a geometrical method for removing edge effects from kernel type nonparametric regression estimators abstract we introduce a simple geometric method for removing edge effects from kernel type nonparametric regression estimators it involves reflecting the data set in two estimated points thereby generating a new data set with three times the range of the original data the usual kernel type estimator may be applied to the new enlarged data set without any danger of edge effects this technique is applicable generally to both regularly spaced and randomly spaced designs and admits a natural analog of leave one out cross validation the new cross validation algorithm may be extended to the very ends of the design interval unlike its more conventional counterpart which must be downweighted at the ends of the interval to avoid edge effects

basic soules matrices and their applications abstract in this paper we study the properties of the basic soules matrices in r n n which are a special subclass of the n n soules matrices generated via the basic soules basis the basic soules basis has the sign pattern n and it corresponds to the vector e n r n of all 1 s the basic soules matrices are up to a multiple by a positive scalar symmetric and doubly stochastic we begin by investigating the permanents of basic soules matrices next for a nonsingular basic soules matrix a r n n we show that the matrix a a 1 which is known to be a nonsingular m matrix has a basic soules basis of eigenvectors furthermore we obtain explicit formulas for the eigenvalues of a a 1 in terms of the eigenvalues of a finally let a r n n be a basic soules matrix of spectral radius 1 and set q i a by investigating the sign pattern of the off diagonal entries of the group inverse q of q we determine when the perron root is a concave function in each of the off diagonal entries at a

on the estimation of a probability density function by the maximum penalized likelihood method abstract a class of probability density estimates can be obtained by penalizing the likelihood by a functional which depends on the roughness of the logarithm of the density the limiting case of the estimates as the amount of smoothing increasing has a natural form which makes the method attractive for data analysis and which provides a rationale for a particular choice of roughness penalty the estimates are shown to be the solution of an unconstrained convex optimization problem and mild natural conditions are given for them to exist rates of consistency in various norms and conditions for asymptotic normality and approximation by a gaussian process are given thus breaking new ground in the theory of maximum penalized likelihood density estimation author

chromatic polynomials for j h i strip graphs and their asymptotic limits abstract we calculate the chromatic polynomials p for n vertex strip graphs of the form j l 1 m h i where j and i are various subgraphs on the left and right ends of the strip whose bulk is comprised of m fold repetitions of a subgraph h the strips have free boundary conditions in the longitudinal direction and free or periodic boundary conditions in the transverse direction this extends our earlier calculations for strip graphs of the form l 1 m h i we use a generating function method from these results we compute the asymptotic limiting function w lim n p 1 n for q z this has physical significance as the ground state degeneracy per site exponent of the ground state entropy of the q state potts antiferromagnet on the given strip in the complex q plane w is an analytic function except on a certain continuous locus b in contrast to the l 1 m h i strip graphs where b i is independent of i and ii consists of arcs and possible line segments that do not enclose any regions in the q plane we find that for some j l 1 m h i strip graphs b i does depend on i and j and ii can enclose regions in the q plane our study elucidates the effects of different end subgraphs i and j and of boundary conditions on the infinite length limit of the strip graphs

on single and double soules matrices abstract until now the concept of a soules basis matrix of sign pattern n consisted of an orthogonal matrix r r n n generated in a certain way from a positive n vector which has the property that for any diagonal matrix diag 1 n with 1 n 0 the symmetric matrix a r r t has nonnegative entries only in the present paper we introduce the notion of a pair of double soules basis matrices of sign pattern n which is a pair of matrices p q each in r n n which are not necessarily orthogonal and which are generated in a certain way from two positive vectors but such that pq t i and such that for any of the aforementioned diagonal matrices the matrix a p q t also has nonnegative entries only we investigate the interesting properties which such matrices a have as a preamble to the above investigation we show that the iterates a k r k r k t generated in the course of the qr algorithm when it is applied to a r r t where r is a soules basis matrix of sign pattern n are again symmetric matrices generated by the soules basis matrices r k of sign pattern n which are themselves modified as the algorithm progresses our work here extends earlier works by soules and elsner et al

transfer matrices and partition function zeros for antiferromagnetic potts models iii triangular lattice chromatic polynomial we study the chromatic polynomial pg q for m n triangular lattice strips of widths m 12p 9f with periodic or free transverse boundary conditions respectively and arbitrary lengths n with free longitudinal boundary conditions the chromatic polynomial gives the zero temperature limit of the partition function for the q state potts antiferromagnet we compute the transfer matrix for such strips in the fortuin kasteleyn representation and obtain the corresponding accumulation sets of chromatic zeros in the complex q plane in the limit n we recompute the limiting curve obtained by baxter in the thermodynamic limit m n and find new interesting features with possible physical consequences finally we analyze the isolated limiting points and their relation with the beraha numbers

asymptotic limits and zeros of chromatic polynomials and ground state entropy of potts antiferromagnets we study the asymptotic limiting function w g q lim n to infty p g q 1 n where p g q is the chromatic polynomial for a graph g with n vertices we first discuss a subtlety in the definition of w g q resulting from the fact that at certain special points q s the following limits do not commute lim n to infty lim q to q s p g q 1 n ne lim q to q s lim n to infty p g q 1 n we then present exact calculations of w g q and determine the corresponding analytic structure in the complex q plane for a number of families of graphs g including circuits wheels biwheels bipyramids and cyclic and twisted ladders we study the zeros of the corresponding chromatic polynomials and prove a theorem that for certain families of graphs all but a finite number of the zeros lie exactly on a unit circle whose position depends on the family using the connection of p g q with the zero temperature potts antiferromagnet we derive a theorem concerning the maximal finite real point of non analyticity in w g q denoted q c and apply this theorem to deduce that q c sq 3 and q c hc 3 sqrt 5 2 for the square and honeycomb lattices finally numerical calculations of w hc q and w sq q are presented and compared with series expansions and bounds

density estimation under random censorship and order restrictions from asymptotic to small samples do a random censorship and or order restrictions e g nonnegativity monotonicity convexity affect estimation of a smooth density under mean integrated squared error mise under mild assumptions the known asymptotic results which are concerned only with rates answer no this answer especially for censored data contradicts practical experience and statistical intuition so what can be said about constants of mise convergence it is shown that asymptotically a censorship does affect the constant and this allows one to find a relationship between sample sizes of directly observed and censored datasets that implies the same precision of estimation and b an order restriction does not affect the constant and thus no isotonic estimation is needed intensive monte carlo simulations show that the lessons of the sharp asymptotics are valuable for small sample sizes also the estimator developed is illustrated both on simulated data and a dataset of lifetimes of conveyer blades used at wastewate

families of graphs with w r g q functions that are nonanalytic at 1 q 0 denoting p g q as the chromatic polynomial for coloring an n vertex graph g with q colors and considering the limiting function w g q lim n to infty p g q 1 n a fundamental question in graph theory is the following is w r g q q 1 w g q analytic or not at the origin of the 1 q plane where the complex generalization of q is assumed this question is also relevant in statistical mechanics because w g q exp s 0 k b where s 0 is the ground state entropy of the q state potts antiferromagnet on the lattice graph g and the analyticity of w r g q at 1 q 0 is necessary for the large q series expansions of w r g q although w r is analytic at 1 q 0 for many g there are some g for which it is not for these w r has no large q series expansion it is important to understand the reason for this nonanalyticity here we give a general condition that determines whether or not a particular w r g q is analytic at 1 q 0 and explains the nonanalyticity where it occurs we also construct infinite families of graphs with w r functions that are non analytic at 1 q 0 and investigate the properties of these functions our results are consistent with the conjecture that a sufficient condition for w r g q to be analytic at 1 q 0 is that g is a regular lattice graph lambda this is known not to be a necessary condition

inequalities for permanents involving perron complements abstract let a r n n and let and be nonempty complementary subsets of 1 n of increasing integers for a we define the generalized perron complement of a in a at as the matrix p a a a a i a 1 a for the classes of the nonnegative matrices and of the positive semidefinite matrices we study the relationship between the permanents of the whole matrices and the permanents of their perron complement our conditions which hold in many cases of interest are such that the value of the permanent increases as we pass from the whole matrix to its generalized perron complement for nonnegative and irreducible matrices we also study the relationship between the maximum circuit geometric mean of the entire matrix and the maximum circuit geometric mean of its perron complements

the symmetric nonnegative inverse eigenvalue problem for 5 5 matrices abstract the symmetric nonnegative inverse eigenvalue problem sniep asks when a list 1 2 n of n real numbers is the spectrum of an n n symmetric nonnegative matrix this problem is completely solved only for n 4 our main goal here is to contribute to the solution of sniep for n 5 we also give a sufficient condition for a list to be realized as the spectrum of a symmetric positive matrix

asymptotic normality of the l_1 error of the grenander estimator groeneboom introduced a jump process that can be used among other things to study the asymptotic properties of the grenander estimator of a monotone density in this paper we derive the asymptotic normality of a suitably rescaled version of the l 1 error of the grenander estimator using properties of this jump process

estimating a density under order restrictions nonasymptotic minimax risk on considere la classe de toutes les densites unimodales definies sur un intervalle de longueur l et borne par h on etudie le risque minimax sur cette classe lorsqu on estime en utilisant n observations i i d la perte etant mesuree par la distance l 1 entre l estimateur et la densite reelle

orthogonal bases that lead to symmetric nonnegative matrices abstract in a paper dating back to 1983 soules constructs from a positive vector x an orthogonal matrix r which has the property that for any nonnegative diagonal matrix with nonincreasing diagonal entries the matrix r rt has all its entries nonnegative independently fiedler in 1988 showed that any symmetric irreducible nonsingular matrix whose powers are all m matrices and hence an mma matrix in the language of friedland hershkowitz and schneider must have an orthogonal matrix of eigenvectors r which has similar properties to those of r here for a given positive n vector x we investigate the structure of all orthogonal matrices r for which for any nonnegative diagonal matrix as above the matrices r rt are nonnegative up to a permutation of its columns each such r corresponds to a binary tree whose vertices are subsets of the set 1 2 n with the property that each vertex has either no successor or exactly two disjoint successors for such orthogonal matrices r and such nonsingular diagonal matrices we show that the set of matrices of the form r rt and the set of inverse mma matrices i e matrices whose inverses are mma matrices coincide using this result we establish a relation between strictly ultrametric matrices and inverse mma matrices finally we show that the qr factorization of r rt for certain such r s has a special sign pattern

chromatic polynomials of planar triangulations the tutte upper bound and chromatic zeros tutte proved that if gpt is a planar triangulation and p gpt q is its chromatic polynomial then p gpt 1 1 n 5 where and n is the number of vertices in gpt here we study the ratio r gpt p gpt 1 1 n 5 for a variety of planar triangulations we construct infinite recursive families of planar triangulations gpt m depending on a parameter m linearly related to n and show that if p gpt m q only involves a single power of a polynomial then r gpt m approaches zero exponentially fast as n we also construct infinite recursive families for which p gpt m q is a sum of powers of certain functions and show that for these r gpt m may approach a finite nonzero constant as n the connection between the tutte upper bound and the observed chromatic zero s near to 1 is investigated we report the first known graph for which the zero s closest to 1 is not real but instead is a complex conjugate pair finally we discuss connections with the nonzero ground state entropy of the potts antiferromagnet on these families of graphs

uncoupling the perron eigenvector problem abstract for a nonnegative irreducible matrix a with spectral radius this paper is concerned with the determination of the unique normalized perron vector which satisfies a 62 0 j j 1 it is explained how to uncouple a large matrix a into two or more smaller matrices say p 11 p 22 p kk such that this sequence of smaller matrices has the following properties 1 each p ii is also nonnegative and irreducible so that each pii has a unique perron vector i 2 each pii has the same spectral radius as a 3 it is possible to determine the i s completely independently of each other so that one can execute the computation of the i s parallel 4 it is easy to couple the smaller perron vectors i back together in order to produce the perron vector for the original matrix a

the grenader estimator a nonasymptotic approach on examine certaines proprietes non asymtotiques de l estimateur de grenander d une densite decroissante f cet estimation est defini comme la pente du plus petit majorant concave de la fonction de repartition empirique on montre que son risque mesure avec perte l 1 est borne par une certaine fonctionnelle qui depend de f et du nombre d observations n le comportement asymptotique de la fonctionnelle quand n tend vers l infini est en accord avec les performances asymptotiques connues de l estimation de grenander

derivatives of the perron root at an essentially nonnegative matrix and the group inverse of an m matrix abstract the results of the work in this paper can be divided into two major parts in the first part we develop formulae using group inverses of irreducible m matrices for the second order partial derivatives with respect to the entries of the perron root r at an essentially nonnegative and irreducible matrix a we use these formulae to obtain explicit expressions for the matrices 2 r a ij 2 and 2 r a ii jj and examine these expressions at some special cases when for example in addition to a being essentially nonnegative and irreducible a is row stochastic or a is of rank 1 and so on in the second major part of this paper we investigate for a singular and irreducible m matrix q a certain matrix g which is a positive diagonal scaling of the group inverse q of q we show for instance that z gz 0 for all real vectors z and use this result and some of the formulae obtained in the first part of the paper to provide a matrix theoretic proof of a theorem due to j e cohen math proc cambridge philos soc 86 1979 343 350 which states that for an essentially nonnegative and irreducible matrix a the perron root is a convex function of the main diagonal of a

generalized helmert ledermann orthogonal matrices and rom simulation abstract deterministic rectangular orthonormal matrices satisfying a hyper plane constraint plays a central role in random orthogonal matrix rom simulation the multivariate skewness and kurtosis sampling properties are encrypted in a given orthonormal matrix we consider a subclass of generalized helmert ledermann ghl orthogonal matrices that have fixed last column are generated by the cayley transform and satisfy the required hyper plane constraint the algebraic structure of ghl orthogonal matrices is determined simple and convenient skewness and kurtosis formulas are obtained we exhibit an asymptotically growing range of variation for skewness and kurtosis which points to an increased flexibility in rom simulation

on the risk of histograms for estimating decreasing densities on cherche a concevoir un estimateur simple et a obtenir des bornes superieures pour son risque qui soient dans un facteur 10 de la borne inferieure de birge 1987

best asymptotic normality of the kernel density entropy estimator for smooth densities in the random sampling setting we estimate the entropy of a probability density distribution by the entropy of a kernel density estimator using the double exponential kernel under mild smoothness and moment conditions we show that the entropy of the kernel density estimator equals a sum of independent and identically distributed i i d random variables plus a perturbation which is asymptotically negligible compared to the parametric rate n sup 1 2 an essential part in the proof is obtained by exhibiting almost sure bounds for the kullback leibler divergence between the kernel density estimator and its expected value the basic technical tools are doob s submartingale inequality and convexity jensen s inequality

estimation of unimodal densities without smoothness assumptions the grenander estimator of a decreasing density which is defined as the derivative of the concave envelope of the empirical c d f is known to be a very good estimator of an unknown decreasing density on the half line mathbb r when this density is not assumed to be smooth it is indeed the maximum likelihood estimator and one can get precise upper bounds for its risk when the loss is measured by the mathbb l 1 distance between densities moreover if one restricts oneself to the compact subsets of decreasing densities bounded by h with support on 0 l the risk of this estimator is within a fixed factor of the minimax risk the same is true if one deals with the maximum likelihood estimator for unimodal densities with known mode when the mode is unknown the maximum likelihood estimator does not exist any more we shall provide a general purpose estimator together with a computational algorithm for estimating nonsmooth unimodal densities its risk is the same as the risk of the grenander estimator based on the knowledge of the true mode plus some lower order term it can also cope with small departures from unimodality

exact potts model partition functions on ladder graphs abstract we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex ladder graphs i e strips of the square lattice with width ly 2 and arbitrary length lx with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of these ladder graphs and the thermodynamics is discussed by comparison with strip graphs of other widths we analyze how the singularities at the zero temperature critical point of the ferromagnet on infinite length finite width strips depend on the width we point out and study the following noncommutativity at certain special values q s lim n lim q q s z 1 n lim q q s lim n z 1 n it is shown that the potts antiferromagnet on both the infinite length line and ladder graphs with cyclic or mobius boundary conditions exhibits a phase transition at finite temperature if 0 b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros certain properties of the complex temperature phase diagrams are shown to exhibit close connections with those of the model on the square lattice showing that exact solutions on infinite length strips provide a way of gaining insight into these complex temperature phase diagrams

maximum likelihood estimation of smooth monotone and unimodal densities we study the nonparametric estimation of univariate monotone and unimodal densities using the maximum smoothed likelihood approach the monotone estimator is the derivative of the least concave majorant of the distribution corresponding to a kernel estimator we prove that the mapping on distributions with density the derivative of the least concave majorant of is a contraction in all l p norms 1 p and some other distances such as the hellinger and kullback leibler distances the contractivity implies error bounds for monotone density estimation almost the same error bounds hold for unimodal estimation

on reduced rank nonnegative matrix factorization for symmetric nonnegative matrices abstract let v r m n be a nonnegative matrix the nonnegative matrix factorization nnmf problem consists of finding nonnegative matrix factors w r m r and h r r n such that v wh lee and seung proposed two algorithms one of which finds nonnegative w and h such that v wh f is minimized after examining the case in which r 1 about which a complete characterization of the solution is possible we consider the case in which m n and v is symmetric we focus on questions concerning when the best approximate factorization results in the product wh being symmetric and on cases in which the best approximation cannot be a symmetric matrix finally we show that the class of positive semidefinite symmetric nonnegative matrices v generated via a soules basis admit for every 1 r rank v a nonnegative factorization wh which coincides with the best approximation in the frobenius norm to v in r n n of rank not exceeding r an example of applications in which nnmf factorizations for nonnegative symmetric matrices v arise is video and other media summarization technology where v is obtained from a distance matrix

on the minimisation of l sp p error in mode estimation we show that for l p convergence of the mode of a nonparametric density estimator to the mode of an unknown probability density finiteness of the pth moment of the underlying distribution is both necessary and sufficient the basic requirement of existence of finite variance has been overlooked by statisticians who have earlier considered mean square convergence of nonparametric mode estimators they have focussed on mean squared error of the asymptotic distribution rather than on asymptotic mean squared error the effect of bandwidth choice on the rate of l p convergence is analysed and smoothed bootstrap methods are used to develop an empirical approximation to the l p measure of error the resulting bootstrap estimator of l p error may be minimised with respect to the bandwidth of the nonparametric density estimator and in this way an empirical rule may be developed for selecting the bandwidth for mode estimation particular attention is devoted to the problem of selecting the appropriate amount of smoothing in the bootstrap algorithm

chromatic polynomials of large triangular lattices evaluating the q colourings of a lattice is equivalent to solving the q state zero temperature antiferromagnetic potts model this has recently been done exactly for an infinite triangular lattice with q real here the results are extended to the full complex q plane giving the limiting distribution of the zeros of the chromatic polynomial the results are compared with finite lattice calculations and the occurrence of isolated real zeros converging on the beraha numbers is noted

double soules pairs and matching soules bases abstract we consider two generalizations of the notion of a soules basis matrix a pair of nonsingular n n matrices p q is called a double soules pair if the first columns of p and q are positive pq t i and p q t is nonnegative for every n n nonnegative diagonal matrix with nonincreasing diagonal elements in a paper by chen han and neumann an algorithm for generating such pairs was given here we characterize all such pairs and discuss some implications of the characterization we also consider pairs of matrices u v such that u and v each consists of k orthonormal columns the first of which is positive and u v is nonnegative for every k k nonnegative diagonal matrix with nonincreasing diagonal elements we call such pairs matching soules pairs we characterize all such pairs and make some observations regarding the nonnegative matrices and generalized inverses generated by them

quantile and probability curves without crossing this paper proposes a method to address the longstanding problem of lack of monotonicity in estimation of conditional and structural quantile functions also known as the quantile crossing problem bassett and koenker 1982 the method consists in sorting or monotone rearranging the original estimated non monotone curve into a monotone rearranged curve we show that the rearranged curve is closer to the true quantile curve than the original curve in finite samples establish a functional delta method for rearrangement related operators and derive functional limit theory for the entire rearranged curve and its functionals we also establish validity of the bootstrap for estimating the limit law of the entire rearranged curve and its functionals our limit results are generic in that they apply to every estimator of a monotone function provided that the estimator satisfies a functional central limit theorem and the function satisfies some smoothness conditions consequently our results apply to estimation of other econometric functions with monotonicity restrictions such as demand production distribution and structural distribution functions we illustrate the results with an application to estimation of structural distribution and quantile functions using data on vietnam veteran status and earnings

stochastic complementation uncoupling markov chains and the theory of nearly reducible systems a concept called stochastic complementation is an idea which occurs naturally although not always explicitly in the theory and application of finite markov chains this paper brings this idea to the forefront with an explicit definition and a development of some of its properties applications of stochastic complementation are explored with respect to problems involving uncoupling procedures in the theory of markov chains furthermore the role of stochastic complementation in the development of the classical simon ando theory of nearly reducible system is presented

ground state degeneracy of potts antiferromagnets cases with noncompact w boundaries having multiple points at we present exact calculations of the zero temperature partition function and ground state degeneracy per site for the q state potts antiferromagnet on a number of families of graphs for which the boundary of regions of analyticity of w in the complex q plane is noncompact and has the properties that i in the plane the point z 0 is a multiple point on and ii includes support for these families are generated by the method of homeomorphic expansion our results give further insight into the conditions for the validity of large q series expansions for the reduced function

the limit distribution of the l error of grenander type estimators let f be a nonincreasing function defined on 0 1 under standard regularity conditions we derive the asymptotic distribution of the supremum norm of the difference between f and its grenander type estimator on sub intervals of 0 1 the rate of convergence is found to be of order n log n 1 3 and the limiting distribution to be gumbel

structural properties of potts model partition functions and chromatic polynomials for lattice strips abstract the q state potts model partition function equivalent to the tutte polynomial for a lattice strip of fixed width l y and arbitrary length l x has the form z g q v j 1 n z g c z g j z g j l x where v is a temperature dependent variable the special case of the zero temperature antiferromagnet v 1 is the chromatic polynomial p g q using coloring and transfer matrix methods we give general formulas for c x g j 1 n x g c x g j for x z p on cyclic and mobius strip graphs of the square and triangular lattice combining these with a general expression for the unique coefficient c z g j of degree d in q c d u 2d q 2 where u n x is the chebyshev polynomial of the second kind we determine the number of z g j s with coefficient c d in z g q v for these cyclic strips of width l y to be n z l y d 2d 1 l y d 1 1 2l y l y d for 0 d l y and zero otherwise for both cyclic and mobius strips of these lattices the total number of distinct eigenvalues z g j is calculated to be n z l y 2l y l y results are also presented for the analogous numbers n p l y d and n p l y for p g q we find that n p l y 0 n p l y 1 1 m l y 1 motzkin number n z l y 0 c l y the catalan number and give an exact expression for n p l y our results for n z l y and n p l y apply for both the cyclic and mobius strips of both the square and triangular lattices we also point out the interesting relations n z l y 2 n da tri l y and n p l y 2 n da sq l y where n da n denotes the number of directed lattice animals on the lattice we find the asymptotic growths n z l y l y 1 2 4 l y and n p l y l y 1 2 3 l y as l y some general geometric identities for potts model partition functions are also presented

on mardia skewness and kurtosis of soules basis matrices in rom simulation abstract relying on an interesting connection between generalized helmert ledermann matrices and soules basis matrices as well as a complete binary tree classification of the latter a recursive determination of the mardia skewness and kurtosis coefficients of soules basis matrices is proposed it is based on a convenient combinatorial representation of soules basis matrices by restricted binary partitions of a natural number into two ordered summands with recursive repetition of this binary partitioning on the summands the obtained simple composition and invariance formulas enable the recursive calculation of these skewness and kurtosis coefficients