on properly embedded minimal surfaces with three ends we classify all complete embedded minimal surfaces in r with three ends of genus g and at least 2g 2 symmetries the surfaces in this class are the costa hoffman meeks surfaces which have 4g 4 symmetries in the case of a flat middle end the proof consists of using the symmetry assumptions to deduce the possible weierstras data and then to study the period problems in all cases to handle the 1 dimensional period problems we develop a new general method to prove convexity results for period quotients the 2 dimensional period problems are reduced to the 1 dimensional case by an extremal length argument 1991 mathematics subject classification primary 53a10 secondary 53c42

inference for multiple change points in time series via likelihood ratio scan statistics summary we propose a likelihood ratio scan method for estimating multiple change points in piecewise stationary processes using scan statistics reduces the computationally infeasible global multiple change point estimation problem to a number of single change point detection problems in various local windows the computation can be efficiently performed with order o npt log n consistency for the estimated numbers and locations of the change points are established moreover a procedure is developed for constructing confidence intervals for each of the change points simulation experiments and real data analysis are conducted to illustrate the efficiency of the likelihood ratio scan method

a low dimension portmanteau test for non linearity abstract a new test for non linearity in the conditional mean is proposed using functions of the principal components of regressors the test extends the non linearity tests based on kolmogorov gabor polynomials thursby and schmidt 1977 tsay 1986 terasvirta et al 1993 but circumvents problems of high dimensionality is equivariant to collinearity and includes exponential functions so is a portmanteau test with power against a wide range of possible alternatives a monte carlo analysis compares the performance of the test to the optimal infeasible test and to alternative tests the relative performance of the test is encouraging the test has the appropriate size and has high power in many situations

transitions of the 3d medial axis under a one parameter family of deformations the instabilities of the medial axis of a shape under deformations have long been recognized as a major obstacle to its use in recognition and other applications these instabilities or transitions occur when the structure of the medial axis graph changes abruptly under deformations of shape the recent classification of these transitions in 2d for the medial axis and for the shock graph was a key factor in the development of an object recognition system where the classified instabilities were utilized to represent deformation paths the classification of generic transitions of the 3d medial axis could likewise potentially lead to a similar representation in 3d in this paper these transitions are classified by examining the order of contact of spheres with the surface leading to an enumeration of possible transitions which are then examined on a case by case basis some cases are ruled out as never occurring in any family of deformations while others are shown to be nongeneric in a one parameter family of deformations finally the remaining cases are shown to be viable by developing a specific example for each our work is inspired by that of bogaevsky who obtained the transitions as part of an investigation of viscosity solutions of hamilton jacobi equations our contribution is to give a more down to earth approach bringing this work to the attention of the computer vision community and to provide explicit constructions for the various transitions using simple surfaces we believe that the classification of these transitions is vital to the successful regularization of the medial axis in its use in real applications

testing time reversibility without moment restrictions in this paper we propose a class of new tests for time reversibility it is shown that this test has an asymptotic normal distribution under the null hypothesis and non trivial power under local alternatives a novel feature of this test is that it does not have any moment restriction in contrast with other time reversibility and linearity tests our simulations also confirm that the proposed test is very robust when data do not possess proper moments an empirical study of stock market indices is also included to illustrate the usefulness of the new test

asymptotic distributions of maximum likelihood tests for change in the mean summary for the maximum likelihood tests for a change in the mean of independent random variables it is proved that the limit distribution is a double exponential i e gumbel distribution our results also explain why the maximum likelihood methods are very powerful on the tails the time intervals between explosions in british coal mines between 1851 and 1950 in which more than ten people were killed were analysed by maguire pearson wynn 1952 who concluded that the data had an exponential distribution with constant mean over time cox lewis 1966 ch 3 jarrett 1979 and worsley 1986 have reanalysed the data and found strong evidence that the mean did not remain constant hsu 1979 found a significant change in stock market returns by analysing the weekly closing values of the dow jones industrial average from july 1 1971 to august 2 1974 cobb 1978 examined measurements of the annual volume of discharge from the nile river at aswan for the years 1871 to 1970 and found a decrease in annual volume after 1898

a dynamic approach to the statistical analysis of point processes summary an approach to the analysis of point processes based on sequential analysis of successive intervals assuming a piecewise constant intensity rate is presented bayesian inference for processes with explanatory variables is derived in terms of on line estimation filtering and prediction limiting equations for the time continuous intensity rate are obtained and solved for the no covariates case examples are provided and the approach is applied in a numerical example similar approaches include the work of koch spreij 1983 and snyder 1975 6 which also use a dynamic model with a system equation these approaches based on filtering theory for time continuous processes provide good probabilistic descriptions their main problem is the difficulty in applying the results in the presence of covariates even in simple problems the purpose of this paper is to provide tractable yet reasonable solutions to the inference problem of point processes with explanatory variables azzalini 1982 warns about the complications of exact calculations required by filtering theory even in the absence of covariates the approach is based on and to some extent generalizes work on survival data gamerman 1991 inference is considered in terms of on line and smoothed or filtered estimation of model parameters and prediction of future occurrences based on past information in 2 the model is stated and inference procedures derived in 3 the model with no covariates is considered in detail and limiting equations for any continuous rate

full predictivistic modeling of stock market data application to change point problems in change point problems in general we should answer three questions how many changes are there where are they and what is the distribution of the data within the blocks in this paper we develop a new full predictivistic approach for modeling observations within the same block of observation and consider the product partition model ppm for treating the change point problem the ppm brings more flexibility into the change point problem because it considers the number of changes and the instants when the changes occurred as random variables a full predictivistic characterization of the model can provide a more tractable way to elicit the prior distribution of the parameters of interest once prior opinions will be required only about observable quantities we also present an application to the problem of identifying multiple change points in the mean and variance of a stock market return time series

capturing the distributional behaviour of the maximum likelihood estimator of a changepoint we consider the problem of estimating the unknown changepoint in a sequence of time ordered observations upper and lower bounds are derived for the asymptotic distribution of the maximum likelihood estimator and methods of approximation are suggested a computationally efficient algorithm is presented for deriving the bounds and approximations for the asymptotic probabilities of the maximum likelihood estimator when the parameters before and after the changepoint are unknown we also show an essentially exponential rate of convergence of the probability distribution of the maximum likelihood estimator from finite samples to the case of infinite samples we apply the algorithm to the cases of normal and exponential distributions for the exponential distribution the lower and upper bounds for the right tail probabilities of the maximum likelihood estimator and the two approximations are identical this is not the case for the normal distribution finally we apply our changepoint analysis for the case of the exponential distribution to data on explosions in british coal mines keywords maximum likelihood estimator maximum of a random walk negative drift parameter change

testing linearity against smooth transition autoregressive models summary we study a general univariate smooth transition autoregressive star model it contains as a special case the self exciting threshold autoregressive setar model we present three tests for testing linearity against star models and discuss their properties the power of the tests in small samples is investigated by simulation when the alternative is the logistic star model one of the tests is identical to tsay s 1986 test statistic and is recommended only in a special case of the two remaining tests with wider applicability one seems superior to the other in small samples it is also more powerful than the cusum test recently proposed for testing linearity against setar models

detection of random change point in one parameter exponential families abstract we consider a sequence x 1 x 2 x n of independent random variables which are susceptible to changing their distribution after the nt first observations where t is a random variable of a distribution known with support in 0 1 the object of this work is to detect the eventual change of distribution for that we study the performance of a test based on the statistic of log likelihood it shows that when the number of observations gets larger the distribution of the statistic log likelihood behaves like that of an affine function of the brownian motion this allows by using the concept of contiguity in the sense of lecam the evaluation of the asymptotic power function of the test

weiss weinstein bound on multiple change points estimation in the context of multiple change points estimation performance analysis of estimators such as the maximum likelihood is often difficult to assess since the regularity assumptions are not met focusing on the estimators variance one can however use lower bounds on the mean square error in this paper we derive the so called weiss weinstein bound wwb that is known to be an efficient tool in signal processing to obtain a fair overview of the estimation behavior contrary to several works about performance analysis in the change point literature our study is adapted to multiple changes first useful formulas are given for a general estimation problem whatever the considered distribution of the data second closed form expressions are given in the cases of gaussian observations with changes in the mean and or the variance and changes in the mean rate of a poisson distribution furthermore a semidefinite programming formulation of the minimization procedure is given in order to compute the tightest wwb specifically it consists of finding the unique minimum volume covering the set constituted by hyperellipsoid elements that are generated using the derived candidate wwb matrices w r t the so called loewner partial ordering finally simulation results are provided to show the good behavior of the proposed bound

a tukey nonadditivity type test for time series nonlinearity the past few years have seen a revived interest in the study of nonlinear stationary time series the most general form of a nonlinear stationary process is that referred to as a volterra expansion this is to a linear process what a higher order polynomial is to a linear function the so called bilinear model is a special case which is quite broad and is defined by a small number of parameters volterra expansions involve more than secondorder theory and require higher order cumulant spectra various tests for linearity have been proposed although none is totally satisfactory because of the similarity of volterra expansions to polynomials an analogue of tukey s 1949 one degree of freedom for nonadditivity test seems quite reasonable as a diagnostic for linearity versus a second order volterra expansion such a test would be time domain based and computationally less complex than the frequency domain based alternatives directly generalizable to higher order than second and possibly suggestive of a power transformation toward linearity

confidence regions and tests for a change point in a sequence of exponential family random variables summary maximum likelihood methods are used to test for a change in a sequence of independent exponential family random variables with particular emphasis on the exponential distribution the exact null and alternative distributions of the test statistics are found and the power is compared with a test based on a linear trend statistic exact and approximate confidence regions for the change point are based on the values accepted by a level x likelihood ratio test and a modification of the method proposed by cox spj0tvoll 1982 the methods are applied to a classical data set on the time intervals between coal mine explosions and the change in variation of stock market returns in both cases the confidence regions for the change point cover historical events that may have caused the changes

confidence sets for the date of a single break in linear time series regressions this paper considers the problem of constructing confidence sets for the date of a single break in a linear time series regression we establish analytically and by small sample simulation that the current standard method in econometrics for constructing such confidence intervals has a coverage rate far below nominal levels when breaks are of moderate magnitude given that breaks of moderate magnitude are a theoretically and empirically relevant phenomenon we proceed to develop an appropriate alternative we suggest constructing confidence sets by inverting a sequence of tests each of the tests maintains a specific break date under the null hypothesis and rejects when a break occurs elsewhere by inverting a certain variant of a locally best invariant test we ensure that the asymptotic critical value does not depend on the maintained break date a valid confidence set can hence be obtained by assessing which of the sequence of test statistics exceeds a single number

small sample properties of forecasts from autoregressive models under structural breaks this paper develops a theoretical framework for the analysis of smallsample properties of forecasts from general autoregressive models under structural breaks finite sample results for the mean squared forecast error of one step ahead forecasts are derived both conditionally and unconditionally and numerical results for different types of break specifications are presented it is established that forecast errors are unconditionally unbiased even in the presence of breaks in the autoregressive coefficients and or error variances so long as the unconditional mean of the process remains unchanged insights from the theoretical analysis are demonstrated in monte carlo simulations and on a range of macroeconomic time series from g7 countries the results are used to draw practical recommendations for the choice of estimation window when forecasting from autoregressive models subject to breaks

financial time series modeling using the hurst exponent abstract this study aims to enhance the understanding of logarithmic asset returns in particular more emphasis is given to the long memory property of financial returns a well documented stylized fact however in the presence of structural breaks other studies suggest that statistical tools such as the autocorrelation function acf can wrongly indicate long memory we propose an insensitive to structural breaks method to test for dependence between distant observations furthermore a model which combines memory in returns and memory in absolute returns is developed in two stages first return series are segmented with respect to changes in the volatility and then the two parameters of the model are estimated to assess the capabilities of the model historical prices of the standard and poor 500 index s p500 financial time stocks exchange 100 index ftse100 deutsche boerse ag german stock index dax and crude oil are used 1 given the estimated parameters and the volatility within each regime 10000 vectors are generated and compared to the original data in terms of the kolmogorov smirnov k s statistical test the obtained results suggest that long memory is present and provide evidence that the additional memory information captured by the model improves financial returns modeling

nonlinearity tests for time series summary this paper considers two nonlinearity tests for stationary time series the idea of tukey s one degree of freedom for nonadditivity test is generalized to the time series setting the case of concurrent nonlinearity is discussed in detail simulation results show that the proposed tests are more powerful than that of keenan 1985

the space of embedded minimal surfaces of fixed genus in a 3 manifold v fixed genus this paper is the fifth and final in a series on embedded minimal surfaces following our earlier papers on disks we prove here two main structure theorems for itnonsimply connected embedded minimal surfaces of any given fixed genus the first of these asserts that any such surface without small necks can be obtained by gluing together two oppositely oriented double spiral staircases the second gives a pair of pants decomposition of any such surface when there are small necks cutting the surface along a collection of short curves after the cutting we are left with graphical pieces that are defined over a disk with either one or two sub disks removed a topological disk with two sub disks removed is called a pair of pants both of these structures occur as different extremes in the two parameter family of minimal surfaces known as the riemann examples

nonparametric tests of linearity for time series summary we introduce tests of linearity for time series based on nonparametric estimates of the conditional mean and the conditional variance the tests are compared to a number of parametric tests and to nonparametric tests based on the bispectrum asymptotic expressions give bad approximations and the null distribution under linearity is constructed using resampling of the best linear approximation the new tests perform well on the examples tested

a powerful portmanteau test of lack of fit for time series a new portmanteau test for time series more powerful than the tests of ljung and box and monti is proposed the test is based on the mth root of the determinant of the mth autocorrelation matrix it is shown that the proposed statistic is a function of all of the squared multiple correlation coefficients of the regressions of the residuals on their lags when the number of lags goes from 1 to m it can also be written as a function of the first m partial autocorrelation coefficients the asymptotic distribution of the test statistic is a linear combination of chi squared distributions and can be approximated by a gamma distribution it is shown depending on the model and sample size that this test can be up to 50 more powerful than the ljung and box and monti tests the test is applied to the detection of several types of nonlinearity by using the autocorrelation matrix of the squared residuals and it is shown that in general the new test is more powerful than the test of mcleod and li an example

the power of likelihood ratio and cumulative sum tests for a change in a binomial probability summary a simple iterative procedure is found for the exact null and alternative distributions of likelihood ratio cumulative sum and related statistics for testing for a change in probability of a sequence of independent binomial random variables it is concluded that the likelihood ratio test is slightly less powerful than the cumulative sum test near the middle of the sequence but that the likelihood ratio test is much more powerful near the ends an example from epidemiology is used to illustrate the results under certain conditions the cumulative sum test is identical to a two sample kolmogorov smirnov test and so the iterative procedure can be used to find the null distribution of kolmogorov smirnov and related statistics

growth motion and 1 parameter families of symmetry sets associated to every plane curve there is the locus of centres of circles bitangent to that curve the so called symmetry set of the curve we can view this set as the spine of our curve which can be recovered by taking the envelope of circles of varying radii along this spine varying the symmetry set in some isotopy while keeping the radius function fixed may be viewed as crudely modelling motion of the original curve viewed as a biological object fixing the symmetry set and varying the radius function can be considered to model growth crudely in this paper we shall describe the generic changes in the curves which take place in the process of growth and motion and outline the corresponding results for spheres centred on a space curve we also use the idea of a stratified morse function to describe the generic changes which occur in one parameter families of full bifurcation sets in the plane applying this to the bifurcation set of distance squared functions we find all the transitions of a symmetry set and evolute which occur in a generic isotopy of a plane curve

maximally selected chi square statistics two samples can be compared by selecting a cut point and then forming a 2 x 2 table of the numbers of observations above and below the cut point in each sample when the cut point is selected so as to maximize the standard chi square statistic the chi square percentile points are inappropriate actual significance levels are computed for large samples and correct percentile poillts are tabulated

testing and locating variance changepoints with application to stock prices abstract this article explores testing and locating multiple variance changepoints in a sequence of independent gaussian random variables assuming known and common mean this type of problem is very common in applied economics and finance a binary procedure combined with the schwarz information criterion sic is used to search all of the possible variance changepoints existing in the sequence the simulated power of the proposed procedure is compared to that of the cusum procedure used by inclan and tiao to cope with variance changepoints the sic and unbiased sic for this problem are derived to obtain the percentage points of the sic criterion the asymptotic null distribution of a function of the sic is obtained and then the approximate percentage points of the sic are tabulated finally the results are applied to the weekly stock prices the unknown but common mean case is also outlined at the end

functional coefficient autoregressive models in this article we propose a new class of models for nonlinear time series analysis investigate properties of the proposed model and suggest a modeling procedure for building such a model the proposed modeling procedure makes use of ideas from both parametric and nonparametric statistics a consistency result is given to support the procedure for illustration we apply the proposed model and procedure to several data sets and show that the resulting models substantially improve postsample multi step ahead forecasts over other models

exact percentage points of the likelihood ratio test for a change point hazard rate model we wish to test that the hazard rate of survival or failure time data is constant against the alternative of a change in hazard after an unspecified time the likelihood ratio is unbounded but the exact null distribution of a restricted likelihood ratio test statistic is found this distribution is not affected by type ii censoring but it does depend very strongly on the interval in which the unknown change point is assumed to lie some exact percentage points are given which are much larger than simulated points that have been reported in the literature

new weighted portmanteau statistics for time series goodness of fit testing we exploit ideas from high dimensional data analysis to derive new portmanteau tests that are based on the trace of the square of the mth order autocorrelation matrix the resulting statistics are weighted sums of the squares of the sample autocorrelation coefficients that unlike many other tests appearing in the literature are numerically stable even when the number of lags considered is relatively close to the sample size the statistics behave asymptotically as a linear combination of chi squared random variables and their asymptotic distribution can be approximated by a gamma distribution the proposed tests are modified to check for nonlinearity and to check the adequacy of a fitted nonlinear model simulation evidence indicates that the proposed goodness of fit tests tend to have higher power than other tests appearing in the literature particularly in detecting long memory nonlinear models the efficacy of the proposed methods is demonstrated by investigating nonlinear effects in apple inc an

optimal detection of changepoints with a linear computational cost in this article we consider the problem of detecting multiple changepoints in large datasets our focus is on applications where the number of changepoints will increase as we collect more data for example in genetics as we analyze larger regions of the genome or in finance as we observe time series over longer periods we consider the common approach of detecting changepoints through minimizing a cost function over possible numbers and locations of changepoints this includes several established procedures for detecting changing points such as penalized likelihood and minimum description length we introduce a new method for finding the minimum of such cost functions and hence the optimal number and location of changepoints that has a computational cost which under mild conditions is linear in the number of observations this compares favorably with existing methods for the same problem whose computational cost can be quadratic or even cubic in simulation studies we show that our new method can

limit theorem for maximum of standardized u statistics with an application we show that the maximally selected standardized u statistic goes in distribution to an infinite sum of weighted chi square random variables in the degenerate case the result is applied to the detection of possible changes in the distribution of a sequence observation

powerful tests for structural changes in volatility abstract detecting structural changes in volatility is important for understanding volatility dynamics and stylized facts observed for financial returns such as volatility persistence we propose modified cusum and lm tests that are built on a robust estimator of the long run variance of squared series we establish conditions under which the new tests have standard null distributions and diverge faster than standard tests under the alternative the theory allows smooth and abrupt structural changes that can be small the smoothing parameter is automatically selected such that the proposed test has good finite sample size and meanwhile achieves decent power gain

testing for nonlinearity with partially observed time series we have implemented a lagrange multiplier test specifically for the alternative of a nonlinear continuous time autoregressive model with the instantaneous mean having one degree of nonlinearity the test is then extended to testing for the alternative of general nonlinear continuous time autoregressive models with multiple degrees of nonlinearity the performance of the test in the finite sample case is compared with several existing tests for nonlinearity including keenan s 1985 test petruccelli davies 1986 test and tsay s 1986 1989 tests the comparison is based on simulated data from some linear autoregressive models self exciting threshold autoregressive models bilinear models and the nonlinear continuous time autoregressive models which the lagrange multiplier test is designed to detect the lagrange multiplier test outperforms the other tests in detecting the model for which it is designed compared with the other tests the test has excellent power in detecting bilinear models but seems less powerful in detecting self exciting threshold autoregressive nonlinearity the test is further illustrated with the hong kong beach water quality data

ridges crests and sub parabolic lines of evolving surfaces the ridge lines on a surface can be defined either via contact of the surface with spheres or via extrema of principal curvatures along lines of curvature certain subsets of ridge lines called crest lines have been singled out by some authors for medical imaging applications there is a related concept of sub parabolic line on a surface also defined via extrema of principal curvatures in this paper we study in detail the structure of the ridge lines crest lines and sub parabolic lines on a generic surface and on a surface which is evolving in a generic one parameter family the mathematical details of this study are in bruce et al 1994c

the effect of serial correlation on tests for parameter change at unknown time it is shown that serial correlation can produce striking effects in distributions of change point statistics failure to account for these effects is shown to invalidate change point tests either through increases in the type 1 error rates if low frequency spectral mass predominates in the spectrum of the noise process or through diminution of the power of the tests when high frequency mass predominates these effects are characterized by the expression 2i f o fjr f a da where f is the spectral density of the noise process in sample survey work this is known as the design effect or deff simple precise adjustments to change point test statistics which account for serial correlation are provided the same adjustment applies to all commonly used regression models residual processes are derived for both stationary time series satisfying a moment condition and for general linear regression models with stationary error structure 1 introduction stochastic models for time sequenced data are generally characterized by several unknown parameters these parameters may change over time and if the changes when they occur do so unannounced and at unknown time points then the associated inferential problem is referred to as the change point problem various important application areas of statistics involve change detection in a central way two of these areas are quality assurance and environmental monitoring most of the statistics commonly applied to the change point problem involve cumulative sums or partial sums of regression residuals the distribution theory for these statistics has been computed under the assumption that the error process for the regression model is white noise in this paper we consider linear regression of a random variable against general nonstochastic functions of time but with error variables that form a serially correlated time series we then examine the large sample properties of the stochastic processes defined by the partial sums of the regression residuals large sample distribution theory for fixed sample size statistics used to detect changes in regression parameters is usually derived by computing the distributions of various functionals on

a bayesian analysis for change point problems a sequence of observations undergoes sudden changes at unknown times we model the process by supposing that there is an underlying sequence of parameters partitioned into contiguous blocks of equal parameter values the beginning of each block is said to be a change point observations are then assumed to be independent in different blocks given the sequence of parameters in a bayesian analysis it is necessary to give probability distributions to both the change points and the parameters we use product partition models barry and hartigan 1992 which assume that the probability of any partition is proportional to a product of prior cohesions one for each block in the partition and that given the blocks the parameters in different blocks have independent prior distributions given the observations a new product partition model holds with posterior cohesions for the blocks and new independent block posterior distributions for parameters the product model thus provides a convenient machinery for allo

additivity tests for nonlinear autoregression summary additivity is commonly used in the statistical literature to simplify data analysis especially in analysis of variance and in multivariate smoothing in this paper we propose three procedures for testing additivity in nonlinear time series analysis the first procedure combines some smoothing techniques with analysis of variance the second is a lagrange multiplier test using nonparametric estimation and the third is a permutation test which uses smoothing techniques to obtain the test statistic and its reference distribution we investigate properties of the proposed tests and use simulation to check their performance in finite samples applications of the tests to nonlinear time series analysis are discussed and illustrated by real examples

hierarchic voronoi skeletons abstract robust and time efficient skeletonization of a planar shape which is connectivity preserving and based on euclidean metrics can be achieved by first regularizing the voronoi diagram vd of a shape s boundary points i e by removal of noise sensitive parts of the tessellation and then by establishing a hierarchic organization of skeleton constituents each component of the vd is attributed with a measure of prominence which exhibits the expected invariance under geometric transformations and noise the second processing step a hierarchic clustering of skeleton branches leads to a multiresolution representation of the skeleton termed skeleton pyramid

tree structure for contractible regions in r 3 for contractible regions in 3 with generic smooth boundary we determine the global structure of the blum medial axis m we give an algorithm for decomposing m into irreducible components which are attached to each other along fin curves the attaching cannot be described by a tree structure as in the 2d case however a simplified but topologically equivalent medial structure m with the same irreducible components can be described by a two level tree structure the top level describes the simplified form of the attaching and the second level tree structure for each irreducible component specifies how to construct the component by attaching smooth medial sheets to the network of y branch curves the conditions for these structures are complete in the sense that any region whose blum medial axis satisfies the conditions is contractible

the classical theory of minimal surfaces we present here a survey of recent spectacular successes in classical minimal surface theory we highlight this article with the theorem that the plane the helicoid the catenoid and the one parameter family frtgt2 0 1 of riemann minimal examples are the only complete properly embedded minimal planar domains in r 3 the proof of this result depends primarily on work of colding

approximations to inverse moments of double indexed weighted sums abstract in this paper we investigate some approximations to inverse moments of double indexed weighted sums of random variables let z n be a sequence of nonnegative independent random variables and w n i be a triangular array of nonnegative non random weights with the finite first moment we establish that e a i 1 n w n i z i a i 1 n w n i e z i holds for all a 0 and 0 by the finite r th moment r 2 the convergence rate of approximation between e a i 1 n w n i z i and a i 1 n w n i e z i is presented on the other hand we obtain some approximations for the estimators of e i 1 n w n i z i a i 1 n z i e i 1 n w n i z i a i 1 n z i 2 and var i 1 n w n i z i a i 1 n z i for all a 0 finally some examples and simulations are illustrated

evaluating forecasting performance for interval data from the overlapping parts and the non overlapping parts of the actual intervals and the forecast intervals it should be defined a criterion which is more efficient to evaluate forecasting performance for interval data in this paper we present evaluation techniques for interval time series forecasting the forecast results are compared by the mean squared error of the interval mean relative interval error and mean ratio of exclusive or simulation and empirical studies show that our proposed evaluation techniques for interval forecasting can provide a more objective decision space in interval forecasting to policymakers

perestroikas of shocks and singularities of minimum functions the shock discontinuities generically present in inviscid solutions of the forced burgers equation and their bifurcations happening in the course of time perestroikas are classified in two and three dimensions the one dimensional case is well known this classification is a result of selecting among all the perestroikas occurring for minimum functions depending generically on time the ones permitted by the convexity of the hamiltonian of the burgers dynamics topological restrictions on the admissible perestroikas of shocks are obtained the resulting classification can be extended to the so called viscosity solutions of a hamilton jacobi equation provided the hamiltonian is convex

determining the geometry of boundaries of objects from medial data we consider a region in r2 or r3 with generic smooth boundary b and blum medial axis m on which is defined a multivalued radial vector field u from points x on m to the points of tangency of the sphere at x with b we introduce a radial shape operator srad and an edge shape operator se which measure how u bends along m these are not traditional differential geometric shape operators nonetheless we derive all local differential geometric invariants of b from these operators this allows us to define from m u a geometric medial map on m which corresponds via a radial map from m to b to the differential geometric properties of b the geometric medial map also includes a description of the relative geometry of b this is defined using the relative critical set of the radius function r on m this set consists of a network of curves on m which describe where b is thickest and thinnest it is computed using the covariant derivative of the tangential component of the unit radial vector field we further determine how these invariants are related to the differential geometric invariants of m and how these invariants change under deforming diffeomorphisms of m

testing and modeling threshold autoregressive processes abstract the threshold autoregressive model is one of the nonlinear time series models available in the literature it was first proposed by tong 1978 and discussed in detail by tong and lim 1980 and tong 1983 the major features of this class of models are limit cycles amplitude dependent frequencies and jump phenomena much of the original motivation of the model is concerned with limit cycles of a cyclical time series and indeed the model is capable of producing asymmetric limit cycles the threshold autoregressive model however has not received much attention in application this is due to a the lack of a suitable modeling procedure and b the inability to identify the threshold variable and estimate the threshold values the primary goal of this article therefore is to suggest a simple yet widely applicable model building procedure for threshold autoregressive models based on some predictive residuals a simple statistic is proposed to test for threshold nonlinearity and specify the

use of cumulative sums of squares for retrospective detection of changes of variance abstract this article studies the problem of multiple change points in the variance of a sequence of independent observations we propose a procedure to detect variance changes based on an iterated cumulative sums of squares icss algorithm we study the properties of the centered cumulative sum of squares function and give an intuitive basis for the icss algorithm for series of moderate size i e 200 observations and beyond the icss algorithm offers results comparable to those obtained by a bayesian approach or by likelihood ratio tests without the heavy computational burden required by these approaches simulation results comparing the icss algorithm to other approaches are presented

heteroscedasticity and autocorrelation robust structural change detection the assumption of weak stationarity is crucial for the validity of most of the conventional tests of structure change in time series under complicated nonstationary temporal dynamics we argue that traditional testing procedures result in mixed structural change signals of the first and second order and hence could lead to biased testing results the article proposes a simple and unified bootstrap testing procedure that provides consistent testing results under general forms of smooth and abrupt changes in the temporal dynamics of the time series monte carlo experiments are performed to compare our testing procedure with various traditional tests our robust bootstrap test is applied to testing changes in an environmental and a financial time series and our procedure is shown to provide more reliable results than the conventional tests

the asymptotic behavior of properly embedded minimal surfaces of finite topology while advances have been made in recent years in the study of properly embedded minimal surfaces of finite topology in r3 3 5 8 9 11 12 13 23 progress has depended in an essential manner on the special structure of such surfaces with finite total curvature a properly immersed minimal surface with finite total curvature is conformally a compact riemann surface punctured in a finite number of points and its gauss map extends to the compact surface as a meromorphic function 22 in particular all the topological ends are conformally equivalent to a punctured disk and there is a well defined limit tangent plane at each end outside of a sufficiently large compact set such an end is a multisheeted graph over the limit tangent plane and if the end is embedded it is asymptotic to either a plane or a half catenoid see e g schoen 24 for a survey of results on properly immersed surfaces of finite total curvature see 9 1 without the assumption of finite total curvature these results are not true in general as an indication of the relative lack of information consider that it is unknown whether or not the helicoid is the only simply connected minimal surface that is properly embedded and nonplanar a question raised by osserman the helicoid has infinite total curvature and a single annular end if a surface has finite topology then all of its ends are annular any general theory of properly embedded minimal surfaces of finite topology must include an understanding of the behavior of these ends in this paper we prove

determination of changes in streamflow variance by means of a wavelet based test 1 changes in the variance of streamflow over time may possibly be caused by land cover changes or climate changes knowledge of these possible changes in the time series structure is desirable for validation of modeling efforts detection is not always straightforward however we present the use of a wavelet based statistical test for the detection of changes in the variance of a time series applied to records of streamflow from 12 rivers in the united states from the period 1954 1999 the test is able to show during which time periods and at what wavelet scales changes in the variance are taking place it is found that the major similarities in variance change for all the streamflow time series exist when the largest time domain wavelet is used and also when a wavelet of the second largest frequency scale is used sampling considerations prevent exact localization of the time and frequency of the change s results of the test indicate that for the data analyzed in this study a change in the variance of the streamflow time series most likely did take place over the time period examined

testing for neglected nonlinearity in regression models based on the theory of random fields abstract within a flexible regression model j d hamilton econometrica 69 3 2001 537 we offer a battery of new lagrange multiplier statistics that circumvent the problem of unidentified nuisance parameters under the null hypothesis of linearity and that are robust to the specification of the covariance function that defines the random field these advantages are the result of i switching from the l 2 to the l 1 norm and ii assuming that the random field is sufficiently smooth for its covariance function to be locally approximated by a high order taylor expansion a monte carlo simulation suggests that our statistics have superior power performance on detecting bilinear neural network and smooth transition autoregressive specifications we also provide an application to the industrial production index of sixteen oecd countries

on the robustness of nonlinearity tests to moment condition failure abstract this paper investigates the consequences of testing nonlinearities in data that suffer from moment condition failure simulation experiments indicate that tests designed to have maximal power against misspecification of the conditional variance seem to be especially sensitive to the nonexistence of moments the relationship between findings of nonlinearity and moment condition failure is further investigated in a sample of daily stock returns the empirical study shows that evidence of nonlinearity in stock returns cannot all be attributed to the nonrobustness of nonlinearity tests to moment condition failure however it is shown that most tests of nonlinearity are not reliable in testing situations involving heavy tailed data

the space of embedded minimal surfaces of fixed genus in a 3 manifold iv locally simply connected this paper is the fourth in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3manifold the key is to understand the structure of an embedded minimal disk in a ball in r3 this was undertaken in cm3 cm4 and the global version of it will be completed here see the discussion around figure 12 for the local case and cm15 for some more details our main results are theorem 0 1 the lamination theorem and theorem 0 2 the one sided curvature estimate the lamination theorem is stated in the global case where the lamination is in fact a foliation the first four papers of this series show that every embedded minimal disk is either a graph of a function or is a double spiral staircase where each staircase is a multivalued graph this is done by showing that if the curvature is large at some point and hence the surface is not a graph then it is a double spiral staircase like the helicoid to prove that such a disk is a double spiral staircase we showed in the first three papers of the series that it is built out of n valued graphs where n is a fixed number in this paper we will deal with how the multi valued graphs fit together and in particular prove regularity of the set of points of large curvature the axis of the double spiral staircase the first theorem is the global version of the statement that every embedded minimal disk is a double spiral staircase

multiscale change point inference summary we introduce a new estimator the simultaneous multiscale change point estimator smuce for the change point problem in exponential family regression an unknown step function is estimated by minimizing the number of change points over the acceptance region of a multiscale test at a level the probability of overestimating the true number of change points k is controlled by the asymptotic null distribution of the multiscale test statistic further we derive exponential bounds for the probability of underestimating k by balancing these quantities will be chosen such that the probability of correctly estimating k is maximized all results are even non asymptotic for the normal case on the basis of these bounds we construct asymptotically honest confidence sets for the unknown step function and its change points at the same time we obtain exponential bounds for estimating the change point locations which for example yield the minimax rate up to a log term finally the simultaneous multiscale change point estimator achieves the optimal detection rate of vanishing signals as n even for an unbounded number of change points we illustrate how dynamic programming techniques can be employed for efficient computation of estimators and confidence regions the performance of the multiscale approach proposed is illustrated by simulations and in two cutting edge applications from genetic engineering and photoemission spectroscopy

the space of embedded minimal surfaces of fixed genus in a 3 manifold iii planar domains this paper is the third in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3 manifold in cm3 cm5 we describe the case where the surfaces are topologically disks on any fixed small scale although the focus of this paper general planar domains is more in line with cm6 we will prove a result here namely corollary iii 3 5 below which is needed in cm5 even for the case of disks roughly speaking there are two main themes in this paper the first is that stability leads to improved curvature estimates this allows us to find large graphical regions these graphical regions lead to two possibilities

specification estimation and evaluation of smooth transition autoregressive models abstract this article considers the application of two families of nonlinear autoregressive models the logistic lstar and exponential estar autoregressive models this includes the specification of the model based on simple statistical tests linearity testing against smooth transition autoregression determining the delay parameter and choosing between lstar and estar models are discussed estimation by nonlinear least squares is considered as well as evaluating the properties of the estimated model the proposed techniques are illustrated by examples using both simulated and real time series

multivariate contemporaneous threshold autoregressive models this paper proposes a contemporaneous threshold multivariate smooth transition autoregressive c mstar model in which the regime weights depend on the ex ante probabilities that latent regime specific variables exceed certain threshold values a key feature of the model is that the transition function depends on all the parameters of the model as well as on the data since the mixing weights are also a function of the regime specific noise covariance matrix the model can account for contemporaneous regime specific co movements of the variables the stability and distributional properties of the proposed model are discussed as well as issues of estimation testing and forecasting the practical usefulness of the c mstar model is illustrated by examining the relationship between us stock prices and interest rates

treatment of kurtosis in financial markets abstract since mandelbrot 1963 2 highlighted the fact that data on the yield of financial assets exhibit leptokurtosis different distributions have been presented as alternatives to the normal distribution so far little consideration has been given to the capacity that these distributions have to recover the kurtosis of the sample data our work aims to present distributions which given the broad range of their kurtosis have the capacity to perform adjustment on many occasions where other distributions fail while also being capable of recovering the peakedness of the empirical data another key characteristic of these distributions is that they are defined within a bounded domain in the same way as the sample data an empirical application of these distributions is presented within the financial field by using daily returns

nonparametric tests for nonstandard change point problems we consider independent random elements x 1 x n n n with values in a measurable space h b so that x 1 x n have a common distribution 1 and the remaining x n 1 x n have a common distribution 2 1 for some 0 1 the change point as well as the distributions are unknown a family of tests is introduced for the nonstandard change point problem h 0 0 versus h 1 0 where 0 is an arbitrary subset of 0 1 the tests are shown to be asymptotic level tests and to be consistent on a large class of alternatives the same holds for the corresponding bootstrap versions of the tests moreover we present a detailed investigation of the local power

testing for homogeneity of variance in time series long memory wavelets and the nile river 1 we consider the problem of testing for homogeneity of variance in a time series with long memorystructure wedemonstratethat atest whose nullhypothesis isdesigned tobewhite noisecan in fact be applied on a scale by scale basis to the discrete wavelet transform of long memory processes in particular we show that evaluating a normalized cumulative sum of squares test statistic using critical levels for the null hypothesis of white noise yields approximately the same null hypothesis rejection rates when applied to the discrete wavelet transform of samples from a fractionally differenced process the point at which the test statistic using a nondecimated version of the discrete wavelet transform achieves its maximum value can be used to estimate the time of the unknown variance change we apply our proposed test statistic on five time series derived from the historical record of nile river yearly minimum water levels covering 622 1922 a d each series exhibiting various degrees of serial correlation including long memory in the longest subseries spanning 622 1284 a d the test confirms an inhomogeneity of variance at short time scales and identifies the change point around 720 a d which coincides closely with the construction of a new device around 715 a d for measuring the nile river the test also detects a change in variance for a record of only 36 years index terms 1829 hydrology groundwater hydrology 1869 hydrology stochastic processes 3299 mathematical geophysics general or miscellaneous

a semiparametric changepoint model a semiparametric changepoint model is considered and the empirical likelihood method is applied to detect the change from a distribution to a weighted distribution in a sequence of independent random variables the maximum likelihood changepoint estimator is shown to be consistent the empirical likelihood ratio test statistic is proved to have the same limit null distribution as that with parametric models a data based test for the validity of the models is also proposed simulation shows the sensitivity and robustness of the semiparametric approach the methods are applied to some classical datasets such as the nile river data and stock price data

a bayesian robust detection of shift in the risk structure of stock market returns abstract in this article we provide a statistical procedure for the analysis of stock market prices that is robust toward departures from the normal distribution assumption and that can detect and evaluate a shift of parameters at an unknown time point the method is an adaptation of a bayesian inferential procedure developed by box and tiao that allows data to deviate moderately from the normal distribution model it is applied to a set of u s stock market prices for 1971 1974 in addition to the detection of shift in distribution parameters the procedure is also applied to the examination of shift of the beta coefficients that represent the degree of undiversifiable systematic risk of individual securities implications of the empirical findings for financial theories and their applications are sketched

hypothesis testing for nonlinearity detection based on an ma model in this correspondence we propose two hypothesis testing ht for nonlinearity detection these ht are based on an moving average ma model allowing us to model signals with null spectral density values unlike autoregressive ar models used in the usual parametric tests for nonlinearity detection these indexes are tested on simulated nonlinear and linear time series performances and drawbacks of these indexes are discussed with respect to the robustness of the indexes and to the difference between the theoretical and estimated laws under the hypothesis of linearity

the maximum likelihood method for testing changes in the parameters of normal observations 1 4 ha 1 oh2 2 o n and ali 2 a nt n for some r e 0 1 has received considerable attention in the literature under ha we can have change only in the mean the variances remain the same page 1954 1955 considered a very simple procedure for testing ho against h sen and srivastava 1975a b srivastava and worsley 1986 yao and davis 1986 james james and siegmund 1987 and worsley 1986 derived tests for ho against ha using the likelihood ratio yao and davis 1986 obtained the asymptotic distribution of the likelihood ratio for further results the asymptotic distributions of these and related statistics we refer to csorgo and horvath 1988 and gombay and horvath 1990 the maximum likelihood method can be used when we want to test ho against the more general alternative ha it is easy to show that the likelihood

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

miscellanea a note on tests for nonlinearity in a vector time series a multivariate extension of the univariate nonlinearity test of tsay 1986 is presented simulation results show that the multivariate test is more powerful than its univariate counterpart especially for series having nonlinear structure involving several components of the vector process and weakly or moderately cross correlated process error terms for illustration the test is applied to a set of seasonally adjusted quarterly capital expenditures and appropriations in u s manufacturing and a vector nonlinear model for the data is constructed

a formal classification of 3d medial axis points and their local geometry this paper proposes a novel hypergraph skeletal representation for 3d shape based on a formal derivation of the generic structure of its medial axis by classifying each skeletal point by its order of contact we show that genetically the medial axis consists of five types of points which are then organized into sheets curves and points 1 sheets manifolds with boundary which are the locus of bitangent spheres with regular tangency a sub 1 sup 2 a sub k sup n notation means n distinct k fold tangencies of the sphere of contact as explained in the text two types of curves 2 the intersection curve of three sheets and the locus of centers of tritangent spheres a sub 1 sup 3 and 3 the boundary of sheets which are the locus of centers of spheres whose radius equals the larger principal curvature i e higher order contact a sub 3 points and two types of points 4 centers of quad tangent spheres a sub 1 sup 4 and 5 centers of spheres with one regular tangency and one higher order tangency a sub 1 a sub 3 the geometry of the 3d medial axis thus consists of sheets a sub 1 sup 2 bounded by one type of curve a sub 3 on their free end which corresponds to ridges on the surface and attached to two other sheets at another type of curve a sub 1 sup 3 which support a generalized cylinder description the a sub 3 curves can only end in a sub 1 a sub 3 points where they must meet an a sub 1 sup 3 curve the a sub 1 sup 3 curves meet together in fours at an a sub 1 sup 4 point this formal result leads to a compact representation for 3d shape referred to as the medial axis hypergraph representation consisting of nodes a sub 1 sup 4 and a sub 1 a sub 3 points links between pairs of nodes a sub 1 sup 3 and a sub 3 curves and hyperlinks between groups of links a sub 1 sup 2 sheets the description of the local geometry at nodes by itself is sufficient to capture qualitative aspects of shapes in analogy to 2d we derive a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph together with the radius function thus this information completely characterizes 3d shape and lays the theoretical foundation for its use in recognition morphing design and manipulation of shapes

some picard theorems for minimal surfaces this paper deals with the study of those closed subsets f r 3 for which the following statement holds if s is a properly immersed minimal surface in r 3 of finite topology that is eventually disjoint from f then s has finite total curvature the same question is also considered when the conclusion is finite type or parabolicity

the likelihood ratio test for a change point in simple linear regression summary we consider likelihood ratio tests to detect a change point in simple linear regression a when the alternative specifies that only the intercept changes and b when the alternative permits the intercept and the slope to change approximations for the significance level are obtained under reasonably general assumptions about the empirical distribution of the independent variable the approximations are compared with simulations in order to assess their accuracy for the model in which only the intercept is allowed to change a confidence region for the change point and an approximate joint confidence region for the change point the difference in intercepts and the slope are obtained by inversion of the appropriate likelihood ratio tests

use of fuzzy statistical technique in change periods detection of nonlinear time series many papers have been presented on the study of change points detection nonetheless we would like to point out that in dealing with the time series with switching regimes we should also take the characteristics of change periods into account because many patterns of change structure in time series exhibit a certain kind of duration those phenomena should not be treated as a mere sudden turning at a certain time in this paper we propose a procedure about change periods detection for nonlinear time series the detecting statistical method is an application of fuzzy classification and a generalization of inclan and tiao s result j am statist assoc 89 1994 913 simulation results show that the performance of the proposed procedure is efficient and successful finally an empirical application about change periods detecting for taiwan monthly visitor s arrival is demonstrated