transformations in regression a robust analysis we consider two approaches to robust estimation for the box cox power transformation model one approach maximizes weighted modified likelihoods a second approach bounds a measure of gross error sensitivity among our primary concerns is the performance of these estimators on actual data in examples that we study there seem to be only minor differences between these two robust estimators but they behave rather differently than the maximum likelihood estimator or estimators that bound only the influence of the residuals these examples show that model selection determination of the transformation parameter and outlier identification are fundamentally interconnected

the masking breakdown point of multivariate outlier identification rules abstract in this article we consider simultaneous outlier identification rules for multivariate data generalizing the concept of so called outlier identifiers as presented by davies and gather for the case of univariate samples such multivariate outlier identifiers are based on estimators of location and covariance therefore it seems reasonable that characteristics of the estimators influence the behavior of outlier identifiers several authors mentioned that using estimators with low finite sample breakdown point is not recommended for identifying outliers to give a formal explanation we investigate how the finite sample breakdown points of estimators used in these identification rules influence the masking behavior of the rules

extremal laplacian energy of directed trees unicyclic digraphs and bicyclic digraphs abstract let a g and d g be the adjacency matrix of a digraph g with n vertices and the diagonal matrix of vertex outdegrees of g respectively then the laplacian matrix of the digraph g is l g d g a g the laplacian energy of a digraph g is defined as l e g i 1 n i 2 by using second spectral moment where 1 2 n are all the eigenvalues of l g of g in this paper by using arc shifting operation and out star shifting operation we determine the directed trees unicyclic digraphs and bicyclic digraphs which attain maximal and minimal laplacian energy among all digraphs with n vertices respectively

the skew energy of a digraph we are interested in the energy of the skew adjacency matrix of a directed graph d which is simply called the skew energy of d in this paper properties of the skew energy of d are studied in particular a sharp upper bound for the skew energy of d is derived in terms of the order of d and the maximum degree of its underlying undirected graph an infinite family of digraphs attaining the maximum skew energy is constructed moreover the skew energy of a directed tree is independent of its orientation and interestingly it is equal to the energy of the underlying undirected tree skew energies of directed cycles under different orientations are also computed some open problems are presented

ordering of oriented unicyclic graphs by skew energies ordering of the oriented unicyclic graphs with n vertices according to their minimal skew energies is investigated we derive the first 22 oriented unicyclic graphs for n 1784 and a series of oriented unicyclic graphs for 3 n 1783

travelling solitons in the externally driven nonlinear schrodinger equation we consider the undamped nonlinear schr dinger equation driven by a periodic external force in the absence of damping solitons do not have undulations on their tails yet we show that they can bind into stationary multisoliton complexes using two previously known stationary solitons and two newly found stationary complexes as starting points we obtain classes of localized travelling waves by the numerical continuation in the parameter space two families of stable solitons are identified one family is stable for sufficiently low velocities while solitons from the second family stabilize when travelling faster than a certain critical speed the stable solitons of the former family can also form stably travelling bound states

asymptotic behaviour of s estimates of multivariate location parameters and dispersion matrices it is shown under appropriate conditions that rousseeuw s minimum volume estimator and other s estimators of multivariate location and dispersion parameters are consistent under certain differentiability conditions the estimates are asymptotically normally distributed with a norming factor of n 1 2

fast very robust methods for the detection of multiple outliers abstract a few repeats of a simple forward search from a random starting point are shown to provide sufficiently robust parameter estimates to reveal masked multiple outliers the stability of the patterns obtained is exhibited by the stalactite plot the robust estimators used are least median of squares for regression and the minimum volume ellipsoid for multivariate outliers the forward search also has potential as an algorithm for calculation of these parameter estimates for large problems parallel computing provides appreciable reduction in computational time

on the relation between s estimators and m estimators of multivariate location and covariance we discuss the relation between s estimators and m estimators of multivariate location and covariance as in the case of the estimation of a multiple regression parameter s estimators are shown to satisfy first order conditions of m estimators we show that the influence function if x s f of s functionals exists and is the same as that of corresponding m functionals also we show that s estimators have a limiting normal distribution which is similar to the limiting normal distribution which is similar to the limiting normal distribution of m estimators finally we compare asymptotic variances and breakdown point of both types of estimators

a note on tyler s modification of the mad for the stahel donoho estimator in 1994 tyler proposed a modification of the mad in order to achieve the maximal possible finite sample breakdown point for the stahel donoho estimator based upon the median and the modified mad a proof is not given however and as far as we know has not been published yet moreover there seems to be an error concerning the exact definition of the modified mad which is needed to achieve this goal for the sake of theoretical completeness and in order to correct this error we calculate the finite sample breakdown point of modifications of the mad and point out which of these modifications lead to a stahel donoho estimator with maximal finite sample breakdown point

finite sample breakdown points of projection based multivariate location and scatter statistics finite sample breakdown points are obtained for two classes of projection based multivariate location and scatter statistics the stahel donoho statistics and the maronna yohai statistics the definition of these multivariate statistics are dependent on the value of location and scale statistics for all univariate projections of the data and consequently their properties depend on the nature of the corresponding univariate location and scale statistics used on the projected data the finite sample breakdown points of the multivariate statistics though are not directly related to those of the corresponding univariate location and scale statistics a uniform finite sample breakdown point concept is needed the median and the median absolute deviation about the median m a d are one possible choice for the univariate location and scale statistics respectively for sparse data sets in high dimensions though it is not recommended that the m a d be used as the univariate scale statistic for the projected data since its uniform finite sample breakdown point is shown to be much less than optimum a simple modification to the m a d however is shown to alleviate this problem for various reasons one may wish to consider univariate location and scale statistics other than the median and the m a d respectively a very broad and natural class of univariate location and scale statistics to consider for the projected data is the simultaneous m estimates of location and scale new results on their breakdown properties are given in this paper implicit formulas for the breakdown points of monotonic simultaneous m estimates of location and scale are known and they tend to imply rather low breakdown points for smooth choices of the defining weight functions it is shown here that this phenomenon does not occur for a large class of nonmonotonic simultaneous m estimates of location and scale furthermore explicit rather than implicit expressions for the uniform finite sample breakdown points are given for these nonmonotonic m estimates

the behavior of the stahel donoho robust multivariate estimator abstract the stahel donoho estimators t v of multivariate location and scatter are defined as a weighted mean and a weighted covariance matrix with weights of the form w r where w is a weight function and r is a measure of outlyingness obtained by considering all univariate projections of the data it has a high breakdown point for all dimensions and order n consistency the asymptotic bias of v for point mass contamination for suitable weight functions is compared with that of rousseeuw s minimum volume ellipsoid mve estimator a simulation shows that for a suitable w t and v exhibit high efficiency for both normal and cauchy distributions and are better than their competitors for normal data with point mass contamination the performances of the estimators for detecting outliers are compared for both a real and a synthetic data set

oriented unicyclic graphs with the first n 92 largest skew energies abstract let g be an oriented graph obtained by assigning an orientation to the edge set of a simple undirected graph g such that g becomes a directed graph let s g be the skew adjacency matrix of g the skew energy of g is defined as the sum of the absolute values of all eigenvalues of s g in this paper we provide a new method to compare the skew energies of two oriented graphs whose skew characteristic polynomials satisfy a given recurrence relation and determine the oriented unicyclic graphs of order n with the first n 9 2 largest skew energies for n 31

the breakdown points of the mean combined with some rejection rules in the past methods for rejection of outliers have been investigated mostly without regard to the quantitative consequences for subsequent estimation or testing procedures moreover although rejection of outliers with subsequent application of least squares methods is one of the oldest and most widespread classes of robust procedures until recently no comparison was made with other robust methods in this article the simplest situation namely estimation of a location parameter in the potential presence of outliers is treated by means of a monte carlo study this study yields monte carlo variances of the arithmetic mean after rejection of outliers according to several classical and recent formal rules the results are also compared with those for other robust estimators of location parameters it turns out that a simple summary and theoretical explanation of the monte carlo results is provided by the breakdown points of the combined rejection estimation procedures as a by product the concept of br

robustness properties of s estimators of multivariate location and shape in high dimension author s rocke david abstract for the problem of robust estimation of multivariate location and shape defilllng s estimators using scale transformations of a fixed p function regardless of the dimension as is usually done leads to a perverse outcome estimators in high dimension can have a breakdown point approaching 50 but still fail to reject as outliers points that are large distances from the main mass of points this leads to a form of nonrobustness that has important practical consequences in this paper estimators are defined that improve on known s estimators in having alll of the following properties 1 maximal breakdown for the given sample size and dimension 2 ability completely to reject as outliers points that are far from the main mass of points 3 convergence to good solutions with a modest amount of computation from a nonrobust starting point for large though not near 50 contamination however to attain maximal breakdown these estimates like other known maximal breakdown estimators require large amounts of computational effort this greater ability of the new estimators to reject outliers comes at a modest cost in efficiency and gross error sensitivity and at a greater but finite cost in local shift sensitivity

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

4 regular oriented graphs with optimum skew energy abstract let g be a simple undirected graph and g be an oriented graph of g with the orientation and skew adjacency matrix s g the skew energy of the oriented graph g denoted by e s g is defined as the sum of the absolute values of all the eigenvalues of s g in this paper we characterize the underlying graphs of all 4 regular oriented graphs with optimum skew energy and give orientations of these underlying graphs such that the skew energy of the resultant oriented graphs indeed attain optimum it should be pointed out that there are infinitely many 4 regular connected optimum skew energy oriented graphs while the 3 regular case only has two graphs k 4 the complete graph on 4 vertices and q 3 the hypercube

energy of matrices abstract let m n c denote the space of n n matrices with entries in c we define the energy of a m n c as 1 e a k 1 n k t r a n where 1 n are the eigenvalues of a tr a is the trace of a and z denotes the modulus of z c if a is the adjacency matrix of a graph g then e a is precisely the energy of the graph g introduced by gutman in 1978 in this paper we compare the energy e with other well known energies defined over matrices then we find upper and lower bounds of e which extend well known results for the energies of graphs and digraphs also we obtain new results on energies defined over the adjacency laplacian and signless laplacian matrices of digraphs

on the skew energy of orientations of hypercubes abstract let d be a digraph of order n and 1 2 n denote all the eigenvalues of the skew adjacency matrix of d the skew energy e s d of d is defined as e s d i 1 n i in this paper it is proved that for any positive integer k 3 there exists a k regular graph of order n having an orientation d with e s d n k this work positively answers a problem proposed by adiga et al c adiga r balakrishnan wasin so the skew energy of a digraph linear algebra appl 432 2010 1825 1835 in addition a digraph is also constructed such that its skew energy is the same as the energy of its underlying graph

bicyclic oriented graphs with the second largest skew energy abstract let g be an oriented graph and p s g x det x i s g be the skew characteristic polynomial of its skew adjacency matrix s g the skew energy of g is defined to be the sum of the absolute values of eigenvalues of s g in this paper we firstly find a novel relation between the coefficients of skew characteristic polynomial of unicyclic graph with the third largest skew energy and bicyclic oriented graphs and then use it along with other techniques to characterize the bicyclic oriented graphs with the second largest skew energy