Testing hypotheses on the “drift” of parameters in ARMA and ARCH models

For an ARMA model, we test the hypothesis that the coefficients of this model remain constant in time and satisfy the stationarity condition against the alternative that the coefficients change (“drift”) in time. We propose asymptotically distribution free tests for such hypothesis based on sequential residual processes. A similar problem is solved for the ARCH model.      

Locally most powerful sequential tests for stochastic processes

For a continuous time stochastic process with distribution P_? depending on a one-dimensional parameter ? the problem of sequentially testing ? = 0 against ? > 0 is treated. We assume that the process of likelihood ratios has a certain representation which allows to obtain identities of the Wald type for stopping times. These identities are then used to derive a result on locally most powerful tests for which a problem of optimal stopping is solved.     

A computational study of graph partitioning

LetG = (N, E) be an edge-weighted undirected graph. The graph partitioning problem is the problem of partitioning the node setN intok disjoint subsets of specified sizes so as to minimize the total weight of the edges connecting nodes in distinct subsets of the partition. We present a numerical study on the use of eigenvalue-based techniques to find upper and lower bounds for this problem. Results for bisecting graphs with up to several thousand nodes are given, and for small graphs some trisection results are presented. We show that the techniques are very robust and consistently produce upper and lower bounds having a relative gap of typically a few percentage points.Corresponding author.     

Factorization of linear elliptic boundary value problems in non-cylindrical domains

We present a method of factorization for linear elliptic boundary value problems considered in non-cylindrical domains. We associate a control problem to the boundary value problem which regularizes it. The technique of change of variables is used to study this problem.     

Dynamic Detection of Change Points in Long Time Series

We consider the problem of detecting change points (structural changes) in long sequences of data, whether in a sequential fashion or not, and without assuming prior knowledge of the number of these change points. We reformulate this problem as the Bayesian filtering and smoothing of a non standard state space model. Towards this goal, we build a hybrid algorithm that relies on particle filtering and Markov chain Monte Carlo ideas. The approach is illustrated by a GARCH change point model.     

Optimal speed of detection in generalized Wiener disorder problems

We define a general Wiener disorder problem in which a sudden change in a time profile of unknown size has to be detected in white noise of small intensity. Since both the time of the change and its size are unknown, this problem is considerably harder than standard Wiener disorder problems where the size of the change is assumed to be known a priori. We formulate the problem within the Bayesian framework of nonlinear filtering theory, and use Strassen's functional law of the iterated logarithm to bound stochastic measures which arise in the nonlinear filtering equations. This leads to explicit expressions for the detection delay in the optimal statistics for small noise intensities, and we indicate how these can be used to analyse the detection delays of recursive suboptimal detection algorithms for this problem.     

A new strategy for the undirected two-commodity maximum flow problem

We address the two-commodity maximum flow problem on undirected networks. As a result of a change of variables, we introduce a new formulation that solves the problem through classical maximum flow techniques with only one-commodity. Therefore, a general strategy, based on this change of variables, is defined to deal with other undirected multi-commodity problems. Finally, we extend the single objective problem to a bicriteria environment. We show that the set of efficient solutions of the biobjective undirected two-commodity maximum flow problem is the set of alternative optimum solutions of the undirected two-commodity maximum flow problem. In addition, we prove that the set of efficient extreme points in the objective space has, at most, cardinality two.     

Pseudo-moderate deviations in the euler method for real diffusion process

We consider the problem of the non-sequential detection of a change in the drift coefficient of a stochastic differential equation, when a misspecified model is used. We formulate the generalized likelihood ratio (GLR) test for this problem, and we study the behaviour of the associated error probabilities (false alarm and nodetection) in the small noise asymptotics. We obtain the following robustness result: even though a wrong model is used, the error probabilities go to zero with exponential rate, and the maximum likelihood estimator (MLE) of the change time is consistent, provided the change to be detected is larger (in some sense) than the misspecification error. We give also computable bounds for selecting the threshold of the test so as to achieve these exponential rates.     

On computer-assisted proofs for solutions of linear complementarity problems

In this paper we consider the linear complementarity problem where the components of the input data M and q are not exactly known but can be enclosed in intervals. We compare three tests to each other each of which can be used by a computer that supports interval arithmetic to give guaranteed bounds for a solution of the LCP defined by M and q.     

Asymptotically optimal estimators for moments of change

We consider the problem of finding asymptotically optimal estimators for many moments of change in the case of incomplete information on distributions. We prove that if the maximum-likelihood estimator is asymptotically optimal, then, under certain conditions, it preserves this property after the replacement of actual values by density estimators. We solve the problem for the case of one moment of change and generalize the results obtained to the case of several moments of change. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 406–416, March, 2006.     

On the solvability of the Tricomi problem with a generalized Frankl transmission condition

We study the solvability of the Tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions in the elliptic part of the domain. On the type change line of the equation, the solution gradient is subjected to a condition that is usually referred to as the generalized Frankl transmission condition. We show that the inhomogeneous Tricomi problem either has a unique solution or is conditionally solvable and the homogeneous problem has only the trivial solution. We write out an integral representation of the solution of this problem.     

Optimal algorithms for page migration in dynamic networks

We present an extension of a classical data management subproblem, the page migration. The problem is investigated in dynamic networks, where costs of communication between different nodes may change with time. We construct asymptotically optimal online algorithms for this problem, both in deterministic and randomized scenarios.     

Investigation of an analog of the Tricomi problem for the case in which the type change line does not coincide with coordinate axes

We consider the Tricomi problem with a type change line that does not coincide with any of the coordinate axes, with a circular sector in the ellipticity domain, and with a triangle formed by the type change line and segments of characteristics in the hyperbolicity domain. We also consider the adjoint problem. The solution of both problems is obtained in the form of a series. For a more general case with a curve of arbitrary shape in the ellipticity domain, we present the maximum principle for both problems.     

Numerical solutions of the m-membranes problem

In this paper numerical approximation for the m-membrane problem is considered. We make a change of variables that leads to a different expression of the quadratic functional that allows after discretizing the problem to reformulate it as finite dimensional bound constrained quadratic problem. To our knowledge this is the first paper on numerical approximation of the m-membrane problem. We reformulate the m-membrane problem as a bound constraint quadratic minimization problem. The bound constraint quadratic form is solved with the gradient projection method.     

Semi-empirical likelihood ratio confidence intervals for the difference of two sample means

We all know that we can use the likelihood ratio statistic to test hypotheses and construct confidence intervals in full parametric models. Recently, Owen (1988,Biometrika,75, 237–249; 1990,Ann. Statist.,18, 90–120) has introduced the empirical likelihood method in nonparametric models. In this paper, we combine these two likelihoods together and use the likelihood ratio to construct confidence intervals in a semiparametric problem, in which one model is parametric, and the other is nonparametric. A version of Wilks's theorem is developed.     

Online strip packing with modifiable boxes

In the strip packing problem the goal is to pack a set of rectangles into a vertical strip so as to minimize the total height of the strip needed. We consider a modified version of the strip packing problem. In this version it is allowed to change the form of the rectangles by lengthening them, keeping the area fixed. We introduce online algorithms to solve this modified problem. Moreover a lower bound is presented, as well.     

Measuring the subprime crisis contagion: Evidence of change point analysis of copula functions

In this paper, we first determine the existence of structural changes in the dependence between time series of equity index returns of two markets using the change point testing method. The method is based on Archimedean copula functions, which are able to comprehensively describe dependence characteristics of random variables. The degree of financial contagion between markets is subsequently estimated using the tail dependence coefficient of copula functions before and after the change point. We empirically test our method by investigating financial contagion during the subprime crisis between the US S&P 500 index and five Asian markets, namely China, Japan, Korea, Hong Kong and Taiwan. Our results show that a statistically significant change point exists in the dependence between the US market and all Asian stock markets except Taiwan. The upper tail dependence is larger after the time of change, implying the existence of contagion during the banking crisis between the US and the Asian economies. The degree of financial contagion is also estimated and found to be consistent with market events and media reports during that period.     

Gellerstedt problem with a generalized Frankl matching condition on the type change line with data on external characteristics

We study the solvability of the Gellerstedt problem for the Lavrent’ev–Bitsadze equation with nonclassical matching conditions for the gradient of the solution (in the sense of Frankl) on the type change line of the equation. We prove that the inhomogeneous Gellerstedt problem with data on the external characteristics of the equation is solvable either uniquely or modulo a nontrivial solution of the homogeneous problem. We obtain integral representations of the solution of the problem in both the elliptic and the hyperbolic parts of the domain. The solution proves to be regular.     

A class of rank-based test for left-truncated and right-censored data

A class of rank-based tests is proposed for the two-sample problem with left-truncated and right-censored data. The class contains as special cases the extension of log-rank test and Gehan test. The asymptotic distribution theory of the test is presented. The small-sample performance of the test is investigated under a variety of situations by means of Mone Carlo simulations.     

The inverse maximum dynamic flow problem

We consider the inverse maximum dynamic flow (IMDF) problem. IMDF problem can be described as: how to change the capacity vector of a dynamic network as little as possible so that a given feasible dynamic flow becomes a maximum dynamic flow. After discussing some characteristics of this problem, it is converted to a constrained minimum dynamic cut problem. Then an efficient algorithm which uses two maximum dynamic flow algorithms is proposed to solve the problem.