a latent class approach to fitting the weighted euclidean model clascal a weighted euclidean distance model for analyzing three way proximity data is proposed that incorporates a latent class approach in this latent class weighted euclidean model the contribution to the distance function between two stimuli is per dimension weighted identically by all subjects in the same latent class this model removes the rotational invariance of the classical multidimensional scaling model retaining psychologically meaningful dimensions and drastically reduces the number of parameters in the traditional indscal model the probability density function for the data of a subject is posited to be a finite mixture of spherical multivariate normal densities the maximum likelihood function is optimized by means of an em algorithm a modified fisher scoring method is used to update the parameters in the m step a model selection strategy is proposed and illustrated on both real and artificial data

avoiding degeneracy in multidimensional unfolding by penalizing on the coefficient of variation multidimensional unfolding methods suffer from the degeneracy problem in almost all circumstances most degeneracies are easily recognized the solutions are perfect but trivial characterized by approximately equal distances between points from different sets a definition of an absolutely degenerate solution is proposed which makes clear that these solutions only occur when an intercept is present in the transformation function many solutions for the degeneracy problem have been proposed and tested but with little success so far in this paper we offer a substantial modification of an approach initiated bythat introduced a normalization factor based on thevariance in the usual least squares loss function heiser unpublishedthesis 1981 and showed that the normalization factor proposed by kruskal and carroll was not strong enough to avoid degeneracies the factor proposed in the present paper based on the coefficient of variation discourages or penalizes nonmetric transformations of the proximities with small variation so that the procedure steers away from solutions with small variation in the interpoint distances an algorithm is described for minimizing the re adjusted loss function based on iterative majorization the results of a simulation study are discussed in which the optimal range of the penalty parameters is determined two empirical data sets are analyzed by our method clearly showing the benefits of the proposed loss function

the dependency of timbre on fundamental frequency the dependency of the timbre of musical sounds on their fundamental frequency f0 was examined in three experiments in experiment i subjects compared the timbres of stimuli produced by a set of 12 musical instruments with equal f0 duration and loudness there were three sessions each at a different f0 in experiment ii the same stimuli were rearranged in pairs each with the same difference in f0 and subjects had to ignore the constant difference in pitch in experiment iii instruments were paired both with and without an f0 difference within the same session and subjects had to ignore the variable differences in pitch experiment i yielded dissimilarity matrices that were similar at different f0 s suggesting that instruments kept their relative positions within timbre space experiment ii found that subjects were able to ignore the salient pitch difference while rating timbre dissimilarity dissimilarity matrices were symmetrical suggesting further that the absolute displacement of the set of i

ruin probability with claims modeled by a stationary ergodic stable process for a random walk with negative drift we study the exceedance probability ruin probability of a high threshold the steps of this walk claim sizes constitute a stationary ergodic stable process we study how ruin occurs in this situation and evaluate the asymptotic behavior of the ruin probability for a large variety of stationary ergodic stable processes our findings show that the order of magnitude of the ruin probability varies significantly from one model to another in particular ruin becomes much more likely when the claim sizes exhibit long range dependence the proofs exploit large deviation techniques for sums of dependent stable random variables and the series representation of a stable process as a function of a poisson process

a thurstonian pairwise choice model with univariate and multivariate spline transformations a probabilistic choice model is developed for paired comparisons data about psychophysical stimuli the model is based on thurstone s law of comparative judgment case v and assumes that each stimulus is measured on a small number of physical variables the utility of a stimulus is related to its values on the physical variables either by means of an additive univariate spline model or by means of multivariate spline model in the additive univariate spline model a separate univariate spline transformation is estimated for each physical dimension and the utility of a stimulus is assumed to be an additive combination of these transformed values in the multivariate spline model the utility of a stimulus is assumed to be a general multivariate spline function in the physical variables the use of b splines for estimating the transformation functions is discussed and it is shown how b splines can be generalized to the multivariate case by using as basis functions tensor products of the univariate basis functions a maximum likelihood estimation procedure for the thurstone case v model with spline transformation is described and applied for illustrative purposes to various artificial and real data sets finally the model is extended using a latent class approach to the case where there are unreplicated paired comparisons data from a relatively large number of subjects drawn from a heterogeneous population an em algorithm for estimating the parameters in this extended model is outlined and illustrated on some real data

restricted multidimensional scaling models for asymmetric proximities restricted multidimensional scaling models bentler weeks 1978 allowing constraints on parameters are extended to the case of asymmetric data separate functions are used to model the symmetric and antisymmetric parts of the data the approach is also extended to the case in which data are presumed to be linearly related to squared distances examples of several models are provided using journal citation data possible extensions of the models are considered

monotone spline transformations for dimension reduction consider a set of data consisting of measurements ofn objects with respect top variables displayed in ann p matrix a monotone transformation of the values in each column represented as a linear combination of integrated basis splines is assumed determined by a linear combination of a new set of values characterizing each row object two different models are used one an eckart young decomposition model and the other a multivariate normal model examples for artificial and real data are presented the results indicate that both methods are helpful in choosing dimensionality and that the eckart young model is also helpful in displaying the relationships among the objects and the variables also results suggest that the resulting transformations are themselves illuminating

large deviation estimates for exceedance times of perpetuity sequences and their dual processes in a variety of problems in pure and applied probability it is relevant to study the large exceedance probabilities of the perpetuity sequence yn b1 a1b2 a1 an 1 bnyn b1 a1b2 a1 an 1 bn where ai bi 0 r ai bi 0 r estimates for the stationary tail distribution of yn yn have been developed in the seminal papers of kesten acta math 131 1973 207 248 and goldie ann appl probab 1 1991 126 166 specifically it is well known that if m supnynm supnyn then p m u cmu p m u cmu as u u while much attention has been focused on extending such estimates to more general settings little work has been devoted to understanding the path behavior of these processes in this paper we derive sharp asymptotic estimates for the normalized first passage time tu logu 1inf n yn u tu log u 1inf n yn u we begin by showing that conditional on tu tu tu tu as u u for a certain positive constant we then provide a conditional central limit theorem for tu tu and study p tu g p tu g as u u for sets g 0 g 0 if g 0 g 0 then we show that p tu g ui g c g p tu g ui g c g as u u for a certain large deviation rate function ii and constant c g c g on the other hand if g g then we show that the tail behavior is actually quite complex and different asymptotic regimes are possible we conclude by extending our results to the corresponding forward process understood in the sense of letac in random matrices and their applications brunswick maine 1984 1986 263 273 amer math soc namely to the reflected process m n max anm n 1 bn 0 mn max anmn 1 bn 0 n z n z using siegmund duality we relate the first passage times of yn yn to the finite time exceedance probabilities of m n mn yielding a new result concerning the convergence of m n mn to its stationary distribution

precise large deviations for dependent regularly varying sequences we study a precise large deviation principle for a stationary regularly varying sequence of random variables this principle extends the classical results of nagaev theory probab appl 14 51 64 193 208 1969 and nagaev ann probab 7 745 789 1979 for iid regularly varying sequences the proof uses an idea of jakubowski stoch proc appl 44 291 327 1993 68 1 20 1997 in the context of central limit theorems with infinite variance stable limits we illustrate the principle for stochastic volatility models real valued functions of a markov chain satisfying a polynomial drift condition and solutions of linear and non linear stochastic recurrence equations

stable limits of martingale transforms with application to the estimation of garch parameters in this paper we study the asymptotic behavior of the gaussian quasi maximum likelihood estimator of a stationary garch process with heavy tailed innovations this means that the innovations are regularly varying with index 2 2 4 then in particular the marginal distribution of the garch process has infinite fourth moment and standard asymptotic theory with normal limits and p n rates breaks down this was recently observed by hall and yao econometrica 71 2003 285 317 it is the aim of this paper to indicate that the limit theory for the parameter estimators in the heavy tailed case nevertheless very much parallels the normal asymptotic theory in the light tailed case the limit theory is based on the clt for stationary ergodic finite variance martingale difference sequences in the heavy tailed case such a general result does not exist but an analogous result with infinite variance stable limits can be shown to hold under certain mixing conditions which are satisfied for garch processes it is the aim of the paper to give a general structural result for infinite variance limits which can also be applied in situations more general than garch

a quasi nonmetric method for multidimensional scaling via an extended euclidean model an extended two way euclidean multidimensional scaling mds model which assumes both common and specific dimensions is described and contrasted with the standard two way mds model in this extended two way euclidean model then stimuli or other objects are assumed to be characterized by coordinates onr common dimensions in addition each stimulus is assumed to have a dimension or dimensions specific to it alone the overall distance between objecti and objectj then is defined as the square root of the ordinary squared euclidean distance plus terms denoting the specificity of each object the specificity s j can be thought of as the sum of squares of coordinates on those dimensions specific to objecti all of which have nonzero coordinatesonly for objecti in practice we may think of there being just one such specific dimension for each object as this situation is mathematically indistinguishable from the case in which there are more than one we further assume that ij f d ij e ij where ij is the proximity value e g similarity or dissimilarity of objectsi andj d ij is the extended euclidean distance defined above whilee ij is an error term assumed i i d n 0 2 f is assumed either a linear function in the metric case or a monotone spline of specified form in the quasi nonmetric case a numerical procedure alternating a modified newton raphson algorithm with an algorithm for fitting an optimal monotone spline or linear function is used to secure maximum likelihood estimates of the paramstatistics can be used to test hypotheses about the number of common dimensions and or the existence of specific in addition tor common dimensions this approach is illustrated with applications to both artificial data and real data on judged similarity of nations

heavy tailed distributions correlations kurtosis and taylor s law of fluctuation scaling pillai meng pillai meng 2016 ann stat 44 20892097 p 2091 speculated that the dependence among random variables rvs can be overwhelmed by the heaviness of their marginal tails we giv

large deviations for solutions to stochastic recurrence equations under kesten s condition in this paper we prove large deviations results for partial sums constructed from the solution to a stochastic recurrence equation we assume kesten s condition acta math 131 1973 207 248 under which the solution of the stochastic recurrence equation has a marginal distribution with power law tails while the noise sequence of the equations can have light tails the results of the paper are analogs to those obtained by a v nagaev theory probab appl 14 1969 51 64 193 208 and s v nagaev ann probab 7 1979 745 789 in the case of partial sums of i i d random variables in the latter case the large deviation probabilities of the partial sums are essentially determined by the largest step size of the partial sum for the solution to a stochastic recurrence equation the magnitude of the large deviation probabilities is again given by the tail of the maximum summand but the exact asymptotic tail behavior is also influenced by clusters of extreme values due to dependencies in the sequence we apply the large deviation results to study the asymptotic behavior of the ruin probabilities in the model

cosmetic crossing changes of fibered knots an open question in classical knot theory is the question of when a crossing change on a knot changes the isotopy class of the knot the purpose of this paper is to answer this question for fibered knots a crossing disc for a knot k s3 is an embedded disc d s3 such that k intersects int d twice with zero algebraic intersection number a crossing change on k can be achieved by twisting d or equivalently by performing appropriate dehn surgery of s3 along the crossing circle d the crossing is called nugatory if and only if d bounds an embedded disc in the complement of k this disc and d form a 2 sphere that decomposes k into a connected sum where some of the summands may be trivial clearly changing a nugatory crossing doesn t change the isotopy class of a knot problem 1 58 of 8 asks whether the converse is true see also 19 for related conjectures that is if a crossing change on a knot k yields a knot isotopic to k is the crossing nugatory in the case that k is the trivial knot an affirmative answer follows from work of gabai 7 an affirmative answer is also known in the case of 2 bridge knots 19 in this paper we will show the following

clustering of critical points in lefschetz fibrations and the symplectic szpiro inequality we prove upper bounds for the number of critical points in semi stable symplectic lefschetz fibrations we also obtain a new lower bound for the number of nonseparating vanishing cycles in lefschetz pencils and reprove the known lower bounds for the commutator lengths of dehn twists

the limit distribution of the maximum increment of a random walk with dependent regularly varying jump sizes we investigate the maximum increment of a random walk with heavy tailed jump size distribution here heavy tailedness is understood as regular variation of the finite dimensional distributions the jump sizes constitute a strictly stationary sequence using a continuous mapping argument acting on the point processes of the normalized jump sizes we prove that the maximum increment of the random walk converges in distribution to a frechet distributed random variable

point process and partial sum convergence for weakly dependent random variables with infinite variance let j be a strictly stationary sequence of random variables with regularly varying tail probabilities we consider via point process methods weak convergence of the partial sums s n 1 n suitably normalized when j satisfies a mild mixing condition we first give a characterization of the limit point processes for the sequence of point processes n n with mass at the points j a n j 1 n where a n is the 1 n 1 quantile of the distribution of 1 then for 0 t n 0 p s n t n np 1 t n tends to a constant which can in general be different from 1 applications of our main results to self norming sums m dependent sequences and linear processes are also given

analysis of pairwise preference data using integrated b splines pairwise preference data are represented as a monotone integral transformation of difference on the underlying stimulus object or utility scale the class of monotone transformations considered is that in which the kernel of the integral is a linear combination of b splines two types of data are analyzed binary and continuous the parameters of the transformation and the underlying scale values or utilities are estimated by maximum likelihood with inequality constraints on the transformation parameters various hypothesis tests and interval estimates are developed examples of artificial and real data are presented

large deviations for random walks under subexponentiality the big jump domain for a given one dimensional random walk sn with a subexponential step size distribution we present a unifying theory to study the sequences xn for which as n uniformly for x xn we also investigate the stronger local analogue our theory is self contained and fits well within classical results on domains of partial attraction and local limit theory when specialized to the most important subclasses of subexponential distributions that have been studied in the literature we reproduce known theorems and we supplement them with new results

large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy tailed innovations in this paper we consider the stochastic recurrence equation y t a t y t 1 b t for an i i d sequence of pairs a t b t of nonnegative random variables where we assume that b t is regularly varying with index k 0 and ea k t x c 1 np y 1 x as x holds for some constant c 1 0 for k 1 we also study the large deviation probabilities p s n es n x x x n for some sequence x n whose growth depends on the heaviness of the tail of the distribution of y 1 we show that the relation p s n es n x c 2 np y 1 x holds uniformly for x x n and some constant c 2 0 then we apply the large deviation results to derive bounds for the ruin probability u p sup n 1 s n es n n u for any 0 we show that u c 3 up y 1 u 1 1 1 for some constant c 3 0 in contrast to the case of i i d regularly varying y t s when the above results hold with c 1 c 2 c 3 1 the constants c 1 c 2 and c 3 are different from 1

a functional limit theorem for dependent sequences with infinite variance stable limits under an appropriate regular variation condition the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non gaussian stable random variable a functional version of this is known to be true as well the limit process being a stable levy process the main result in the paper is that for a stationary regularly varying sequence for which clusters of high threshold excesses can be broken down into asymptotically independent blocks the properly centered partial sum process still converges to a stable levy process due to clustering the levy triple of the limit process can be different from the one in the independent case the convergence takes place in the space of cadlag functions endowed with skorohod s m 1 topology the more usual j 1 topology being inappropriate as the partial sum processes may exhibit rapid successions of jumps within temporal clusters of large values collapsing in the limit to a single jump the result rests on a new limit theorem for point processes which is of independent interest the theory is applied to moving average processes squared garch 1 1 processes and stochastic volatility models

sw gr from the seiberg witten equations to pseudo holomorphic curves the purpose of this article is to explain how pseudo holomorphic curves in a symplectic 4 manifold can be constructed from solutions to the seiberg witten equations as such the main theorem proved here theorem 1 3 is an existence theorem for pseudo holomorphic curves this article thus provides a proof of roughly half of the main theorem in the announcement t1 that theorem theorem 4 1 asserts an equivalence between the seiberg witten invariants for a symplectic manifold and a certain gromov invariant which counts with signs the number of pseudoholomorphic curves in a given homology class the seiberg witten invariants were introduced to mathematicians by witten w based on his joint work with nat seiberg sw1 sw2 a description of these invariants is given in section 1 see also km1 t1 suffice it to say here that when x is a compact oriented 4 dimensional manifold with

every mapping class group is generated by 6 involutions abstract let mod g b denote the mapping class group of a surface of genus g with b punctures luo asked in torsion elements in the mapping class group of a surface math gt 0004048 v1 8apr2000 if there is a universal upper bound independent of genus for the number of torsion elements needed to generate mod g b we answer luo s question by proving that 3 torsion elements suffice to generate mod g 0 we also prove the more delicate result that there is an upper bound independent of genus not only for the number of torsion elements needed to generate mod g b but also for the order of those elements in particular our main result is that 6 involutions i e orientation preserving diffeomorphisms of order two suffice to generate mod g b for every genus g 3 b 0 and g 4 b 1

bounded cohomology and non uniform perfection of mapping class groups using the existence of certain symplectic submanifolds in symplectic 4 manifolds we prove an estimate from above for the number of singular fibers with separating vanishing cycles in minimal lefschetz fibrations over surfaces of positive genus this estimate is then used to deduce that mapping class groups are not uniformly perfect and that the map from their second bounded cohomology to ordinary cohomology is not injective

graphical representations and odds ratios in a distance association model for the analysis of cross classified data although rc m association models have become a generally useful tool for the analysis of cross classified data the graphical representation resulting from such an analysis can at times be misleading the relationships present between row category points and column category points cannot be interpreted by inter point distances but only through projection in order to avoid incorrect interpretation by a distance rule joint plots should be made that either represent the row categories or the column categories as vectors in contrast the present study proposes models in which the distances between row and column points can be interpreted directly with a large small distance corresponding to a small large value for the association the models provide expressions for the odds ratios in terms of distances which is a feature that makes the proposed models attractive reparametrizations to the usual rc m parametrization comparisons to existing data analysis techniques plus an overview of related models and their connections are also provided

limit theory for the sample autocorrelations and extremes of a garch 1 1 process the asymptotic theory for the sample autocorrelations and extremes of a garch i 1 process is provided special attention is given to the case when the sum of the arch and garch parameters is close to 1 that is when one is close to an infinite variance marginal distribution this situation has been observed for various financial log return series and led to the introduction of the igarch model in such a situation the sample autocorrelations are unreliable estimators of their deterministic counterparts for the time series and its absolute values and the sample autocorrelations of the squared time series have nondegenerate limit distributions we discuss the consequences for a foreign exchange rate series

stable limits for sums of dependent infinite variance random variables the aim of this paper is to provide conditions which ensure that the affinely transformed partial sums of a strictly stationary process converge in distribution to an infinite variance stable distribution conditions for this convergence to hold are known in the literature however most of these results are qualitative in the sense that the parameters of the limit distribution are expressed in terms of some limiting point process in this paper we will be able to determine the parameters of the limiting stable distribution in terms of some tail characteristics of the underlying stationary sequence we will apply our results to some standard time series models including the garch 1 1 process and its squares the stochastic volatility models and solutions to stochastic recurrence equations

sections of surface bundles and lefschetz fibrations we investigate the possible self intersection numbers for sections of surface bundles and lefschetz fibrations over surfaces when the fiber genus g and the base genus h are positive we prove that the adjunction bound 2h 2 is the only universal bound on the self intersection number of a section of any such genus g bundle and fibration as a side result in the mapping class group of a surface with boundary we calculate the precise value of the commutator lengths of all powers of a dehn twist about a boundary component concluding that the stable commutator length of such a dehn twist is 1 2 we furthermore prove that there is no upper bound on the number of critical points of genus g lefschetz fibrations over surfaces with positive genera admitting sections of maximal self intersection for g 2

confidence regions for indscal using the jackknife and bootstrap techniques bootstrap and jackknife techniques are used to estimate ellipsoidal confidence regions of group stimulus points derived from indscal the validity of these estimates is assessed through monte carlo analysis asymptotic estimates of confidence regions based on a multiscale solution are also evaluated our findings suggest that the bootstrap and jackknife techniques may be used to provide statements regarding the accuracy of the relative locations of points in space our findings also suggest that multiscale asymptotic estimates of confidence regions based on small samples provide an optimistic view of the actual statistical reliability of the solution

multidimensional scaling models for reaction times and same different judgments a method for joint analysis of reaction times and same different judgments is discussed a set of stimuli is assumed to have some parametric representation which uniquely defines dissimilarities between the stimuli those dissimilarities are then related to the observed reaction times and same different judgments through a model of psychological processes three representation models of dissimilarities are considered the minkowski power distance model the linear model and tversky s feature matching model maximum likelihood estimation procedures are developed and implemented in the form of a fortran program an example is given to illustrate the kind of analyses that can be performed by the proposed method

a large deviations approach to limit theory for heavy tailed time series in this paper we propagate a large deviations approach for proving limit theory for generally multivariate time series with heavy tails we make this notion precise by introducing regularly varying time series we provide general large deviation results for functionals acting on a sample path and vanishing in some neighborhood of the origin we study a variety of such functionals including large deviations of random walks their suprema the ruin functional and further derive weak limit theory for maxima point processes cluster functionals and the tail empirical process one of the main results of this paper concerns bounds for the ruin probability in various heavy tailed models including garch stochastic volatility models and solutions to stochastic recurrence equations

the sample autocorrelations of heavy tailed processes with applications to arch we study the sample acvf and acf of a general stationary sequence under a weak mixing condition and in the case that the marginal distributions are regularly varying this includes linear and bilinear processes with regularly varying noise and arch processes their squares and absolute values we show that the distributional limits of the sample acf can be random provided that the variance of the marginal distribution is infinite and the process is nonlinear this is in contrast to infinite variance linear processes if the process has a finite second but infinite fourth moment then the sample acp is consistent with scaling rates that grow at a slower rate than the standard root n consequently asymptotic confidence bands are wider than those constructed in the classical theory we demonstrate the theory in full detail far an arch 1 process

an invariance principle for sums and record times of regularly varying stationary sequences we prove a sequence of limiting results about weakly dependent stationary and regularly varying stochastic processes in discrete time after deducing the limiting distribution for individual clusters of extremes we present a new type of point process convergence theorem it is designed to preserve the entire information about the temporal ordering of observations which is typically lost in the limit after time scaling by going beyond the existing asymptotic theory we are able to prove a new functional limit theorem its assumptions are satisfied by a wide class of applied time series models for which standard limiting theory in the space d of cadlag functions does not apply to describe the limit of partial sums in this more general setting we use the space e of so called decorated cadlag functions we also study the running maximum of partial sums for which a corresponding functional theorem can be still expressed in the familiar setting of space d we further apply our method to analyze record times in a sequence of dependent stationary observations even when their marginal distribution is not necessarily regularly varying under certain restrictions on dependence among the observations we show that the record times after scaling converge to a relatively simple compound scale invariant poisson process

the cluster index of regularly varying sequences with applications to limit theory for functions of multivariate markov chains we introduce the cluster index of a multivariate stationary sequence and characterize the index in terms of the spectral tail process this index plays a major role in limit theory for partial sums of sequences we illustrate the use of the cluster index by characterizing infinite variance stable limit distributions and precise large deviation results for sums of multivariate functions acting on a stationary markov chain under a drift condition

on the asymptotic distributions of partial sums of functionals of infinite variance moving averages this paper investigates the asymptotic distribution of the partial sum s n sum n 1 n k x n ek x n as n to infty where x n is a moving average stable process and k is a bounded and measurable function the results show that s n follows a central or non central limit theorem depending on the rate at which the moving average coefficients tend to 0