examples given by pericchi and smith 1992 and pericchi and walley 1991

calibration and empirical bayes variable selection for the problem of variable selection for the normal linear model selection criteria such as aic c p bic and ric have fixed dimensionality penalties such criteria are shown to correspond to selection of maximum posterior models under implicit hyperparameter choices for a particular hierarchical bayes formulation based on this calibration we propose empirical bayes selection criteria that use hyperparameter estimates instead of fixed choices for obtaining these estimates both marginal and conditional maximum likelihood methods are considered as opposed to traditional fixed penalty criteria these empirical bayes criteria have dimensionality penalties that depend on the data their performance is seen to approximate adaptively the performance of the best fixed penalty criterion across a variety of orthogonal and nonorthogonal set ups including wavelet regression empirical bayes shrinkage estimators of the selected coefficients are also proposed

one step sparse estimates in nonconcave penalized likelihood models fan and li propose a family of variable selection methods via penalized likelihood using concave penalty functions the nonconcave penalized likelihood estimators enjoy the oracle properties but maximizing the penalized likelihood function is computationally challenging because the objective function is nondifferentiable and nonconcave in this article we propose a new unified algorithm based on the local linear approximation lla for maximizing the penalized likelihood for a broad class of concave penalty functions convergence and other theoretical properties of the lla algorithm are established a distinguished feature of the lla algorithm is that at each lla step the lla estimator can naturally adopt a sparse representation thus we suggest using the one step lla estimator from the lla algorithm as the final estimates statistically we show that if the regularization parameter is appropriately chosen the one step lla estimates enjoy the oracle properties with good initial estimators computationally the one step lla estimation methods dramatically reduce the computational cost in maximizing the nonconcave penalized likelihood we conduct some monte carlo simulation to assess the finite sample performance of the one step sparse estimation methods the results are very encouraging

eccentric distance sum a novel graph invariant for predicting biological and physical properties abstract eccentric distance sum a novel graph invariant with vast potential in structure activity property relationships has been conceptualized in the present study this graph invariant displayed high discriminating power with respect to both biological activity and physical properties the structure activity relationship of eccentric distance sum was investigated with regard to anti hiv activity of dihydroseselins the values of eccentric distance sum of each analogue in the data set were computed and active range identified subsequently biological activity was assigned to each analogue in the data set which was then compared with the reported anti hiv activity of dihydroseselin analogues surprisingly the accuracy of prediction was found to be more than 88 with regard to anti hiv activity on the other hand investigations pertaining to quantitative structure property relationship of the novel graph invariant with regard to various physical properties of diverse nature for data sets consisting of primary amines secondary amines and alcohols revealed correlation percentages ranging from 93 to 99 the over all results with regard to structure activity and quantitative structure property studies using eccentric distance sum were better than the corresponding values obtained using wiener s index

mixtures of g priors for bayesian variable selection zellner s g prior remains a popular conventional prior for use in bayesian variable selection despite several undesirable consistency issues in this article we study mixtures of g priors as an alternative to default g priors that resolve many of the problems with the original formulation while maintaining the computational tractability that has made the g prior so popular we present theoretical properties of the mixture g priors and provide real and simulated examples to compare the mixture formulation with fixed g priors empirical bayes approaches and other default procedures please see arnold zellner s letter and the author s response

bayesian lasso regression summary the lasso estimate for linear regression corresponds to a posterior mode when independent double exponential prior distributions are placed on the regression coefficients this paper intro duces new aspects of the broader bayesian treatment of lasso regression a direct characterization of the regression coefficients posterior distribution is provided and computation and inference under this characterization is shown to be straightforward emphasis is placed on point estima tion using the posterior mean which facilitates prediction of future observations via the posterior predictive distribution it is shown that the standard lasso prediction method does not neces sarily agree with model based bayesian predictions a new gibbs sampler for bayesian lasso regression is introduced

graph distance measures based on topological indices revisited graph distance measures based on topological indices have been already explored by dehmer et al also inequalities for those graph distance measures have been proved in this paper we continue studying such comparative graph measures based on the well known wiener index graph energy and randic index respectively we prove extremal properties of the graph distance measures for some special classes of graphs to demonstrate useful properties of the measures we also discuss numerical results to conclude the paper we state some open problems

exact and approximate posterior moments for a normal location parameter the forms of first and second posterior moments for a normal location parameter are identified for a rather general class of prior distributions exact and approximate illustrations are given where the prior distribution is double exponential or student t respectively

bayesian regression using a prior on the model fit the r2 d2 shrinkage prior prior distributions for high dimensional linear regression require specifying a joint distribution for the unobserved regression coefficients which is inherently difficult we instead propose a ne

some extremal results on the connective eccentricity index of graphs abstract the connective eccentricity index cei of a graph g is defined as ce g v i v g d v i e v i where e v i and d v i are the eccentricity and the degree of vertex v i respectively in g in this paper we obtain some lower and upper bounds on the connective eccentricity index for all trees of order n and with matching number and characterize the corresponding extremal trees and the maximal graphs of order n and with matching number and n edges have been determined which maximize the connective eccentricity index also the extremal graphs with maximal connective eccentricity index are completely characterized among all connected graphs of order n and with matching number moreover we establish some relations between connective eccentricity index and eccentric connectivity index as another eccentricity based invariant of graphs

application of graph theory relationship of eccentric connectivity index and wiener s index with anti inflammatory activity abstract graph theory was successfully applied in developing a relationship between chemical structure and biological activity the relationship of two graph invariants the eccentric connectivity index and the wiener s index was investigated with regard to anti inflammatory activity for a data set consisting of 76 pyrazole carboxylic acid hydrazide analogues the values of the eccentric connectivity index and the wiener s index of each analogue in the data set were computed and active ranges were identified subsequently each analogue was assigned a biological activity that was compared with the anti inflammatory activity reported as percent reduction in paw swelling prediction with an accuracy of 90 was obtained using the eccentric connectivity index as compared to 84 in the case of wiener s index

on the extremal properties of the average eccentricity the eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity ecc g of a graph g is the mean value of eccentricities of all vertices of g the average eccentricity is deeply connected with a topological descriptor called the eccentric connectivity index defined as a sum of products of vertex degrees and eccentricities in this paper we analyze extremal properties of the average eccentricity introducing two graph transformations that increase or decrease ecc g furthermore we resolve four conjectures obtained by the system autographix about the average eccentricity and other graph parameters the clique number and the independence number refute one autographix conjecture about the average eccentricity and the minimum vertex degree and correct one autographix conjecture about the domination number

fracture analysis of nonhomogeneous materials via a moduli perturbation approach abstract this paper presents a first order moduli perturbation algorithm for fracture analysis of nonhomogeneous materials the formulation is based on the bueckner rice weight function theory in the perturbation procedure a homogeneous body is chosen as the reference so that non homogeneous quantities are treated as being perturbed from the reference solutions it is shown that the perturbation formulae can be derived from the potential energy bounds for nonhomogeneous materials but they generally do not give bounds for estimating stress intensity factors the perturbation algorithm is applied to calculate the stress intensity factors for several crack problems involving spatially varying material moduli comparisons with a few exact solutions indicate that the perturbation results give reasonable predictions over a substantial range of moduli variation the solution for a cracked body with sinusoidally varying shear modulus is obtained from perturbation analysis and then used to construct general solutions for arbitrarily varying modulus via fourier analysis

weight functions for external circular cracks abstract in this paper the three dimensional 3d weight functions are derived for external circular cracks the solution method used is similar to that developed by bueckner int j solids structures23 57 93 1987 for internal circular cracks lying in infinite elastic solids a papkovitch neuber potential is used to represent the tensile mode weight function field this potential derived from some known solutions to a mixed boundary value potential problem by galin contact problems in the theory of elasticity school of physical sciences and applied mathematics north carolina state college 1953 also uniquely determines the shear mode weight function fields the results for internal circular cracks by bueckner are presented for comparison and for completeness for external circular cracks different forms of the weight functions exist corresponding to different displacement boundary conditions at infinity the neuber fields denoting the elastic fields of an external circular crack due to remote forces and or moments are used to determine the weight functions under various boundary conditions the crack face weight functions defined as the intensity factors induced by a pair of equal oppositely sensed unit point forces acting on the upper and lower crack faces are presented in closed formulae in the appendices the present results are checked against some existing solutions e g intensity factor solutions due to the point forces acting along the central axis normal to the crack plane

empirical bayes estimates for large scale prediction problems classical prediction methods such as fisher s linear discriminant function were designed for small scale problems in which the number of predictors n is much smaller than the number of observations n modern scientific devices often reverse this situation a microarray analysis for example might include n 100 subjects measured on n 10 000 genes each of which is a potential predictor this article proposes an empirical bayes approach to large scale prediction where the optimum bayes prediction rule is estimated employing the data from all of the predictors microarray examples are used to illustrate the method the results demonstrate a close connection with the shrunken centroids algorithm of tibshirani et al 2002 a frequentist regularization approach to large scale prediction and also with false discovery rate theory

on the connective eccentricity index of trees and unicyclic graphs with given diameter abstract the connective eccentricity index cei of a graph g is defined as ce g v v g d v e v where e v d v denote the eccentricity and the degree of v respectively in this paper we investigate the connective eccentricity index of trees and unicyclic graphs on n vertices with diameter d we also characterize the extremal graphs

on scale mixtures of normal distributions summary the exponential power family of distributions of box tiao 1973 is shown to be a subset of the class of scale mixtures of normals the corresponding mixing distributions are explicitly obtained identifying a close relationship between the exponential power family and a further class of normal scale mixtures namely the stable distributions

an alternating technique for evaluating weight functions for 3 d surface flaws in finite solid bodies an efficient method based on the schwarz neumann alternating technique is presented for computing weight functions of a general solid 3 d as well as 2 d with embedded or surface flaw configurations the total rate of change of the crack opening displacements due to simple perturbations of crack dimension characteristics is conveniently decomposed into the infinite domain and boundary correction parts the former is determined from available analytical solutions of ideal shaped cracks whereas the latter is computed numerically by imposing nil boundary traction requirements for the displacement field corresponding to the weight functions numerical examples with solutions for 3 d weighted average and local stress intensity factors indicate that the proposed method is very accurate and efficient

mixtures envelopes and hierarchical duality summary we develop a connection between mixture and envelope representations of objective functions that arise frequently in statistics we refer to this connection by using the term hierarchical duality our results suggest an interesting and previously underexploited relationship between marginalization and profiling or equivalently between the fenchel moreau theorem for convex functions and the bernstein widder theorem for laplace transforms we give several different sets of conditions under which such a duality result obtains we then extend existing work on envelope representations in several ways including novel generalizations to variance mean models and to multivariate gaussian location models this turns out to provide an elegant missing data interpretation of the proximal gradient method which is a widely used algorithm in machine learning we show several statistical applications in which the framework proposed leads to easily implemented algorithms including a robust version of the fused lasso non linear quantile regression via trend filtering and the binomial fused double pareto model code for the examples is available on github at https github com jgscott hierduals

the horseshoe estimator for sparse signals this paper proposes a new approach to sparsity called the horseshoe estimator the horseshoe is a close cousin of other widely used bayes rules arising from for example double exponential and cauchy priors in that it is a member of the same family of multivariate scale mixtures of normals but the horseshoe enjoys a number of advantages over existing approaches including its robustness its adaptivity to dierent sparsity patterns and its analytical tractability we prove two theorems that formally characterize both the horseshoe s adeptness at large outlying signals and its super ecient rate of convergence to the correct estimate of the sampling density in sparse situations finally using a combination of real and simulated data we show that the horseshoe estimator corresponds quite closely to the answers one would get by pursuing a full bayesian model averaging approach using a discrete mixture prior to model signals and noise

the bayesian bridge summary we propose the bayesian bridge estimator for regularized regression and classification two key mixture representations for the bayesian bridge model are developed a scale mixture of normal distributions with respect to an stable random variable a mixture of bartlett fejer kernels or triangle densities with respect to a two component mixture of gamma random variables both lead to markov chain monte carlo methods for posterior simulation and these methods turn out to have complementary domains of maximum efficiency the first representation is a well known result due to west and is the better choice for collinear design matrices the second representation is new and is more efficient for orthogonal problems largely because it avoids the need to deal with exponentially tilted stable random variables it also provides insight into the multimodality of the joint posterior distribution which is a feature of the bridge model that is notably absent under ridge or lasso type priors we prove a theorem that extends this representation to a wider class of densities representable as scale mixtures of beta distributions and we provide an explicit inversion formula for the mixing distribution the connections with slice sampling and scale mixtures of normal distributions are explored on the practical side we find that the bayesian bridge model outperforms its classical cousin in estimation and prediction across a variety of data sets both simulated and real we also show that the markov chain monte carlo algorithm for fitting the bridge model exhibits excellent mixing properties particularly for the global scale parameter this makes for a favourable contrast with analogous markov chain monte carlo algorithms for other sparse bayesian models all methods described in this paper are implemented in the r package bayesbridge an extensive set of simulation results is provided in two on line supplemental files

generalized gamma convolutions and complete monotonicity lete be the class ofpdf sf on 0 such that for eachu 0 f uv f u v is completely monotone as a function ofw v v 1 this class includes many familiarpdf s and is closed with respect to multiplication and division of independentrv s further e t wheret is the class of generalized gamma convolutions ggc introduced by o thorin moreover e coincides with the class ofpdf s of the form c cdot x beta 1 cdot mathop pi limits i 1 n 1 c i x y i all parameters positive or limits thereof the laplace transform of aggc is characterized by complete monotonicity of uv u v as a function ofw this characterization has many consequences and applications it follows that also the classt has simple multiplicative properties