analysis of a fourth order scheme for a three dimensional convection diffusion model problem we derive closed form expressions for the eigenvalues and discrete solution arising from a 19 point compact discretization of a three dimensional convection diffusion problem it is shown that the coefficient matrix is positive definite when the cell reynolds number is greater than some critical value by analyzing the terms composing the discrete solution we prove that an oscillation free discrete solution is guaranteed whenever the cell reynolds number exceeds a value which is grid size dependent an interesting result is that as the mesh size is refined this value approaches the golden mean

notes on some new kinds of pseudoprimes j browkin defined in his recent paper math comp 73 2004 pp 1031 1037 some new kinds of pseudoprimes called sylow p pseudoprimes and elementary abelian p pseudoprimes he gave examples of strong pseudoprimes to many bases which are not sylow p pseudoprime to two bases only where p 2 or 3 in this paper in contrast to browkin s examples we give facts and examples which are unfavorable for browkin s observation to detect compositeness of odd composite numbers in section 2 we tabulate and compare counts of numbers in several sets of pseudoprimes and find that most strong pseudoprimes are also sylow 2 pseudoprimes to the same bases in section 3 we give examples of sylow p pseudoprimes to the first several prime bases for the first several primes p we especially give an example of a strong pseudoprime to the first six prime bases which is a sylow p pseudoprime to the same bases for all p 2 3 5 7 11 13 in section 4 we define n to be a k fold carmichael sylow pseudoprime if it is a sylow p pseudoprime to all bases prime to n for all the first k smallest odd prime factors p of n 1 we find and tabulate all three 3 fold carmichael sylow pseudoprimes 10 16 in section 5 we define a positive odd composite n to be a sylow uniform pseudoprime to bases b 1 b k or a syl upsp b 1 b k for short if it is a syl p psp b 1 b k for all the first w n 1 1 small prime factors p of n 1 where w n 1 is the number of distinct prime factors of n 1 we find and tabulate all the 17 syl upsp 2 3 5 s 10 16 and some syl upsp 2 3 5 7 11 s 10 24 comparisons of effectiveness of browkin s observation with miller tests to detect compositeness of odd composite numbers are given in section 6

on hotelling s formula for the volume of tubes and naiman s inequality abstract motivation by the question of testing for a nonlinear parameter in a regression model with independent homoscedastic normal residuals hotelling 1939 was led to consider the geometric problem of computing the volume of a tube of given radius around a curve in s to the n 1 power the unit sphere in ir to the nth power the answer involves only the arc length of the curve and not its curvature providing the radius of the tube is sufficiently small that there is no self overlap in the tube starting from a somewhat different statistical setting naiman 1986 arrived at the same geometric problem and showed that hotelling s result properly interpreted is an upper bound for the volume of a tube of arbitrary radius the purpose of this note is to give two new derivations of the hotelling naiman results the first involves differential inequalities the second is probabilistic using the concept of upcrossing borrowed from the theory of gaussian processes in the context of gaussian processes knowles 1987 has observed that approximations obtained from hotelling s result and bounds derived via upcrossings are related

convergence of an efficient and compact finite difference scheme for the klein gordon zakharov equation a compact and semi explicit finite difference scheme is proposed and analyzed for the klein gordon zakharov kgz equation the new scheme is decoupled and linearized in practical computation i e at each time step only two tri diagonal systems of linear algebraic equations need to be solved by thomas algorithm so the new scheme is more efficient and more accurate than the classical finite difference schemes unique solvability of the difference solution is proved by using the energy method besides the standard energy method in order to overcome the difficulty in obtaining the a priori estimate an induction argument is introduced to prove that the new scheme is convergent for u x t in the discrete h 1 norm and respectively for m x t in the discrete l 2 norm at the order of o 2 h 4 with time step and mesh size h numerical results are reported to verify the theoretical analysis

optimality of the cusum procedure in continuous time the optimality of cusum under a lorden type criterion setting is considered we demonstrate the optimality of the cusum test for ito processes in a sense similar to lorden s but with a criterion that replaces expected delays by the corresponding kullback leibler divergence

fourth order compact finite difference methods and monotone iterative algorithms for semilinear elliptic boundary value problems in this paper we study numerical methods for a class of two dimensional semilinear elliptic boundary value problems with variable coefficients in a union of rectangular domains a compact finite difference method with an anisotropic mesh is proposed for the problems the existence of a maximal and a minimal compact difference solution is proved by the method of upper and lower solutions and two sufficient conditions for the uniqueness of the solution are also given the optimal error estimate in the discrete l norm is obtained under certain conditions the error estimate shows the fourth order accuracy of the proposed method when two spatial mesh sizes are proportional by using an upper solution or a lower solution as the initial iteration an almost optimal picard type of monotone iterative algorithm is developed for solving the resulting nonlinear discrete system efficiently numerical results are presented to confirm our theoretical analysis

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

change point estimation using nonparametric regression we consider a regression model in which the mean function may have a discontinuity at an unknown point we propose an estimate of the location of the discontinuity based on one side nonparametric regression estimates of the mean function the change point estimate is shown to converge in probability at rate 0 n 1 and to have the same asymptotic distribution as maximum likelihood estimates considered by other authors under parametric regression models confidence regions for the location and size of the change are also discussed

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

tests for a change point summary the problem considered is that of testing a sequence of independent normal random variables with constant known or unknown variance for no change in mean versus alternatives with a single change point various tests such as those based on the likelihood ratio and recursive residuals are studied power approximations are developed by integrating approximations for conditional boundary crossing probabilities a comparison of several tests is made

unbiased testing in exponential family regression let x ij z i i 1 2 k j 1 2 n i be independent observations such that z i is a fixed r x 1 vector r can be 0 no z s observed or 1 2 k 1 and x ij is distributed according to a one parameter exponential family which is log concave with natural parameter i we test the hypothesis that z where 1 k z is the matrix whose ith row is z i and 1 r is a vector of parameters we focus on r 2 and z i 1 z i i 1 2 k z i z i 1 the null hypothesis on hand is thus of the form i 1 2 z i in such a case the model under the null hypothesis becomes logistic regression in the binomial case poisson regression in the poisson case and linear regression in the normal case we consider mostly the one sided alternative that the second order differences of the natural parameters are nonnegative such testing problems test goodness of fit vs alternatives in which the natural parameters behave in a convex way we find classes of tests that are unbiased and that lie in a complete class we also note that every admissible test of constant size is unbiased in some discrete situations we find the minimal complete class of unbiased admissible tests generalizations and examples are given

conditional boundary crossing probabilities with applications to change point problems abstract for normal random walks s sub 1 s sub 2 formed from independent identically distributed random variables x sub 1 x sub 2 we determine the asymptotic behavior under regularity conditions of p s sdub n mg n m for some n m sub m mxi m lambda0 xi g 1 where u sub m x sub one to the 2nd power x sub m to the 2nd power the result is applied to a normal change point problem to approximate null distributions of test statistics and to obtain approximate confidence sets for the change point

finding strong pseudoprimes to several bases ii define m to be the smallest strong pseudoprime to all the first m prime bases if we know the exact value of m we will have for integers n m a deterministic efficient primality testing algorithm which is easy to implement thanks to pomerance et al and jaeschke the m are known for 1 m 8 upper bounds for g 10 and 11 were first given by jaeschke and those for 10 and 11 were then sharpened by the first author in his previous paper math comp 70 2001 863 872 in this paper we first follow the first author s previous work to use biquadratic residue characters and cubic residue characters as main tools to tabulate all strong pseudoprimes spsp s n 1024 to the first five or six prime bases which have the form n pq with p q odd primes and q 1 k p 1 k 4 3 5 2 3 2 6 then we tabulate all carmichael numbers 1020 to the first six prime bases up to 13 which have the form n q1q2q3 with each prime factor qi 3 mod 4 there are in total 36 such carmichael numbers 12 numbers of which are also spsp s to base 17 5 numbers are spsp s to bases 17 and 19 one number is an spsp to the first 11 prime bases up to 31 as a result the upper bounds for 9 10 and 11 are lowered from 20 and 22 decimal digit numbers to a 19 decimal digit number 9 10 11 q11 3825 12305 65464 13051 19 digits 149491 747451 34233211 we conjecture that 9 10 11 3825 12305 65464 13051 and give reasons to support this conjecture the main idea for finding these carmichael numbers is that we loop on the largest prime factor q3 and propose necessary conditions on n to be a strong pseudoprime to the first 5 prime bases comparisons of effectiveness with arnault s bleichenbacher s jaeschke s and pinch s methods for finding carmichael numbers with three prime factors which are strong pseudoprimes to the first several prime bases are given

inferences for the linear errors in variables with changepoint models abstract a new linear structural errors in variables regression with changepoint model is considered in this model we consider the likelihood ratio test based on the maximum hotelling t 2 for the test of no change against the alternative of exactly one change if there is a change either known a priori or by testing then we estimate the unknown changepoint parameter and some other related parameters by maximum likelihood the limiting distribution of the changepoint estimator is investigated and it is shown that the asymptotic efficiency increases as the absolute regression slope coefficient increases a monte carlo study shows that the proposed estimator performs satisfactorily

two kinds of strong pseudoprimes up to 10 36 let n 1 be an odd composite integer write n 1 2 s d with d odd if either b d 1 mod n or b 2r d 1 mod n for some r 0 1 s 1 then we say that n is a strong pseudoprime to base b or spsp b for short define t to be the smallest strong pseudoprime to all the first t prime bases if we know the exact value of t we will have for integers n 10 36 and to give reasons and numerical evidence of k2 and c 3 spsp s 12 where t resp t is the smallest k2 resp c 3 strong pseudoprime to all the first t prime bases for this purpose we describe procedures for computing and enumerating the two kinds of spsp s 10 36 to the first 9 prime bases the entire calculation took about 4000 hours on a pc pentium iv 1 8ghz recall that a k2 spsp is an spsp of the form n pq with p q primes and q 1 2 p 1 and that a c 3 spsp is an spsp and a carmichael number of the form n q 1 q 2 q 3 with each prime factor q i 3 mod 4

decision theoretic optimality of the cusum procedure suppose x 1 x 2 are independent random variables such that for some unknown each of x 1 x 1 is distributed according to f 0 while x x 1 are all distributed according to f 1 we prove a result of moustakides that claims that the cusum procedures are optimal in the sense of lorden we do that by proving that the procedures are bayes for some stochastic mechanism of generating

finding c 3 strong pseudoprimes let q 1 10 9 and so that 12 3186 65857 83403 11511 67461 24 digits 399165290221 798330580441 which was found by the author in an earlier paper we give reasons to support the conjecture the main idea of our method for finding those 21978 c 3 spsp 2 s is that we loop on candidates of signatures and kernels with heights bounded subject those candidates n q 1 q 2 q 3 of c 3 spsp 2 s and their prime factors q 1 q 2 q 3 to miller s tests and obtain the desired numbers at last we speed our algorithm for finding larger c 3 spsp s say up to 10 50 with a given signature to more prime bases comparisons of effectiveness with arnault s and our previous methods for finding c 3 strong pseudoprimes to the first several prime bases are given

skew hermitian based iterations for nine point approximations of convection diffusion problems we study skew hermitian based splitting methods for the iterative solution of nonsymmetric linear systems arising from high order compact hoc approximations of two dimensional convection diffusion problems such discretisations lead to nine point system matrices with block tridiagonal structures the nonsymmetric discretisation matrix for a constant coefficient grid aligned flow problem is shown to be positive definite and thus hermitian and skew hermitian splitting hss methods can be considered for the solution of the corresponding linear system for the solution of this linear system using the hss iteration we derive an analytical expression for the optimal value for the upper bound of the contraction factor comparison of the hss iteration with triangular and skew hermitian splitting tss methods and their corresponding block variants btss are carried out for various test problems cpu timings for the different splitting methods using experimentally determined optimal acceleration parameters are also given our results show that the combination of high order compact discretisations and skew hermitian based iterative methods for solving the corresponding linear systems provide efficient procedures for approximating the solutions of two dimensional convection diffusion equations

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

robustness of the likelihood ratio test for a change in simple linear regression abstract this article examines the robustness of the likelihood ratio tests for a change point in simple linear regression we first summarize the normal theory of kim and siegmund who have considered the likelihood ratio tests for no change in the regression coefficients versus the alternatives with a change in the intercept alone and with a change in the intercept and slope we then discuss the robustness of these tests using the convergence theory of stochastic processes we show that the test statistics converge to the same limiting distributions regardless of the underlying distribution we perform simulations to assess the distributional insensitivity of the test statistics to a weibull a lognormal and a contaminated normal distribution in two different cases fixed and random independent variables numerical examples illustrate that the test has a correct size and retains its power when the distribution is nonnormal we also study the effects of the independent variable s configuration with the

optimal stopping times for detecting changes in distributions it is shown that page s stopping time is optimal for the detection of changes in distributions in a well defined sense this work is a generalization of an existing result where it was shown that page s stopping time is optimal asymptotically

uniform point wise error estimates of semi implicit compact finite difference methods for the nonlinear schrodinger equation perturbed by wave operator abstract this study is devoted to analysis of semi implicit compact finite difference sicfd methods for the nonlinear schrodinger equation nls perturbed by the wave operator nlsw with a perturbation strength described by a dimensionless parameter e 0 1 uniform l norm error bounds of the proposed sicfd schemes are built to give immediate insight on point wise error occurring as time increases and the explicit dependence of the mesh size and time step on the parameter e is also figured out in the small e regime highly oscillations arise in time with o e 2 wavelength this highly oscillatory nature in time as well as the difficulty raised by the compact fd discretization make establishing the l norm error bounds uniformly in e of the sicfd methods for nlsw to be a very interesting and challenging issue the uniform l norm error bounds in e are proved to be of o h 4 and o h 4 2 3 with time step and mesh size h for well prepared and ill prepared initial data finally numerical results are reported to verify the error estimates and show the sharpness of the convergence rates in the respectively parameter regimes

poisson disorder problem with exponential penalty for delay we solve the poisson disorder problem when the delay is penalized exponentially our objective is to detect as quickly as possible the unobservable time of the change or disorder in the intensity of a poisson process the disorder time delimits two different regimes in which one employs distinct strategies e g investment advertising manufacturing we seek a stopping rule that minimizes the frequency of false alarms and an exponential unlike previous formulations which use a linear cost function of the detection delay in the financial applications the exponential penalty is a more apt measure for the delay cost because of the compounding of the investment growth the poisson disorder problem with a linear delay cost was studied by peskir and shiryaev 2002 solving the poisson disorder problem advances in finance and stochastics springer berlin germany 295 312 which is a limiting case of ours

tests for parameter instability and structural change with unknown change point this paper considers tests for parameter instability and structural change with unknown change point the results apply to a wide class of parametric models that are suitable for estimation by generalized method of moments procedures the asymptotic distributions of the test statistics considered here are nonstandard because the change point parameter only appears under the alternative hypothesis and not under the null the tests considered here are shown to have nontrivial asymptotic local power against all alternatives for which the parameters are nonconstant the tests are found to perform quite well in a monte carlo experiment reported elsewhere copyright 1993 by the econometric society

strong pseudoprimes to the first eight prime bases define m to be the smallest strong pseudoprime to the firstm prime bases the exact value of m is known for 1 m 8 z zhang have found a 19decimal digit number q11 3825 12305 65464 13051 which is a strong pseudoprime to the first 11 prime bases and he conjectured that 9 10 11 q11 we prove the conjecture by algorithms

inference for non stationary time series regression with or without inequality constraints summary we consider statistical inference for time series linear regression where the response and predictor processes may experience general forms of abrupt and smooth non stationary behaviours over time meanwhile the regression parameters may be subject to linear inequality constraints a simple and unified procedure for structural stability checks and parameter inference is proposed in the case where the regression parameters are constrained the methodology proposed is shown to be consistent whether or not the true regression parameters are on the boundary of the restricted parameter space via utilizing an asymptotically invariant geometric property of polyhedral cones

average case error estimates for the strong probable prime test consider a procedure that chooses k bit odd numbers independently and from the uniform distribution subjects each number to t independent iterations of the strong probable prime test miller rabin test with randomly chosen bases and outputs the first number found that passes all t tests let pk t denote the probability that this procedure returns a composite number we obtain numerical upper bounds for pk t for various choices of k t and obtain clean explicit functions that bound pk t for certain infinite classes of k t for example we show pioo 10 2 in addition we characterize the worst case numbers with unusually many false witnesses and give an upper bound on their distribution that is probably close to best possible

bayesian switching multiple disorder problems the switching multiple disorder problem seeks to determine an ordered infinite sequence of times of alarms which are as close as possible to the unknown times of disorders or change points at which the observable process changes its probability characteristics we study a bayesian formulation of this problem for an observable brownian motion with switching constant drift rates the method of proof is based on the reduction of the initial problem to an associated optimal switching problem for a three dimensional diffusion posterior probability process and the analysis of the equivalent coupled parabolic type free boundary problem we derive analytic form estimates for the bayesian risk function and the optimal switching boundaries for the components of the posterior probability process

thresholded multivariate principal component analysis for phase i multichannel profile monitoring abstractmonitoring multichannel profiles has important applications in manufacturing systems improvement but it is nontrivial to develop efficient statistical methods because profiles are high dim

detecting relevant changes in time series models summary most of the literature on change point analysis by means of hypothesis testing considers hypotheses of the form h0 1 2versush1 1 2 where 1 and 2 denote parameters of the process before and after a change point the paper takes a different perspective and investigates the null hypotheses of no relevant changes i e h0 1 2 where is an appropriate norm this formulation of the testing problem is motivated by the fact that in many applications a modification of the statistical analysis might not be necessary if the difference between the parameters before and after the change point is small a general approach to problems of this type is developed which is based on the cumulative sum principle for the asymptotic analysis weak convergence of the sequential empirical process must be established under the alternative of non stationarity and it is shown that the resulting test statistic is asymptotically normally distributed the results can also be used to establish similarity of the parameters i e h1 1 2 at a controlled type 1 error and to estimate the magnitude 1 2 of the change with a corresponding confidence interval several applications of the methodology are given including tests for relevant changes in the mean variance parameter in a linear regression model and distribution function among others the finite sample properties of the new tests are investigated by means of a simulation study and illustrated by analysing a data example from portfolio management

nonparametric inference on structural breaks this paper proposes estimators of location and size of structural breaks in a possibly dynamic nonparametric regression model the structural breaks can be located at given periods of time and or they can be explained by the values taken by some regressor as in threshold models no previous knowledge of the underlying regression function is required the paper also studies the case in which several regressors explain the breaks we derive the rate of convergence and provide central limit theorems for the estimators of the location s and size s a monte carlo experiment illustrates the performance of our estimators in small samples

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

optimal point wise error estimate of a compact difference scheme for the klein gordon schrodinger equation abstract in this paper we propose a compact finite difference scheme for computing the klein gordon schrodinger equation kgse with homogeneous dirichlet boundary conditions the proposed scheme not only conserves the total mass and energy in the discrete level but also is linearized in practical computation except for the standard energy method a new technique is introduced to obtain the optimal convergent rate without any restriction on the grid ratios at the order of o h 4 2 in the l norm with time step and mesh size h finally numerical results are reported to test the theoretical results

sequential changepoint detection in quality control and dynamical systems after a brief survey of a large variety of sequential detection procedures that are widely scattered in statistical references on quality control and engineering references on fault detection and signal processing we study some open problems concerning these procedures and introduce a unified theory of sequential changepoint detection this theory leads to a class of sequential detection rules which are not too demanding in computational and memory requirements for on line implementation and yet are nearly optimal under several performance criteria

on block circulant preconditioners for high order compact approximations of convection diffusion problems we study some properties of block circulant preconditioners for high order compact approximations of convection diffusion problems for two dimensional problems the approximation gives rise to a nine point discretisation matrix and in three dimensions we obtain a nineteen point matrix we derive analytical expressions for the eigenvalues of the block circulant preconditioner and this allows us to establish the invertibility of the preconditioner in both two and three dimensions the eigenspectra of the preconditioned matrix in the two dimensional case is described for different test cases our numerical results indicate that the block circulant preconditioning leads to significant reduction in iteration counts and comparisons between the high order compact and upwind discretisations are carried out for the unpreconditioned systems we observe fewer iteration counts for the hoc discretisation but for the preconditioned systems we find similar iteration counts for both finite difference approximations of constant coefficient two dimensional convection diffusion problems

bayesian sequential detection with phase distributed change time and nonlinear penalty a pomdp lattice programming approach we show that the optimal decision policy for several types of bayesian sequential detection problems has a threshold switching curve structure on the space of posterior distributions this is established by using lattice programming and stochastic orders in a partially observed markov decision process pomdp framework a stochastic gradient algorithm is presented to estimate the optimal linear approximation to this threshold curve we illustrate these results by first considering quickest time detection with phase type distributed change time and a variance stopping penalty then it is proved that the threshold switching curve also arises in several other bayesian decision problems such as quickest transient detection exponential delay risk sensitive penalties stopping time problems in social learning and multi agent scheduling in a changing world using blackwell dominance it is shown that for dynamic decision making problems the optimal decision policy is lower bounded by a myopic policy finally it is shown how the achievable cost of the optimal decision policy varies with change time distribution by imposing a partial order on transition matrices

permutation tests for joinpoint regression with applications to cancer rates the identification of changes in the recent trend is an important issue in the analysis of cancer mortality and incidence data we apply a joinpoint regression model to describe such continuous changes and use the grid search method to fit the regression function with unknown joinpoints assuming constant variance and uncorrelated errors we find the number of significant joinpoints by performing several permutation tests each of which has a correct significance level asymptotically each p value is found using monte carlo methods and the overall asymptotic significance level is maintained through a bonferroni correction these tests are extended to the situation with non constant variance to handle rates with poisson variation and possibly autocorrelated errors the performance of these tests are studied via simulations and the tests are applied to u s prostate cancer incidence and mortality rates

nonparametric sequential change point detection by a vertically trimmed box method this paper examines a new method for sequential detection of a sudden and unobservable change in a sequence of independent observations with completely unspecified distribution functions a nonparametric detection rule is proposed which relies on the concept of a moving vertically trimmed box as such it will be coined as the vertical box control chart v box chart its implementation requires merely to count the number of data points which fall into the box attached to the last available observation no a priori knowledge of data distributions is required and proper tuning of the box size provides a quick detection technique this is supported by establishing statistical properties of the method which explain the role of the tuning parameters used in the v box chart these theoretical results are verified by simulation studies which indicate that the v box chart may provide quick detection with zero delay for jumps of moderate sizes its averaged run length to detection is more favorable than the one for the classical ewma method by comparison with the classical shewhart chart which was optimized for normal errors our method provides comparable or better performance

boundary crossing probabilities and statistical applications abstract this paper surveys recent results involving boundary crossing probabilities and related statistical applications the first part is concerned with problems of sequential analysis especially repeated significance tests and their application to sequential clinical trials involving survival data the second part develops the probability theory motivated by the problems of part 1 a method for computing first passage distributions of brownian motion to linear boundaries is introduced and then modified to handle problems in discrete time and those involving nonlinear boundaries the third part is concerned with fixed sample statistical problems especially change point problems which involve boundary crossing probabilities examples are given of problems for which methods of part 2 appear adequate and of problems which require new methods

quickest detection of a minimum of two poisson disorder times a multisource quickest detection problem is considered assume there are two independent poisson processes x 1 and x 2 with disorder times theta 1 and theta 2 respectively i e the intensities of x 1 and x 2 change at random unobservable times theta 1 and theta 2 respectively theta 1 and theta 2 are independent of each other and are exponentially distributed define theta triangleq theta 1 wedge theta 2 min theta 1 theta 2 for any stopping time t that is measurable with respect to the filtration generated by the observations define a penalty function of the form r tau mathbb p tau 0 and tau theta is the positive part of tau theta it is of interest to find a stopping time t that minimizes the above performance index this performance criterion can be useful e g in the following scenario there are two assembly lines that produce products a and b respectively assume that the malfunctioning disorder of the machines producing a and b are independent events later the products a and b are to be put together to obtain another product c a product manager who is worried about the quality of c will want to detect the minimum of the disorder times as accurately as possible in the assembly lines producing a and b another problem to which we can apply our framework is the internet surveillance problem a router receives data from say n channels the channels are independent and the disorder times of channels are theta 1 ldots theta n the router is said to be under attack at theta theta 1 wedge cdots wedge theta n the administrator of the router is interested in detecting theta as quickly as possible since both observations x 1 and x 2 reveal information about the disorder time t even this simple problem is more involved than solving the disorder problems for x 1 and x 2 separately this problem is formulated in terms of a three dimensional sufficient statistic and the corresponding optimal stopping problem is examined the solution is characterized by iterating a suitable functional operator

approximate tail probabilities for the maxima of some random fields abstract hogan and siegmund 1986 adapt the method developed by pickands 1969 qualls and watanabe 1973 and bickel and rosenblatt 1973 to obtain explicit large deviation approximations for the maxima of several gaussian random fields arising in statistics using a special argument for one particular case they suggest a heuristic second order approximation for that case and they show by a monte carlo experiment that the second order approximation frequently gives considerably better numerical results the purpose of this paper is to show that the method developed by woodroofe 1976 1982 for problems in one dimensional time can be adapted to study maxima of random fields overall it involves simpler computations than the previous method and consequently seems potentially capable of delivering a genuine second order approximation should one seem desirable

testing for parameter constancy in linear regressions an empirical distribution function approach this paper proposes some tests for parameter constancy in linear regressions the tests use weighted empirical distribution functions of estimated residuals and are asymptotically distribution free the local power analysis reveals that the proposed tests have nontrivial local power against a wide range of alternatives in particular the tests are capable of detecting error heterogeneity that is not necessarily manifested in the form of changing variances the model allows for both dynamic and trending regressors the residuals may be obtained based on any root n consistent estimator under the null of regression parameters as an intermediate result some weak convergence for stochastically weighted sequential empirical processes is established

bayesian analysis of a poisson process with a change point summary a bayesian approach to estimation and hypothesis testing for a poisson process with a change point is developed and an example given

the cusum test of homogeneity with an application in spontaneous abortion epidemiology we test the hypothesis of homogeneity over time in the rate of an event with a modification of page s cumulative sum cusum procedure the modification draws on an indifference zone approach to testing that is well suited to the epidemiologic study of retrospective data we provide procedures for evaluating the significance level of the largest observed cusum value occurring in a sequence of fixed length we draw inferences unconditionally with a known baseline rate of the event and conditionally on observed margins with unknown baseline rate data collected over a five year period from one new york city hospital serve to illustrate the modified cusum test in a study of fluctuations in the proportions of chromosomally normal trisomic x monosomic and triploid spontaneous abortions

identifying optimal risk windows for self controlled case series studies of vaccine safety in vaccine safety studies subjects are considered at increased risk for adverse events for a period of time after vaccination known as risk window to our knowledge risk windows for vaccine safety studies have tended to be pre defined and not to use information from the current study inaccurate specification of the risk window can result in either including the true control period in the risk window or including some of the risk window in the control period which can introduce bias we propose a data based approach for identifying the optimal risk windows for self controlled case series studies of vaccine safety the approach involves fitting conditional poisson regression models to obtain incidence rate ratio estimates for different risk window lengths for a specified risk window length l the average time at risk t l is calculated when the specified risk window is shorter than the true the incidence rate ratio decreases with 1 t l increasing but there is no explicit relationship when the specified risk window is longer than the true the incidence rate ratio increases linearly with 1 t l increasing theoretically the risk window with the maximum incidence ratio is the optimal risk window because of sparse data problem we recommend using both the maximum incidence rate ratio and the linear relationship when the specified risk window is longer than the true to identify the optimal risk windows both simulation studies and vaccine safety data applications show that our proposed approach is effective in identifying medium and long risk windows

a note on ritov s bayes approach to the minimax property of the cusum procedure 1 introduction page 1954 proposed the cusum procedures as a solu tion to the problem of sequential detection of a change in distribution lorden 1971 introduced a notion of minimax optimality for change point models andshowed that the cusum procedures are asymptotically optimal in his sense moustakides 1986 then showed the minimax optimality of the cusum pro cedures when the initial and nal distribution are both known ritov 1990 used a bayes approach and gave an interesting argument for moustakides result similar to that of lehmann 1959 on the optimality of the sequentialprobability ratio test we transfer his approach to a continuous time modeland discuss the structure of the bayes risk let be an arbitrary but known constant we will consider a process wwithw

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

dynamic linear models with markov switching abstract in this paper hamilton s 1988 1989 markov switching model is extended to a general state space model this paper also complements shumway and stoffer s 1991 dynamic linear models with switching by introducing dependence in the switching process and by allowing switching in both measurement and transition equations building upon ideas in hamilton 1989 cosslett and lee 1985 and harrison and stevens 1976 a basic filtering and smoothing algorithm is presented the algorithm and the maximum likelihood estimation procedure is applied in estimating lam s 1990 generalized hamilton model with a general autoregressive component the estimation results show that the approximation employed in this paper performs an excellent job with a considerable advantage in computation time a state space representation is a very flexible form and the approach taken in this paper therefore allows a broad class of models to be estimated that could not be handled before in addition the algorithm for calculating smoothed inferences on the unobserved states is a vastly more efficient one than that in the literature

quickest detection with exponential penalty for delay the problem of detecting a change in the probability distribution of a random sequence is considered stopping times are derived that optimize the tradeoff between detection delay and false alarms within two criteria in both cases the detection delay is penalized exponentially rather than linearly as has been the case in previous formulations of this problem the first of these two criteria is to minimize a worst case measure of the exponential detection delay within a lower bound constraint on the mean time between false alarms expressions for the performance of the optimal detection rule are also developed for this case it is seen for example that the classical page cusum test can be arbitrarily unfavorable relative to the optimal test under exponential delay penalty the second criterion considered is a bayesian one in which the unknown change point is assumed to obey a geometric prior distribution in this case the optimal stopping time effects an optimal trade off between the expected exponential detection delay and the probability of false alarm finally generalizations of these results to problems in which the penalties for delay may be path dependent are also considered

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

on strong pseudoprimes to several bases with y k denoting the smallest strong pseudoprime to all of the first k primes taken as bases we determine the exact values for 5 q6 q7 q8 and give upper bounds for v 9 w t 1 we discuss the methods and underlying facts for obtaining these results 1 primality tests by means of strong pseudoprimes computer algebra systems as for instance axiom 2 use strong pseudoprimes for testing primality of integers the advantage of such tests is that they are very efficient the disadvantage is that they are only probabilistic tests when the integers are not restricted to certain intervals to make such tests deterministic for integers in prescribed intervals one has to know the exact number of necessary so called strong pseudoprimality tests for this purpose we introduce the numbers v1i v 2 for which we compute lower and upper bounds these numbers are defined and discussed in this section in 2 we derive some facts which are the basis for finding bounds for the numbers v k in 3 we discuss the methods which led to our results in view of fermat s little theorem we know that n is certainly not a prime when we have bn 1 i 1 mod n for an integer b with 1 0 and when n is a composite number then n is called a strong pseudoprime to base b if either

finding strong pseudoprimes to several bases define m to be the smallest strong pseudoprime to all the first m prime bases if we know the exact value of m we will have for integers n m a deterministic primality testing algorithm which is not only easier to implement but also faster than either the jacobi sum test or the elliptic curve test thanks to pornerance et al and jaeschke m are known for 1 m 8 upper bounds for 9 10 and 1 were given by jaeschke in this paper we tabulate all strong pseudoprimes spsp s n 10 24 to the first ten prime bases 2 3 29 which have the form n pq with p q odd primes and q 1 k p 1 k 2 3 4 there are in total 44 such numbers six of which are also spsp 31 and three numbers are spsp s to both bases 31 and 37 as a result the upper bounds for 10 and 11 are lowered from 28 and 29 decimal digit numbers to 22 decimal digit numbers and a 24 decimal digit upper bound for 12 is obtained the main tools used in our methods are the biquadratic residue characters and cubic residue characters we propose necessary conditions for n to be a strong pseudoprime to one or to several prime bases comparisons of effectiveness with both jaeschke s and arnault s methods are given