stein type shrinkage estimators in gamma regression model with application to prostate cancer data gamma regression is applied in several areas such as life testing forecasting cancer incidences genomics rainfall prediction experimental designs and quality control gamma regression models allow for a monotone and no constant hazard in survival models owing to the broad applicability of gamma regression we propose some novel and improved methods to estimate the coefficients of gamma regression model we combine the unrestricted maximum likelihood ml estimators and the estimators that are restricted by linear hypothesis and we present stein type shrinkage estimators ses we then develop an asymptotic theory for ses and obtain their asymptotic quadratic risks in addition we conduct monte carlo simulations to study the performance of the estimators in terms of their simulated relative efficiencies it is evident from our studies that the proposed ses outperform the usual ml estimators furthermore some tabular and graphical representations are given as proofs of our assertions this study is finally ended by appraising the performance of our estimators for a real prostate cancer data

on the theory of the synthesis of offset dual shaped reflectors case examples in an earlier paper by v galindo israel et al see ibid vol ap 35 p 887 96 august 1987 the geometrical optics go principles constraints and requirements of the dual and single offset shaped reflector synthesis problem were collected and developed into a set of nonlinear first order partial differential equations pdes methods of solving these pdes numerically were illustrated as were certain problems that may arise an extension of the methods by which solutions to the pdes can be obtained is presented together with several case examples these examples are independently analyzed by go and physical optics diffraction methods the starting point for the integration over each reflector can be taken on the outer rim at the center or at an intermediate point the intermediate starting point being the more general case the utility of the speed of this synthesis method is demonstrated this makes practical the incorporation of the synthesis into a search algorithm that can optimize one or more parameters of the reflector system as an example the optimization of the mapping equations for low cross polarization is discussed

random generators of the symmetric group diameter mixing time and spectral gap abstract let g h be a random pair of generators of g sym n or g alt n we show that with probability tending to 1 as n a the diameter of g with respect to s g h g 1 h 1 is at most o n 2 log n c and b the mixing time of g with respect to s is at most o n 3 log n c both c and the implied constants are absolute these bounds are far lower than the strongest worst case bounds known in helfgott seress 2013 they roughly match the worst known examples we also give an improved though still non constant bound on the spectral gap our results rest on a combination of the algorithm in babai beals seress 2004 and the fact that the action of a pair of random permutations is almost certain to act as an expander on l tuples where l is an arbitrary constant friedman et al 1998

comparison techniques for random walk on finite groups we develop techniques for bounding the rate of convergence of a symmetric random walk on a finite group to the uniform distribution the techniques gives bounds on the second largest and other eigenvalues in terms of the eigenvalues of a comparison chain with known eigenvalues the techniques yield sharp rates for a host of previously intractable problems on the symmetric group

determination of reflector surfaces from near field scattering data ii numerical solution let o be a nonisotropic point source of light and i m the power intensity of this source in direction m suppose that the light rays emitted by the source o through an aperture d fall on a perfectly reflecting surface r and reflect off it so that the reflected rays illuminate a closed domain t on some plane with intensity l the inverse problem consists of constructing the reflector surface r from given position of the source o the input aperture d function i target set t and output intensity l for example the input intensity may have a bell like shape and we may wish to redistribute the energy uniformly over a prespecified region the analytical formulation of the described above problem leads to a non linear partial differential equation of monge ampere type in our previous paper we proved existence of weak solutions to this inverse problem and in this paper we describe and illustrate with examples an algorithm for its numerical solution the proposed numerical method can be easily modified for the case when t is a closed domain on an arbitrary surface

the structure of a random graph at the point of the phase transition consider the random graph models g n edges m and g n prob edge p with m m n 1 n 1 3 n 2 and p p n 1 an 1 3 n for 1 1 define an i component of a random graph as a component which has exactly i more edges than vertices call an i component with i 1 a complex component for both models we show that when a is constant the expected number of complex components is bounded almost surely a s each of these components if any exist has size of order n2 3 and the maximum value of i is bounded in probability we prove that a s the largest suspended tree in each complex component has size of order n2 3 and deletion of all suspended trees results in a smoothed graph of size of order n 3 with the maximum vertex degree 3 the total number of branching vertices i e of degree 3 is bounded in probability thus each complex component is almost surely topologically equivalent to a 3 regular multigraph of a uniformly bounded size lengths of the shortest cycle and of the shortest path between two branching vertices of a smoothed graph are each of order n 3 we find a relatively simple integral formula for the limit distribution of the numbers of complex components which implies in particular that all values of the complexity spectrum have positive limiting probabilities we also answer questions raised by erdos and renyi back in 1960 it is proven that there exists p a the limiting planarity probability with 0 p a 1 p oo 1 p oo 0 in particular g n m g n p resp is almost surely nonplanar iff m n 2 n 213 00 np i n 1 3 oo resp

eigenvalue distributions of random permutation matrices let m be a randomly chosen n x n permutation matrix for a fixed arc of the unit circle let x be the number of eigenvalues of m which lie in the specified arc we calculate the large n asymptotics for the mean and variance of x and show that x e x var x 1 2 is asymptotically normally distributed in addition we show that for several fixed arcs i 1 i m the corresponding random variables are jointly normal in the large n limit

loop correlations in random wire models we introduce a family of loop soup models on the hypercubic lattice the models involve links on the edges and random pairings of the link endpoints on the sites we conjecture that loop correlations of distant points are given by poisson dirichlet correlations in dimensions three and higher we prove that in a specific random wire model that is related to the classical xy spin system the probability that distant sites form an even partition is given by the poisson dirichlet counterpart

analyzing glauber dynamics by comparison of markov chains a popular technique for studying random properties of a combinatorial set is to design a markov chain monte carlo algorithm for many problems there are natural markov chains connecting the set of allowable configurations which are based on local moves or glauber dynamics typically these single site update algorithms are difficult to analyze so often the markov chain is modified to update several sites simultaneously recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems in this work we use the comparison technique of diaconis and saloff coste to show that several of the natural single point update algorithms are efficient the strategy is to relate the mixing rate of these algorithms to the corresponding nonlocal algorithms which have already been analyzed this allows us to give polynomial time bounds for single point update algorithms for problems such as generating planar tilings and random triangulations of convex polygons we also survey several other comparison techniques along with specific applications which have been used in the context of estimating mixing rates of markov chains

independent process approximations for random combinatorial structures abstract many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables conditioned on the value of a weighted sum of the variables it is interesting to compare the combinatorial structure directly to the independent discrete process without renormalizing the quality of approximation can often be conveniently quantified in terms of total variation distance for functionals which observe part but not all of the combinatorial and independent processes among the examples are combinatorial assemblies e g permutations random mapping functions and partitions of a set multisets e g polynomials over a finite field mapping patterns and partitions of an integer and selections e g partitions of an integer into distinct parts and square free polynomials over finite fields we consider issues common to all the above examples including equalities and upper bounds for total variation distances existence of limiting processes heuristics for good approximations the relation to standard generating functions moment formulas and recursions for computing densities refinement to the process which counts the number of parts of each possible type the effect of further conditioning on events of moderate probability large deviation theory and nonuniform measures on combinatorial objects and the possibility of getting useful results by overpowering the conditioning

preliminary test and stein estimations in simultaneous linear equations abstract in modeling of an economic system there may occur some stochastic constraints that can cause some changes in the estimators and their respective behaviors in this approach we formulate the simultaneous equation models into the problem of estimating the regression parameters of a multiple regression model under elliptical errors we define five different sorts of estimators for the vector parameter their exact risk expressions are also derived under the balanced loss function comparisons are then made to clarify the performance of the proposed estimators it is shown that the shrinkage factor of the stein estimator is robust with respect to departures from normality assumption

upper bound on the characters of the symmetric groups let c be a conjugacy class in the symmetric group s n and be a partition of n let f be the degree of the irreducible representation s c the character of s at c and r c the normalized character c f we prove that there exist constants b 0 and 1 q 0 such that for n 4 for every conjugacy class c in s n and every irreducible representation s of s n r c max q 1 n 1 n b supp c where supp c is the number of non fixed digits under the action of a permutation in c 1 is the size of the largest part in and 1 is the number of parts in the proof is obtained by enumeration of rim hook tableaux the hook formula and probabilistic arguments combinatorial algebraic and statistical applications follow this result in particular we estimate the rate of mixing of random walks on the alternating groups with respect to conjugacy classes

cutoff for conjugacy invariant random walks on the permutation group we prove a conjecture raised by the work of diaconis and shahshahani z wahrscheinlichkeitstheorie verwandte geb 57 2 159 179 1981 about the mixing time of random walks on the permutation group induced by a given conjugacy class to do this we exploit a connection with coalescence and fragmentation processes and control the kantorovich distance by using a variant of a coupling due to oded schramm as well as contractivity of the distance recasting our proof in the language of ricci curvature our proof establishes the occurrence of a phase transition which takes the following form in the case of random transpositions at time cn 2 the curvature is asymptotically zero for c le 1 and is strictly positive for c 1

a phase transition in the random transposition random walk our work is motivated by bourque pevzner s simulation study of the effectiveness of the parsimony method in studying genome rearrangement and leads to a surprising result about the random transposition walk in continuous time on the group of permutations on n elements starting from the identity let d t be the minimum number of transpositions needed to go back to the identity element from the location at time t d t undergoes a phase transition for 0 1 d cn 2 u c n where u is an explicit function satisfying u x x 2 moreover we describe the fluctuations of d cn 2 about its mean at each of the three stages subcritical critical and supercritical the techniques used involve viewing the cycles in the random permutation as a coagulation fragmentation process and relating the behavior to the erd s renyi random graph model

mixing times for random k cycles and coalescence fragmentation chains let sn be the permutation group on n elements and consider a random walk on sn whose step distribution is uniform on k cycles we prove a well known conjecture that the mixing time of this process is 1 k nlog n with threshold of width linear in n our proofs are elementary and purely probabilistic and do not appeal to the representation theory of sn

the poisson dirichlet law is the unique invariant distribution for uniform split merge transformations we consider a markov chain on the space of countable partitions of the interval 0 1 obtained first by size biased sampling twice allowing repetitions and then merging the parts if the sampled parts are distinct or splitting the part uniformly if the same part was sampled twice we prove a conjecture of vershik stating that the poisson dirichlet law with parameter 1 is the unique invariant distribution for this markov chain our proof uses a combination of probabilistic combinatoric and representation theoretic arguments

the random k cycle walk on the symmetric group we study the random walk on the symmetric group s n generated by the conjugacy class of cycles of length k we show that the convergence to uniform measure of this walk has a cut off in total variation distance after frac n k log n steps uniformly in k o n as n rightarrow infty the analysis follows from a new asymptotic estimation of the characters of the symmetric group evaluated at cycles

a probabilistic interpretation of the macdonald polynomials the two parameter macdonald polynomials are a central object of algebraic combinatorics and representation theory we give a markov chain on partitions of k with eigenfunctions the coefficients of the macdonald polynomials when expanded in the power sum polynomials the markov chain has stationary distribution a new two parameter family of measures on partitions the inverse of the macdonald weight rescaled the uniform distribution on cycles of permutations and the ewens sampling formula are special cases the markov chain is a version of the auxiliary variables algorithm of statistical physics properties of the macdonald polynomials allow a sharp analysis of the running time in natural cases a bounded number of steps suffice for arbitrarily large k

a central limit theorem for the determinant of a wigner matrix abstract we establish a central limit theorem for the log determinant log det m n of a wigner matrix m n under the assumption of four matching moments with either the gue or goe ensemble more specifically we show that this log determinant is asymptotically distributed like n log n 1 2 log n 1 2 log n r when one matches moments with gue and n log n 1 4 log n 1 4 log n r when one matches moments with goe

the hyperbolic geometry of random transpositions turn the set of permutations of n objects into a graph g n by connecting two permutations that differ by one transposition and let t be the simple random walk on this graph in a previous paper berestycki and durrett in discrete random walks 2005 17 26 showed that the limiting behavior of the distance from the identity at time cn 2 has a phase transition at c 1 here we investigate some consequences of this result for the geometry of g n our first result can be interpreted as a breakdown for the gromov hyperbolicity of the graph as seen by the random walk which occurs at a critical radius equal to n 4 let t be a triangle formed by the origin and two points sampled independently from the hitting distribution on the sphere of radius an for a constant 0 0 whereas it is always o n thick when a 1 4 we also show that the hitting distribution of the sphere of radius an is asymptotically singular with respect to the uniform distribution finally we prove that the critical behavior of this gromov like hyperbolicity constant persists if the two endpoints are sampled from the uniform measure on the sphere of radius an however in this case the critical radius is a 1 log 2

painting a graph with competing random walks let x1 x2 be independent random walks on zdn d 3 each starting from the uniform distribution initially each site of zdn is unmarked and whenever xi visits such a site it is set irreversibly to i the mean of ai the cardinality of the set ai of sites painted by i once all of zdn has been visited is 12nd by symmetry we prove the following conjecture due to pemantle and peres for each d 3 there exists a constant d such that limn var ai hd n 14 d where h3 n n4 h4 n n4 logn and hd n nd for d 5 we will also identify d explicitly and show that d 1 as d this is a special case of a more general theorem which gives the asymptotics of var ai for a large class of transient vertex transitive graphs other examples include the hypercube and the caley graph of the symmetric group generated by transpositions

the bohnenblust spitzer algorithm and its applications the familiar bijections between the representations of permutations as words and as products of cycles have a natural class of data driven extensions that permit us to use purely combinatorial means to obtain precise probabilistic information about the geometry of random walks in particular we show that the algorithmic bijection of bohnenblust and spitzer can be used to obtain means variances and concentration inequalities for several random variables associated with a random walk including the number of vertices and length of the convex minorant concave majorant and convex hull

invariable generation of the symmetric group abstract we say that permutations 1 r sn invariably generate sn if no matter how one chooses conjugates 1 r of these permutations 1 r generate sn we show that if 1 2 3 are chosen randomly from sn then with probability tending to 1 as n they do not invariably generate sn by contrast it was shown recently by pemantle peres and rivin that four random elements do invariably generate sn with probability bounded away from zero we include a proof of this statement which while sharing many

the cycle structure of random permutations the total variation distance between the process which counts cycles of size 1 2 ldots b of a random permutation of n objects and a process z 1 z 2 ldots z b of independent poisson random variables with mathbb e z i 1 i converges to 0 if and only if b n rightarrow 0 this poisson approximation can be used to give simple proofs of limit theorems and bounds for a wide variety of functionals of random permutations these limit theorems include the erdos turan theorem for the asymptotic normality of the log of the order of a random permutation and the delaurentis pittel functional central limit theorem for the cycle sizes we give a simple explicit upper bound on the total variation distance to show that this distance decays to zero superexponentially fast as a function of n b rightarrow infty a similar result holds for derangements and more generally for permutations conditioned to have given numbers of cycles of various sizes comparison results are included to show that in approximating the cycle structure by an independent poisson process the main discrepancy arises from independence rather than from poisson marginals

on the theory of the synthesis of single and dual offset shaped reflector antennas since kinber radio technika and engineering 1963 and galindo ieee trans antennas propagat 1963 1964 developed the solution to the circular symmetric dual shaped synthesis problem the question of existence and of uniqueness for offset dual or single shaped synthesis has been a point of controversy many researchers thought that the exact offset solutions may not exist later galindo israel and mittra ieee trans antennas propagat 1979 and others formulated the problem exactly and obtained excellent and numerically efficient but approximate solutions using a technique similar to that first developed by schruben for the single reflector problem journal of the optical society 1973 brickell and westcott proc institute of electrical engineering 1981 developed a monge ampere ma second order nonlinear partial differential equation for the dual reflector problem they solved an elliptic form of this equation by a technique introduced by rall 1979 which iterates by a newton method a finite difference linearized ma equation the elliptic character requires a set of finite difference equations to be developed and solved iteratively existence still remained in question although the second order ma equation developed by schruben is elliptic the first order equations from which the ma equation is derived can be integrated progressively e g as for an initial condition problem such as for hyperbolic equations a noniterative and usually more rapid type solution in this paper we have solved numerically the first order equations exact solutions are thus obtained by progressive integration furthermore we have concluded that not only does an exact solution exist but an infinite set of such solutions exists these conclusions are inferred in part from numerical results

counting semiregular permutations which are products of a full cycle and an involution character theoretic methods and the group algebra of the symmetric group are used to derive properties of the number of permutations with only p cycles for an arbitrary but fixed p which are expressible as the product of a full cycle and a fixed point free involution this problem has application to single face embeddings of p regular graphs on surfaces of given genus

functional limit theorems for random regular graphs consider d uniformly random permutation matrices on n labels consider the sum of these matrices along with their transposes the total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices we consider limit theorems for various combinatorial and analytical properties of this graph or the matrix as n grows to infinity either when d is kept fixed or grows slowly with n in a suitable weak convergence framework we prove that the finite but growing in length sequences of the number of short cycles and of cyclically non backtracking walks converge to distributional limits we estimate the total variation distance from the limit using stein s method as an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix a key step in this latter derivation is an extension of the kahn szemeredi argument for estimating the second largest eigenvalue for all values of d and n

a pair of difference differential equations of euler cauchy type we study two classes of linear difference differential equations analogous to euler cauchy ordinary differential equations but in which mul tiple arguments are shifted forward or backward by fixed amounts special cases of these equations have arisen in diverse branches of number theory and combinatorics they are also of use in linear control theory here we study these equations in a general setting building on previous work going back to de bruijn we show how adjoint equations arise naturally in the problem of uniqueness of solutions exploiting the adjoint relationship in a new way leads to a significant strengthening of previous uniqueness results specifically we prove here theorem 11 that the general euler cauchy difference differential equation with advanced arguments has a unique solution up to a multiplica tive constant in the class of functions bounded by an exponential function on the positive real line for the closely related class of equations with re tarded arguments we focus on a corresponding class of solutions locating and classifying the points of discontinuity we also provide an explicit asymptotic expansion at infinity

brownian excursions critical random graphs and the multiplicative coalescent let b t s 0 s be reflecting inhomogeneous brownian motion with drift t s at time s started with b t 0 0 consider the random graph script g sign n n 1 tn 4 3 whose largest components have size of order n 2 3 normalizing by n 2 3 the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of b t corollary 2 the dynamics of merging of components as t increases are abstracted to define the multiplicative coalescent process the states of this process are vectors x of nonnegative real cluster sizes x i and clusters with sizes x i and x j merge at rate x i x j the multiplicative coalescent is shown to be a feller process on l 2 the random graph limit specifies the standard multiplicative coalescent which starts from infinitesimally small clusters at time the existence of such a process is not obvious

transforming cabbage into turnip polynomial algorithm for sorting signed permutations by reversals genomes frequently evolve by reversals r i j that transform a gene order 1 i i 1 j 1 j n into 1 i j 1 i 1 j n reversal distance between permutations and sis the minimum number of reversals to transform into a analysis of genome rearrangements in molecular biology started in the late 1930 s when dobzhansky and sturtevant published a milestone paper presenting a rearrangement scenario with 17 inversions between the species of drosophilia analysis of genomes evolving by inversions leads to a combinatorial problem of sorting by reversals studied in detail recently we study sorting of signed permutations by reversals a problem that adequately models rearrangements in a small genomes like chloroplast or mitochondrial dna the previously suggested approximation algorithms for sorting signed permutations by reversals compute the reversal distance between permutations with an astonishing accuracy for both simulated and biological data we prove a duality theorem explaining this intriguing performance and show that there exists a hidden parameter that allows one to compute the reversal distance between signed permutations in polynomial time

ridge estimation in semi parametric regression models under the stochastic restriction and correlated elliptically contoured errors abstract some linear stochastic constraints may occur during real data set modeling based on either additional information or prior knowledge these stochastic constraints often cause some changes in the behaviors of estimators in this research shrinkage ridge estimators as well as their positive parts are proposed in the semi parametric model when some stochastic constrains are imposed under a multicollinearity setting also it is assumed that the error terms are dependent and distributed according to the elliptically contoured models the bias and risk expressions of the proposed estimators for comparison purposes are derived finally the monte carlo simulation studies and a real application related to electricity consumption data are conducted to support our theoretical discussion

an invariance principle for certain dependent sequences let x 1 x 2 cdots be a strictly stationary second order sequence which is associated i e is such that any two coordinatewise nondecreasing functions of the x i s of finite variance are nonnegatively correlated if sum j operatorname cov x 1 x j infty then the partial sum processes w n t defined in the usual way so that w n m n x 1 cdots x m me x 1 sqrt n for m 1 2 cdots converge in distribution on c lbrack 0 t rbrack to a wiener process this result is based on two general theorems concerning associated random variables which are of independent interest

the evolution of the random reversal graph genomes are typically represented as signed permutations where each integer corresponds to a single gene and its sign indicates the orientation of the gene in this paper we study genome rearrangements via reversals sankoff and blanchette 1999 20 flipping entire segments of genes and changing their orientations this abstraction leads to the notion of the reversal graph in which two signed permutations are neighbors if they differ by one reversal we identify the evolution of multiple genomes within the random reversal graph i e the probability space consisting of subgraphs over signed permutations obtained by selecting edges reversals with independent probability n which is an important framework for the analysis of genome rearrangements in the sense of relating the reversal rate n with the global evolution of genomes our main result shows that the structure of the random reversal graph changes dramatically at n 1 n 1 2 for n 1 n 1 2 the random graph consists of components of size at most o n ln n a s and for 1 n 1 2 there emerges a unique largest component of size 2 n n a s where is the survival probability of a certain branching process this giant component is furthermore dense in the reversal graph

simple regression in view of elliptical models abstract for the simple linear model y 1 x where the error vector follows the elliptically contoured distribution we consider the unrestricted restricted preliminary test and shrinkage estimators for the intercept parameter when it is suspected that the slope parameter may be o the exact bias and mse expressions are derived and the mean square relative efficiency is taken to determine the relative dominance properties of the proposed estimators in comparison in the continuation the optimal level of significance of the preliminary test estimator is tabulated and some graphical result are also displayed

cutoff for general spin systems with arbitrary boundary conditions the cutoff phenomenon describes a sharp transition in the convergence of a markov chain to equilibrium in recent work the authors established cutoff and its location for the stochastic ising model on the d dimensional torus n d for any d 1 the proof used the symmetric structure of the torus and monotonicity in an essential way here we enhance the framework and extend it to general geometries boundary conditions and external fields to derive a cutoff criterion that involves the growth rate of balls and the log sobolev constant of the glauber dynamics in particular we show there is cutoff for the stochastic ising model on any sequence of bounded degree graphs with subexponential growth under arbitrary external fields provided the inverse log sobolev constant is bounded for lattices with homogenous boundary such as all plus we identify the cutoff location explicitly in terms of spectral gaps of infinite volume dynamics on half plane intersections analogous results establishing cutoff are obtained for nonmonotone spin systems at high temperatures including the gas hard core model the potts model the antiferromagnetic potts model and the coloring model 2014 wiley periodicals inc