a symmetric approach to cyclization for systems over c y abstract the cyclization problem from the theory of systems over rings is transformed to a symmetric equivalent problem which is remarkably easier and leads to a solution in the 4 dimensional case for c y the complex polynomial ring

feedback cyclization over certain principal ideal domains definition de classes de paires atteignables sur des domaines principaux ideaux certains pour lesquelles la cyclisation de la retroaction est possible ceci conduit a une condition necessaire pour la cyclisation de la boucle de retour sous forme de resolution de quelques equations algebriques

exact partition functions for potts antiferromagnets on cyclic lattice strips abstract we present exact calculations of the zero temperature partition function of the q state potts antiferromagnet on arbitrarily long strips of the square triangular and kagome lattices with width l y 2 or 3 vertices and with periodic longitudinal boundary conditions from these in the limit of infinite length we obtain the exact ground state entropy s 0 k b ln w these results are of interest since this model exhibits nonzero ground state entropy s 0 0 for sufficiently large q and hence is an exception to the third law of thermodynamics we also include results for homeomorphic expansions of the square lattice strip the analytic properties of w q are determined and related to zeros of the chromatic polynomial for long finite strips

ground state entropy of the potts antiferromagnet on strips of the square lattice abstract we present exact solutions for the zero temperature partition function chromatic polynomial p and the ground state degeneracy per site w exponent of the ground state entropy for the q state potts antiferromagnet on strips of the square lattice of width ly vertices and arbitrarily great length lx vertices the specific solutions are for a ly 4 fbcy pbcx cyclic b ly 4 fbcy tpbcx mobius c ly 5 6 pbcy fbcx cylindrical and d l y 5 fbc y fbc x open where fbc pbc and tpbc denote free periodic and twisted periodic boundary conditions respectively in the lx limit of each strip we discuss the analytic structure of w in the complex q plane the respective w functions are evaluated numerically for various values of q several inferences are presented for the chromatic polynomials and analytic structure of w for lattice strips with arbitrarily great ly the absence of a nonpathological lx limit for real nonintegral q in the interval 0 0 for strips of the square triangular lattice is discussed

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

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

matrix cyclization over complex polynomials abstract let a be an n n b an n m complex polynomial matrix the following open problem is investigated does there exist a complex polynomial m n matrix f such that a bf has a cyclic vector in the image of b an explicit solution of the problem is given for the following generic situation n 4 b ab a n 1 b is rightinvertible necessary the entries of a specific normalized form of a b do not satisfy certain polynomial equations and one specific entry has at least one simple zero outside a certain finite set the exceptional equations are given explicitely and can easily be checked for

oscillatory critical amplitudes in hierarchical models we study the oscillatory critical amplitudes of theq states potts model on a diamond hierarchical lattice we consider an example of the generic case finite critical index as well as the degenerate case essential singularity in both cases we compare the magnitude of the oscillations with geometrical characteristics of the julia set of zeroes of the partition function

exact potts model partition functions on wider arbitrary length strips of the square lattice abstract we present exact calculations of the partition function of the q state potts model for general q and temperature on strips of the square lattice of width l y 3 vertices and arbitrary length l x with periodic longitudinal boundary conditions of the following types i fbc y pbc x cyclic ii fbc y tpbc x mobius iii pbc y pbc x toroidal and iv pbc y tpbc x klein bottle where fbc and t pbc refer to free and twisted periodic boundary conditions results for the l y 2 torus and klein bottle strips are also included in the infinite length limit the thermodynamic properties are discussed and some general results are given for low temperature behavior on strips of arbitrarily great width we determine the submanifold in the c 2 space of q and temperature where the free energy is singular for these strips our calculations are also used to compute certain quantities of graph theoretic interest

a note on feedback cyclicity of c y abstract let a b be an n dimensional linear system with 2 inputs over c y the ring of polynomials in one variable over the field of complex numbers we prove the feedback cyclicity of a b under certain conditions on their entries and deduce that a b is feedback cyclic in an exceptional case left open in w schmale linear algebra appl 275 276 1998 551 562

locally adaptive hazard smoothing we consider nonparametric estimation of hazard functions and their derivatives under random censorship based on kernel smoothing of the nelson 1972 estimator one critically important ingredient for smoothing methods is the choice of an appropriate bandwidth since local variance of these estimates depends on the point where the hazard function is estimated and the bandwidth determines the trade off between local variance and local bias data based local bandwidth choice is proposed a general principle for obtaining asymptotically efficient data based local bandwiths is obtained by means of weak convergence of a local bandwidth process to a gaussian limit process several specific asymptotically efficient bandwidth estimators are discussed we propose in particular an asymptotically efficient method derived from direct pilot estimators of the hazard function and of the local mean squared error this bandwidth choice method has practical advantages and is also of interest in the uncensored case as well as for density estimation

asymptotics of the susceptibility for the ising model on the hierarchical lattices using the gks inequality griffiths and kaufman showed in 12 that the susceptibility in the ising model on the hierarchical lattices is infinite fort tc in the present work we give an analytic calculation of the dominant singularity of the susceptibility whenh 0 fort tc moreover we analytically find the spontaneous magnetization fort tc and prove that fort tc the susceptibility is finite

phase transitions on fractals iii infinitely ramified lattices for pt ii see ibid vol 17 p 435 1984 in the first two papers of this series the authors considered self similar fractal lattices with a finite order of ramification r in the present paper they study physical models defined on a family of fractals with r infinity in order to characterise the geometry of these systems they need the connectivity q and the lacunarity l in addition to the fractal dimensionality d it is found that discrete symmetry spin models on these lattices undergo a phase transition at tc 0 an approximate renormalisation group scheme is constructed and used to find the dependence of tc and the critical exponents on the geometrical factors they also solve the problem of resistor networks on these fractals and discuss its consequences concerning spin models with continuous symmetry

chromatic polynomials for families of strip graphs and their asymptotic limits abstract we calculate the chromatic polynomials p gs m q and from these the asymptotic limiting functions w g s q lim n p g s q 1 n for families of n vertex graphs gs m comprised of m repeated subgraphs h adjoined to an initial graph i these calculations of w gs q for infinitely long strips of varying widths yield important insights into properties of w q for two dimensional lattices in turn these results connect with statistical mechanics since w q is the ground state degeneracy of the q state potts model on the lattice for our calculations we develop and use a generating function method which enables us to determine both the chromatic polynomials of finite strip graphs and the resultant w gs q function in the limit n from this we obtain the exact continuous locus of points b where w gs q is nonanalytic in the complex q plane this locus is shown to consist of arcs which do not separate the q plane into disconnected regions zeros of chromatic polynomials are computed for finite strips and compared with the exact locus of singularities b we find that as the width of the infinitely long strips is increased the arcs comprising b elongate and move toward each other which enables one to understand the origin of closed regions that result for the infinite 2d lattice

a mixture model for time to aids data with left truncation and an uncertain origin summary the use of a mixture model to estimate the probability of developing aids after hiv infection and the distribution of incubation time is investigated the possibility of left truncation and an uncertain time of infection is incorporated into the analysis the efficiency losses associated with uncertain infection times are investigated an example is given

ground state entropy of the potts antiferromagnet on cyclic strip graphs we present exact calculations of the zero temperature partition function chromatic polynomial and the exponent of the ground state entropy s0 for the q state potts antiferromagnet on families of cyclic and twisted cyclic mobius strip graphs composed of p sided polygons our results suggest a general rule concerning the maximal region in the complex q plane to which one can analytically continue from the physical interval where s0 0 the chromatic zeros and their accumulation set exhibit the rather unusual property of including support for re q 0 and provide further evidence for a relevant conjecture

a note on the product limit estimator under right censoring and left truncation summary in many applications involving follow up studies individuals lifetimes may be subject to left truncation in addition to the usual right censoring here we consider the product limit estimator of the survival curve s or an appropriate conditional version of s the asymptotic behaviour of the estimator is briefly described together with an example

fractal structure of zeros in hierarchical models we consider an ising and aq state potts model on a diamond hierarchical lattice we give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq these zeros are just the julia set corresponding to the renormalization group transformation

t 0 partition functions for potts antiferromagnets on square lattice strips with twisted periodic boundary conditions we present exact calculations of the zero temperature partition function for the q state potts antiferromagnet or equivalently the chromatic polynomial for two families of arbitrarily long strip graphs of the square lattice with periodic boundary conditions in the transverse direction and i periodic and ii twisted periodic boundary conditions in the longitudinal direction so that the strip graphs are embedded on i a torus and ii a klein bottle in the limit of infinite length we calculate the exponent of the entropy w q show it to be the same for i and ii and determine its analytic structure

phase transitions on fractals ii sierpinski gaskets for pt i see ibid vol 16 p 1267 1983 the authors construct and investigate a family of fractals which are generalisations of the sierpinski gaskets sgs to all euclidean dimensionalities these fractal lattices have a finite order of ramification and can be considered marginal between one dimensional and higher dimensional geometries physical models defined on them are exactly solvable the authors argue that short range spin models on the sg show no finite temperature phase transitions as examples they solve a few spin models and study the resistor network and percolation problems on these lattices

spanning trees on the sierpinski gasket we present the numbers of spanning trees on the sierpinski gasket sg d n at stage n with dimension d equal to two three and four the general expression for the number of spanning trees on sg d n with arbitrary d is conjectured the numbers of spanning trees on the generalized sierpinski gasket sg d b n with d 2 and b 3 4 are also obtained

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

julia sets and complex singularities in hierarchical ising models we study the analytical continuation in the complex plane of free energy of the ising model on diamond like hierarchical lattices it is known 12 13 that the singularities of free energy of this model lie on the julia set of some rational endomorphismf related to the action of the migdal kadanoff renorm group we study the asymptotics of free energy when temperature goes along hyperbolic geodesics to the boundary of an attractive basin off we prove that for almost all with respect to the harmonic measure geodesics the complex critical exponent is common and compute it

bivariate estimation with right truncated data abstract bivariate estimation with survival data has received considerable attention recently however most of the work has focused on random censoring models another common feature of survival data random truncation is considered in this study truncated data may arise if the time origin of the events under study precedes the observation period in a random right truncation model one observes the iid samples of y t only if y t where y is the variable of interest and t is an independent variable that prevents the complete observation of y suppose that y x is a bivariate vector of random variables where y is subject to right truncation in this study the bivariate reverse hazard vector is introduced and a nonparametric estimator is suggested an estimator for the bivariate survival function is also proposed weak convergence and strong consistency of this estimator are established via a representation by iid variables an expression for the limiting covariance function is provided and a

the estimation of the hazard function from randomly censored data by the kernel method by convolution smoothing of the empirical hazards a kernel estimate of the hazard function from censored data is obtained small and large sample expressions for the mean and the variance of the estimator are given conditions for asymptotic normality are investigated using the hajek projection method

biometrika centenary nonparametrics summary contributions in biometrika to nonparametric statistics are surveyed with particular emphasis on more recent work it is argued that even in the journal s very early years it had interests in smoothing methods for density estimation and in techniques for nonparametric regression in addition to its more obvious focus on classical nonparametric statistics such as ranking methods biometrika s early involvement with permutation testing has links to its contemporary work on bootstrap methods emphases on different types of nonparametric statistics have evolved over time however a rapidly increasing proportion of the journal s pages is devoted to papers that include some use of nonparametric ideas or methods

ground state entropy of the potts antiferromagnet with next nearest neighbor spin spin couplings on strips of the square lattice we present exact calculations of the zero temperature partition function chromatic polynomial and w q the exponent of the ground state entropy for the q state potts antiferromagnet with next nearest neighbor spin spin couplings on square lattice strips of width l y 3 and l y 4 vertices and arbitrarily great length l x vertices with both free and periodic boundary conditions the resultant values of w for a range of physical q values are compared with each other and with the values for the full two dimensional lattice these results give insight into the effect of such nonnearest neighbor couplings on the ground state entropy we show that the q 2 ising and q 4 potts antiferromagnets have zero temperature critical points on the l x ensuremath rightarrow ensuremath infty limits of the strips that we study with the generalization of q from z to c we determine the analytic structure of w q in the q plane for the various cases

potts models on hierarchical lattices and renormalization group dynamics ii examples and numerical results we obtain the exact renormalization map and plots of lee yang and fisher zeros distributions for potts models on a number of hierarchical lattices the diamond hierarchical lattice a lattice we call spider web the sierpinski gasket and cylinders such models are only examples among those we can study in the general framework of hierarchical lattices developed in a previous paper

biometrika centenary survival analysis although papers relating to survival analysis have appeared in biometrika since the very first issue the current emphasis on the topic in this journal is fairly new especially as it relates to applications in biostatistics this survey traces the development of survival analysis throughout the twentieth century as reflected in the pages of biometrika focusing primarily on work published since 1980

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

exact potts model partition function on strips of the triangular lattice abstract in this paper 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 strip graphs of width ly 2 and arbitrary length of the triangular lattice 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 the graphs and the thermodynamics is discussed considering the full generalization to arbitrary complex q and temperature we determine the singular locus 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 comparisons are made with our previous exact calculation of potts model partition functions for the corresponding strips of the square lattice our present calculations yield as special cases several quantities of graph theoretic interest

q plane zeros of the potts partition function on diamond hierarchical graphs we report exact results concerning the zeros of the partition function of the potts model in the complex q plane as a function of a temperature like boltzmann variable v for the m th iterate graphs dm of the diamond hierarchical lattice including the limit m in this limit we denote the continuous accumulation locus of zeros in the q planes at fixed v v0 as bq v0 we apply theorems from complex dynamics to establish the properties of bq v0 for v 1 the zero temperature potts antiferromagnet or equivalently chromatic polynomial we prove that bq 1 crosses the real q axis at i a minimal point q 0 ii a maximal point q 3 iii q 32 27 iv a cubic root that we give with the value q q1 1 638 896 9 and v an infinite number of points smaller than q1 converging to 32 27 from above similar results hold for bq v0 for any 1 0 potts ferromagnet we also provide the computer generated plots of bq v0 at various values of v0 in both the antiferromagnetic and ferromagnetic regimes and compare them to the numerically computed zeros of z d4 q v0

estimating a distribution function with truncated data let mathscr p be a finite population with n geq 1 elements for each e in mathscr p let x e and y e be independent positive random variables with unknown distribution functions f and g and suppose that the pairs x e y e are i i d we consider the problem of estimating f g and n when the data consist of those pairs x e y e for which e in mathscr p and y e leq x e the nonparametric maximum likelihood estimators mles of f and g are described and their asymptotic properties as n rightarrow infty are derived it is shown that the mles are consistent against pairs f g for which f and g are continuous g 1 0 leq f 1 0 and g 1 1 leq f 1 1 sqrt n times estimation error for f converges in distribution to a gaussian process if int infty 0 1 g df infty but may fail to converge if this integral is infinite

continuity of julia set and its hausdorff dimension of yang lee zeros abstract we show the continuity of the julia set and its hausdorff dimension about a family of rational maps concerning 2 dimensional diamond hierarchical potts models about anti ferromagnetic coupling in statistical mechanics

on connectivity of julia sets of yang lee zeros we prove that the julia sets of the yang lee zeros of the potts model on the diamond hierarchical lattice are connected sets on the complex plane

ground state degeneracy of potts antiferromagnets on two dimensional lattices approach using infinite cyclic strip graphs the q state potts antiferromagnet on a lattice lambda exhibits nonzero ground state entropy s0 kb ln w for sufficiently large q and hence is an exception to the third law of thermodynamics an outstanding challenge has been the calculation of w sq q on the square sq lattice we present here an exact calculation of w on an infinite length cyclic strip of the square lattice which embodies the expected analytic properties of w sq q similar results are given for the kagome lattice

hazard function estimators a simulation study kernel based methods for the smooth non parametric estimation of the hazard function have received considerable attention in the statistical literature although the mathematical properties of the kernel based hazard estimators have been carefully studied their statistical properties have not we reviewed various kernel based methods for hazard function estimation from right censored data and compared the statistical properties of these estimators through computer simulations our simulations covered seven distributions three levels of random censoring four types of bandwidth functions two sample sizes and three types of boundary correction we conducted a total of 504 simulation experiments with 500 independent samples each our results confirmed the advantages of two recent innovations in kernel estimation boundary correction and locally optimal bandwidths the median relative improvement decrease in mean square error over fixed bandwidth estimators without boundary correction was 3 per cent for fixed bandwidth estimators with left boundary correction 52 per cent locally optimal bandwidths without boundary correction and 66 per cent for locally optimal bandwidths with left boundary correction the locally optimal bandwidth estimators with left boundary correction also outperformed three previously published and publicly available algorithms with median relative improvements in mean square error of 31 per cent 77 per cent and 80 per cent

feedback cyclization for rings with finite stable range abstract for systems over commutative rings we introduce a property called fc s s 0 which means feedback cyclization with s inputs given a controllable system a b there exist a feedback matrix k and a matrix u with s columns such that a bk bu is controllable clearly fc 1 is the usual fc property the main result of this paper is the following for a ring r with the gcu property whenever a b is controllable there exists a vector u with bu unimodular r satisfies a strong form of the fc s property if and only if r is s stable i e r has s in its stable range this generalizes the known facts that 1 stable gcu rings have the fc property and principal ideal domains which are 2 stable gcu rings satisfy an analogous cyclization property with 2 inputs examples are given of fc s rings for s 1 which are not fc rings

t 0 partition functions for potts antiferromagnets on lattice strips with fully periodic boundary conditions abstract we present exact calculations of the zero temperature partition function for the q state potts antiferromagnet equivalently the chromatic polynomial for families of arbitrarily long strip graphs of the square and triangular lattices with width ly 4 and boundary conditions that are doubly periodic or doubly periodic with reversed orientation i e of torus or klein bottle type these boundary conditions have the advantage of removing edge effects in the limit of infinite length we calculate the exponent of the entropy w q and determine the continuous locus b where it is singular we also give results for toroidal strips involving crossing subgraphs these make possible a unified treatment of torus and klein bottle boundary conditions and enable us to prove that for a given strip the locus b is the same for these boundary conditions

three dimensional feedback cyclization over c y abstract this paper answers in the affirmative the following open question for 3 dimensional linear systems over the ring of complex polynomials in one variable if the system is reachable does there exist a feedback that makes it reachable from a single input channel the result given remains valid for slightly more general coefficient domains

cyclizable matrix pairs over c x and a conjecture on toplitz pencils abstract the 22 year old conjecture is addressed which claims that any reachable matrix pair a b from c x n n m is cyclizable i e allows for f u over c x s t a bf k bu k 0 n 1 is a basis for c y n it is shown that for a whole class of pairs the correctness of the conjecture is a consequence of the correctness of a conjecture on certain toplitz pencils an algebraic computable test for the validity of the latter is given based on these results the validity of the conjecture on cyclizability can be extended up to dimension 5

pole assignment for systems over rings a very simple proof of the pole assignment theorem for systems over a principal ideal domain and other rings is given furthermore an algorithm is presented extensions are also indicated

pole assignability in polynomial rings power series rings and prufer domains the topics treated in this paper have their origins in the area of algebraic systems theory however the paper itself should be classified as pure commutative algebra and we shall present it as such in the body of the text still it is appropriate to give a brief paragraph of motivation if a physical system is governed by a pair f g of matrices then the stability of the system can be determined by examining the eigenvalues of the matrix f if the system is unstable a feedback matrix k can sometimes be employed in such a way that the eigenvalues of the matrix f gk measure the stability of the modified system f gk g in this manner an unstable system can be rendered stable the pole assignability problem over commutative rings is one method of attacking the problem of finding such matrices k the paper is divided into four sections section 1 is given over almost entirely to defining the properties in which we shall be interested it concludes with a theorem about residuating and lifting the properties section 2 is concerned with the preservation of the properties under polynomial ring and power series ring formation section 3 is concerned with feedback cyclization a strong form of pole assignability section 4 is concerned with pole assignability over priifer domains a complete elaboration of our results must await the introduction of the necessary terminology for now we mention the following results in somewhat vague language if r is a zero dimensional ring then the pole assignability problem is solvable in r x and in r xl from a ringtheoretic standpoint almost any class of commutative rings contains members for which the feedback cyclization problem is solvable on the other hand if r is a ring with 1 in its stable range then the feedback cyclization problem is solvable in r if and only if a certain nice matricial property

phase transitions on fractals i quasi linear lattices magnetic spin models and resistor networks are studied on certain self similar fractal lattices which are described as quasi linear because they share a significant property of the line finite portions can be isolated from the rest by removal of two points sites in all cases there is no long range order at finite temperature the transition at zero temperature has a discontinuity in the magnetisation and the associated magnetic exponent is equal to the fractal dimensionality d when the lattice reduces to a non branching curve the thermal exponent v 1 y is equal to d when the lattice is a branching curve y is related respectively to the dimensionality of the single channel segments of the curve for the ising model or to the exponent describing the resistivity for models with continuous spin symmetry

a comparison of hazard rate estimators for left truncated and right censored data summary left truncation and right censoring arise frequently in practice for life data this paper is concerned with the estimation of the hazard rate function for such data two types of nonparametric estimators based on kernel smoothing methods are considered the first one is obtained by convolving a kernel with a cumulative hazard estimator the second one is in the form of a ratio of two statistics local properties including consistency asymptotic normality and mean squared error expressions are presented for both estimators these properties facilitate locally adaptive bandwidth choice the two types of estimators are then compared based on their theoretical and empirical performances the effect of overlooking the truncation factor is demonstrated through the channing house data