polynomially computable bounds for the probability of the union of events we consider the problem of finding upper and lower bounds for the probability of the union of events when the probabilities of the single events and the probabilities of the intersections of up to m events are given it is known that the best possible bounds can be obtained by solving linear programming problems with a number of variables that is exponential in the number of events because of their size and structure these large linear programs are known to be very hard to solve in the literature simpler polynomially sized aggregations are considered and numerous closed form or polynomially computable bounds are derived from those we present here a new approach that introduces additional constraints to the dual linear programming problems in such a way that those become polynomially solvable by using different sets of additional constraints we introduce three new classes of polynomially computable upper and lower bounds we show that they dominate almost all efficiently computable bounds known in t

detecting a shift in location some robust tests abstract the test statistic w suggested by hawkins 1977 and worsley 1979 for testing a sequence of observations for a shift in location is sensitive to the assumption that the distribution of the population being sampled is normal a modification of this statistic is proposed which is robust against heavy tailed distributions we also study the performance of several alternative testing procedures by means of a simulation experiment the farley hinich test 1975 and the homogeneity test discussed in brown durbin and evans 1975 perform well in the simulation experiment against heavy tailed distributions some applications to economic data are also discussed in the paper

variational perturbation theory for markov processes we develop a convergent variational perturbation theory for conditional probability densities of markov processes the power of the theory is illustrated by applying it to the diffusion of a particle in an anharmonic potential

an improved bonferroni inequality and applications summary we present an improved bonferroni inequality which gives an upper bound for the probability of the union of an arbitrary sequence of events the bound is constructed in terms of the joint probability of pairs of events which are represented by edges on a graph examples of applications to periodicity location shift detection kolmogorovsmirnov tests and outlier detection are given

backlund transformations ward solitons and unitons the ward equation also called the modified 2 1 chiral model is obtained by a dimension reduction and a gauge fixing from the self dual yang mills field equation on r it has a lax pair and is an integrable system ward constructed solitons whose extended solutions have distinct simple poles he also used a limiting method to construct 2 solitons whose extended solutions have a double pole ioannidou and zakrzewski and anand constructed more soliton solutions whose extended solutions have a double or triple pole some of the main results of this paper are i we construct algebraic backlund transformations bts that generate new solutions of the ward equation from a given one by an algebraic method ii we use an order k limiting method and algebraic bts to construct explicit ward solitons whose extended solutions have arbitrary poles and multiplicities iii we prove that our construction gives all solitons of the ward equation explicitly and the entries of ward solitons must be rational functions in x y and t iv since stationary ward solitons are unitons our method also gives an explicit construction of all k unitons from finitely many rational maps from c to c

detecting sparse signals in random fields with an application to brain mapping brain mapping data have been modeled as gaussian random fields and local increases in mean are detected by local maxima of a random field of test statistics derived from these data accurate p values for local maxima are readily available for isotropic data based on the expected euler characteristic of the excursion set of the test statistic random field in this article we give a simple method for dealing with nonisotropic data our approach has connections to the model of sampson and guttorp for nonisotropy in which there exists an unknown mapping of the support of the data to a space in which the random fields are isotropic heuristic justification for our approach comes from the nash embedding theorem formally we show that our method gives consistent unbiased estimators for the true p values based on new results of taylor and adler for random fields on manifolds which replace the euclidean metric by the variogram the results are used to detect gender differences in the cortical thickness of the b

approximations for the tail probabilities and moments of the scan statistic the scan statistic is used to test the hypothesis that the observed events occur at random uniformly distributed in time or space versus the hypothesis that they cluster within a moving window of size w to implement the testing procedure based on the scan statistic its tail probabilities have to be effectively evaluated in this article a survey of results on the approximations of the distribution of the scan statistic and its moments is presented numerical results comparing these approximations are also given numerous references with applications in epidemiological studies using the scan statistics are mentioned related scan statistics that have been used in many other interesting applications are listed in the reference section the article concludes with the presentation of unsolved problems related to the scan statistic

when the telegrapher s equation furnishes a better approximation to the transport equation than the diffusion approximation it has been suggested that a solution to the transport equation which includes anisotropic scattering can be approximated by the solution to a telegrapher s equation a j ishimaru appl opt 28 2210 1989 we show that in one dimension the telegrapher s equation furnishes an exact solution to the transport equation in two dimensions we show that since the solution can become negative the telegrapher s equation will not furnish a usable approximation a comparison between simulated data in three dimensions indicates that the solution to the telegrapher s equation is a good approximation to that of the full transport equation at the times at which the diffusion equation furnishes an equally good approximation s1063 651x 97 04205 0

periodic and homoclinic solutions of the modified 2 1 chiral model we use algebraic backlund transformations bts to construct explicit solutions of the modified 2 1 chiral model from t2 r to su n where t2 is a 2 torus algebraic bts are parametrized by z c poles and holomorphic maps from t2 to gr k cn we apply backlund transformations with carefully chosen poles and s to construct infinitely many solutions of the 2 1 chiral model that are i doubly periodic in space variables and periodic in time i e triply periodic ii homoclinic in the sense that the solution u has the same stationary limit u0 as t and is tangent to a stable linear mode of u0 as t and is tangent to an unstable mode of u0 as t

statistical process control for shifts in mean or variance using a changepoint formulation statistical process control spc involves ongoing checks to ensure that neither the mean nor the variability of the process readings has changed conventionally this is done by pairs of charts shewhart x and s or r charts cumulative sum charts for mean and for variance or exponentially weighted moving average charts for mean and variance the traditional methods of calculating the statistical properties of control charts are based on the assumption that the in control true mean and variance were known exactly and use these assumed true values to set center lines control limits and decision intervals the reality however is that true parameter values are seldom if ever known exactly rather they are commonly estimated from a phase i sample the random errors in the estimates lead to uncertain run length distribution of the resulting charts an attractive alternative to the traditional charting methods is a single chart using the unknown parameter likelihood ratio test for a change in mean and o

modified sequentially rejective multiple test procedures abstract suppose that n hypotheses h 1 h 2 h n with associated test statistics t 1 t 2 t n are to be tested by a procedure with experimentwise significance level the probability of rejecting one or more true hypotheses smaller than or equal to some specified value a commonly used procedure satisfying this condition is the bonferroni b procedure which consists of rejecting h i for any i iff the associated test statistic t i is significant at the level n holm 1979 introduced a modified bonferroni procedure with greater power than the b procedure under holm s sequentially rejective bonferroni srb procedure if any hypothesis is rejected at the level n the denominator of for the next test is n 1 and the criterion continues to be modified in a stagewise manner with the denominator of reduced by 1 each time a hypothesis is rejected so that tests can be conducted at successively higher significance levels holm proved that the experimentwise significance level

likelihood ratio tests for a change in the multivariate normal mean abstract a sequence of independent multivariate normal vectors with equal but possibly unknown variance matrices are hypothesized to have equal mean vectors and we wish to test that the mean vectors have changed after an unknown point in the sequence the likelihood ratio test is based on the maximum hotelling t 2 for the sequences before and after the change point the main result is a conservative approximation for its null distribution based on an improved bonferroni inequality if the change is judged significant then further changes are estimated by splitting the two subsequences formed by the first change point the methods can also be used to test for a change in row probabilities of a contingency table allowing for extramultinomial variation the results are used to find changes in a set of geological data previously analyzed by chernoff 1973 by the faces method and to find changes in the frequencies of pronouns in the plays of shakespeare

characteristic times in stock market indices abstract in this study we analyze the standard and poor s 500 index data of the new york stock exchange for more than 32 years using a simple random walk model we demonstrate that the proper variable to look at is the logarithmic return in the statistical analysis we have done fittings to the levy distribution using either the index data as such or pre processing it with arch garch or igarch methods which tend to remove the time dependent variance for short times the truncated levy distribution is found to fit the data quite well since this is not a stable distribution the scaling behavior observed for short times should brake down for longer times we demonstrate that the characteristic time where this cross over starts is of the order of one day

a useful upper bound for the tail probabilities of the scan statistic when the sample size is large abstract although a number of approximations to the distribution under uniformity of the scan statistic are known they are when the sample size is large either difficult to compute or numerically unstable a simple upper bound for the upper tail of the distribution based on a result of hunter 1976 is derived it is found that for large sample sizes and a reasonably wide range of window widths this bound gives a numerically stable approximation that is sufficiently accurate for hypothesis testing purposes

an improved bonferroni procedure for multiple tests of significance summary a modification of the bonferroni procedure for testing multiple hypotheses is presented the method based on the ordered p values of the individual tests is less conservative than the classical bonferroni procedure but is still simple to apply a simulation study shows that the probability of a type i error of the procedure does not exceed the nominal significance level a for a variety of multivariate normal and multivariate gamma test statistics for independent tests the procedure has type i error probability equal to a the method appears particularly advantageous over the classical bonferroni procedure when several highly correlated test statistics are involved

mean time to failure of systems with dependent components we consider first a system of two components that are possibly dependent two simple reliability structures can be formed series system and parallel redundant system the exact formulas and two sided bounds for the mttf of systems are provided we extend some of the results of kotz et al 2003 8 on the effect of redundancy or competing risk when component lifetimes are no longer independent the main results are illustrated by some useful distributions the new r s out of n system is also discussed in addition a new concept of dependence ordering more diagonal dependent between bivariate distributions is introduced and used to compare the system lifetimes finally the extension of the main results to the three component systems and multi component systems are also briefly investigated

diagrammatic computation of the random flight motion abstract we present a perturbation theory by extending a prescription due to feynman for computing the probability density function for the random flight motion the method can be applied to a wide variety of otherwise difficult circumstances the series for the exact moments if not the distribution itself for many important cases can be summed for arbitrary times as expected the behavior at early time regime for the sample processes considered deviate significantly from diffusion theory a fact with important consequences in various applications such as financial physics a half dozen sample problems are solved starting with one posed by feynman who originally solved it only to first order another illustrative case is found to be a physically more plausible substitute for both the uhlenbeck ornstein and the wiener processes the remaining cases we have solved are useful for applications in regime switching and isotropic scattering of light in turbid media it is demonstrated that under isotropic random flight with invariant speed the description of motion in higher dimensions is recursively related to either the one or two dimensional movement also for this motion we show that the solution heretofore presumed correct in one dimension applies only to the case we have named the shooting gallery problem a special case of the full problem a new class of functions dubbed damped exponential integrals are also identified z

testing for a pulse in seasonal event data most tests for seasonal clustering are sensitive to the peak in incidence in some season 90 day interval followed by a trough 6 months later we present two statistics sensitive to a relatively sharp increase in disease incidence for a season superimposed on a constant incidence over the entire year the ratchet scan based on the maximum number of events in k consecutive months and the continuous circular scan the maximum number of events in d consecutive days we tabulate the tail distribution of the ratchet scan for n less than or equal to 35 and k 2 and 3 and find the asymptotic distribution which is applicable for n greater than or equal to 50 when k 2 or 3 we evaluate an approximation for the distribution of the continuous scan the statistics are applied to investigate seasonal clustering of adolescent suicide in the united states an illustration is given of the application of the circular scan and a small simulation study and the example are used to compare the statistics with others proposed in the literature language en

rejecton of a single outiier in two or three way layouts accurate bounds are presented for the fractiles of the maximum normed residual which is often used to test for a single outlier for two and three way layouts it is shown that the second bonferroni bound of the critical value while not conservative is an excellent approximation to the critical value being much more accurate than the first bonferroni upper bound the third bonferroni upper bound which although conservative is expensive to calculate agrees with the second bound to at least four decimal places for all factor combinations considered in this paper

harmonic maps into lie groups classical solutions of the chiral model on considere deux aspects de la structure algebrique de 7b m l espace des applications harmoniques d un domaine a 2 dimensions simplement connexe dans un groupe de lie reel g r la forme reelle d un groupe complexe g on construit une representation du groupe boucle 7b a s 1 g r sur 7b m correspondant a l algebre de lie de kacmoody des deformations infinitesimales on decrit l action sur 7b m et l action d un sous groupe sur l espace des applications harmoniques dans des gaussmanniens dans une seconde partie on se restreint a une theorie qui s applique seulement quand est un domaine riemannien a 2 dimensions simplement connexe et le groupe est g r u n

transition models for change point estimation in logistic regression although a wide variety of change point models are available for continuous outcomes few models are available for dichotomous outcomes this paper introduces transition methods for logistic regression models in which the dose response relationship follows two different straight lines which may intersect or may present a jump at an unknown change point in these models the logit includes a differentiable transition function that provides parametric control of the sharpness of the transition at the change point allowing for abrupt changes or more gradual transitions between the two different linear trends as well as for estimation of the location of the change point linear linear logistic models are particular cases of the proposed transition models we present a modified iteratively reweighted least squares algorithm to estimate model parameters and we provide inference procedures including a test for the existence of the change point these transition models are explored in a simulation study and they are used to evaluate the existence of a change point in the association between plasma glucose after an oral glucose tolerance test and mortality using data from the mortality follow up of the second national health and nutrition examination survey

unbiased split variable selection for random survival forests using maximally selected rank statistics the most popular approach for analyzing survival data is the cox regression model the cox model may however be misspecified and its proportionality assumption may not always be fulfilled an alternative approach for survival prediction is random forests for survival outcomes the standard split criterion for random survival forests is the log rank test statistic which favors splitting variables with many possible split points conditional inference forests avoid this split variable selection bias however linear rank statistics are utilized by default in conditional inference forests to select the optimal splitting variable which cannot detect non linear effects in the independent variables an alternative is to use maximally selected rank statistics for the split point selection as in conditional inference forests splitting variables are compared on the p value scale however instead of the conditional monte carlo approach used in conditional inference forests p value approximations are employed we describe several p value approximations and the implementation of the proposed random forest approach a simulation study demonstrates that unbiased split variable selection is possible however there is a trade off between unbiased split variable selection and runtime in benchmark studies of prediction performance on simulated and real datasets the new method performs better than random survival forests if informative dichotomous variables are combined with uninformative variables with more categories and better than conditional inference forests if non linear covariate effects are included in a runtime comparison the method proves to be computationally faster than both alternatives if a simple p value approximation is used copyright 2017 john wiley sons ltd

breathers as homoclinic geometric wave maps a classical breather solution of the sine gordon equation can be viewed as a homoclinic wave map from periodic minkowski space mp2 into s2 this wave map is an unstable orbit of a simple steady state wave map homoclinic wave maps are also constructed into non round spheres

on the cauchy problem for the integrable system of lie minimal surfaces in this paper we apply the cartan kahler theory of exterior differential systems to solve the cauchy problem for the integrable system of lie minimal surfaces and discuss the underlying geometry one purpose for this work is to show how methods and language from the theory of exterior differential systems may prove to be useful in the study of real analytic initial value problems especially for gaining insight into the geometric aspects of the initial conditions and the solutions

the significance attained by the best fitting regressor variable abstract in the forward selection of independent variables for a linear regression model the next best fitting variable for inclusion is the variable with largest f to enter we consider the evaluation of the p value for this selection

an approximation for the distribution of the scan statistic the scan statistic evaluates whether an apparent cluster of disease in time is due to chance the statistic employs a moving window of length w and finds the maximum number of cases revealed through the window as it scans or slides over the entire time period t computation of the probability of observing a certain size cluster under the hypothesis of a uniform distribution is infeasible when n the total number of events is large and w is of moderate or small size relative to t we give an approximation that is an asymptotic upper bound easy to compute and for the purposes of hypothesis testing more accurate than other approximations presented in the literature the approximation applies both when n is fixed and when n has a poisson distribution we illustrate the procedure on a data set of trisomic spontaneous abortions observed in a two year period in new york city

option pricing from path integral for non gaussian fluctuations natural martingale and application to truncated levy distributions within a path integral formalism for non gaussian price fluctuations we set up a simple stochastic calculus and derive a natural martingale for option pricing from the wealth balance of options stocks and bonds the resulting formula is evaluated for truncated levy distributions

maximally selected rank statistics a common statistical problem is the assessment of the predictive power of a quantitative variable for some dependent variable a maximally selected rank statistic regarding the quantitative variable provides a test and implicitly an estimate of a cutpoint as a simple classification rule restricting the selection to an arbitrary given inner part of the support of the quantitative variable we show that the asymptotic null distribution of the maximally selected rank statistic is the distribution of the supremum of the absolute value of a standardized gaussian process on an interval the asymptotic argument holds also in the case of tied or censored observations we compare monte carlo results with an approximation of the asymptotic distribution under the null hypothesis in addition we investigate the behaviour of the test procedure and of the familiar spearman rank test for independence under some alternatives moreover we discuss some aspects of the problem of estimating an underlying cutpoint

testing for a two phase multiple regression maximum likelihood methods are used to estimate the parameters of two separate multiple regressions that switch at an unknown point in the data normal errors with constant variance are assumed and likelihood ratio statistics are used to test for the presence of two separate regressions our main result is a conservative bound on the null distribution function of the test statistic this bound is based on an improved bonferroni inequality and a simple power series approximation is provided similar bounds are given for likelihood ratio statistics that test for a shift in the constant term of the regression only the accuracies of the bounds and approximations are evaluated on a number of examples

a multivariate change point model for statistical process control multivariate statistical process control spc carries out ongoing checks to ensure that a process is in control these checks on the process are traditionally done by t2 multivariate cusum and multivariate exponentially weighted moving average control charts these traditional spc charts assume that the in control true parameters are known exactly and use these assumed true values to set the control limits the reality however is that true parameter values are seldom if ever known exactly rather they are commonly estimated from a phase i sample it is increasingly recognized that this phase i study needs to involve large samples if the parameter estimates are to provide run behavior matching that of the known parameter situation but apart from the general undesirability of large and thus expensive studies preliminary to actual charting some industrial settings have a paucity of relevant data for estimating the process parameters an attractive alternative to traditional charting methods when mon

the length heuristic for simultaneous hypothesis tests we observe a sequence of test statistics t t 1 t 2 t j each of which is approximately n 0 1 under the null hypothesis h 0 but which are correlated with each other not being certain which t j is best we use the test statistic t max max t 1 t j for a given observed value of t max say c what is the significance probability pr t max c define the length of t to be j l 2 arccos p j 1 j where p j 1 j is the null hypothesis correlation between t j 1 and t j hotelling s theorem on the volume of tubes leads to a length based bound on pr t max c that usually beats the bonferroni bound it is easy to improve hotelling s bound several examples show that in favourable circumstances the improved bounds can be good approximations to the actual value of pr t max c

a path length inequality for the multivariate t distribution with applications to multiple comparisons abstract this article presents a new inequality for the multivariate t distribution which implies a new method for multiple comparisons whose foundation rests on a recent inequality due to naiman the new method is promising in view of the fact that it utilizes information estimator intercorrelations ignored by the most widely used multiple comparison methods yet is not computationally prohibitive requiring only the numerical evaluation of a single one dimensional integral in this article the validity of the new method in the normal theoretic general linear model is established and efficiency studies relative to the methods of scheffe bonferroni sidak and hunter worsley are presented the new method is shown to always improve on scheffe s method the new method is also shown to perform well that is to lead to a smaller critical point than its competitors with low degrees of freedom but the method is not as efficient as the hunter worsley method for high degrees of freedom in addition the me

probability bounds with cherry trees a third order upper bound is presented on the probability of the union of a finite number of events by means of graphs called cherry trees these are graphs that we construct recursively in such a way that every time we pick a new vertex connect it with two already existing vertices if the latters are always adjacent we call the cherry tree a t cherry tree a cherry tree has a weight that provides us with the upper bound on the union any hunter worsley bound can be majorized by a t cherry bound constructed by the use of the hunter worsley tree a cherry tree bound can be identified as a feasible solution to the dual of the boolean probability bounding problem a t cherry tree bound can be identified as the objective function value of the dual vector corresponding to a dual feasible basis in the boolean problem this enables us to improve on the bound algorithmically if we use the dual method of linear programming

generalized maximally selected statistics maximally selected statistics for the estimation of simple cutpoint models are embedded into a generalized conceptual framework based on conditional inference procedures this powerful framework contains most of the published procedures in this area as special cases such as maximally selected chi squared and rank statistics but also allows for direct construction of new test procedures for less standard test problems as an application a novel maximally selected rank statistic is derived from this framework for a censored response partitioned with respect to two ordered categorical covariates and potential interactions this new test is employed to search for a high risk group of rectal cancer patients treated with a neo adjuvant chemoradiotherapy moreover a new efficient algorithm for the evaluation of the asymptotic distribution for a large class of maximally selected statistics is given enabling the fast evaluation of a large number of cutpoints

correction of the p value after multiple coding of an explanatory variable in logistic regression we propose a method and a program to determine a significance level for a series of codings of an explanatory variable in logistic regression dichotomous and box cox transformations are considered three methods of correcting the significance level are studied the bonferroni method efron s method which uses the correlation between successive tests and the exact calculation by numerical integration using all correlations a simulation study has led to a strategy for the choice and number of the different codings of the variable this method is illustrated using the data of a study of the relation between cholesterol and dementia

statistical method for detecting structural change in the growth process due to competition among individual trees and other exogenous factors that change the growth environment each tree grows following its own growth trend with some structural changes in growth over time in the present article a new method is proposed to detect a structural change in the growth process we formulate the method as a simple statistical test for signal detection without constructing any specific model for the structural change to evaluate the p value of the test the tube method is developed because the regular distribution theory is insufficient using two sets of tree diameter growth data sampled from planted forest stands of cryptomeria japonica in japan we conduct an analysis of identifying the effect of thinning on the growth process as a structural change our results demonstrate that the proposed method is useful to identify the structural change caused by thinning we also provide the properties of the method in terms of the size and power of the test

approximations and bounds for the distribution of the scan statistic abstract the scan statistic is used in many areas of science to test the null hypothesis of uniformity against a clustering alternative this article derives approximations and bounds for the distribution of scan statistics an extensive simulation study is carried out to compare the approximations and the bounds derived in this article with other known approximations and bounds the new lower and upper bounds that have been derived are very useful in evaluating the approximations and the simulation a quantity of average number of points in the scanning interval of length d d is introduced to study the approximations for the distribution of the scan statistic for low values of d the approximation derived in this article is the most accurate one the importance of the scan statistics arises from their applications in many areas including nuclear physics geology radio optics photography and epidemiology

a class of overdetermined systems defined by tableaux involutiveness and the cauchy problem this article addresses the question of involutiveness and discusses the initial value problem for a class of overdetermined systems of partial differential equations which arise in the theory of integrable systems and are defined by tableaux