A multiple window scan statistic for time series models

In this article we extend the results derived for scan statistics in Wang and Glaz (2014) for independent normal observations. We investigate the performance of two approximations for the distribution of fixed window scan statistics for time series models. An R algorithm for computing multivariate normal probabilities established in Genz and Bretz (2009) can be used along with proposed approximations to implement fixed window scan statistics for ARMA models. The accuracy of these approximations is investigated via simulation. Moreover, a multiple window scan statistic is defined for detecting a local change in the mean of a Gaussian white noise component in ARMA models, when the appropriate length of the scanning window is unknown. Based on the numerical results, for power comparisons of the scan statistics, we can conclude that when the window size of a local change is unknown, the multiple window scan statistic outperforms the fixed window scan statistics.     

Asymptotic results in segmented multiple regression

This paper studies the asymptotic behavior of the least squares estimators in segmented multiple regression. For a model with more than one partitioning variable, each of which has one or more change-points, we study the asymptotic properties of the estimated change-points and regression coefficients. Using techniques in empirical process theory, we prove the consistency of the least squares estimators and also establish the asymptotic normality of the estimated regression coefficients. For the estimated change-points, we obtain their consistency at the rates of or 1/n, with or without continuity constraints, respectively. The change-points estimated under the continuity constraints are also shown to asymptotically have a multivariate normal distribution. For the case where the regression mean functions are not assumed to be continuous at the change-points, the asymptotic distribution of the estimated change-points involves a step function process, whose distribution does not follow a well-known distribution.     

On the asymptotic distribution of the scan statistic for empirical distributions

Extremes - This paper investigates the asymptotic behavior of several variants of the scan statistic for empirical distributions, which can be applied to detect the presence of an anomalous...     

Asymptotic probabilities for conformal restriction measures for an annulus

We show that for the conformal restriction measure with exponent b in the unit disk on hulls γ connecting eix to 1 the probability of the event that γ avoids the disk of radius q centered at zero decays like exp(−bπx/(1−q)) if either b∈[5/8,1]∪[5/4,∞) and x∈(0,π], or if b∈(1,5/4), x∈(0,π), and bx?π.     

Representation results for jump processes with application to optimal stopping

It is shown that functions, measurable on the past of a jump process up to a stopping time, can be expressed as functions of the jump times and jump locations up to the stopping time. These results lead to formulas for conditional expectations with respect to the past of the process up to the stopping time. The use of these results is illustrated in giving a sufficient condition for optimality for optimal stopping of a partially observed jump Markov process.     

Asymptotic results for random processes

In this paper we establish asymptotic results and a generalized uniform law of the iterated logarithm (LIL) for the increments of a strictly stationary random process, whose results are proved by separating linearly positive quadrant dependent (LPQD) random process and linearly negative quadrant dependent (LNQD) one, respectively.     

Asymptotic expansions for the statistic and risk function of a Bayesian classification rule

Asymptotic expansions for the stastistic and risk function of a Bayesian classification rule for the problem of assigning an observation to one of the two one-dimensional populations with unknown parameters are constructed. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 69–77, Perm, 1991.     

Asymptotic results for polygonal processes related to an autoregression

We establish functional central limit theorems for polygonal process constructed from consecutive estimators of a simple AR(1) model. We consider both stationary and unit root cases. The results are applied to change segment analysis.     

A spatial scan statistic for case event data based on connected components

We propose a new spatial scan statistic based on graph theory as a method for detecting irregularly-shaped clusters of events over space. A graph-based method is proposed for identifying potential clusters in spatial point processes. It relies on linking the events closest than a given distance and thus defining a graph associated to the point process. The set of possible clusters is then restricted to windows including the connected components of the graph. The concentration in each of these possible clusters is measured through classical concentration indices based on likelihood ratio and also through a new concentration index which does not depend on any alternative hypothesis. These graph-based spatial scan tests seem to be very powerful against any arbitrarily-shaped cluster alternative, whatever the dimension of the data. These results have applications in various fields, such as the epidemiological study of rare diseases or the analysis of astrophysical data.     

Asymptotic robustness of the normal theory likelihood ratio statistic for two-level covariance structure models

Data in social and behavioral sciences are often hierarchically organized. Special statistical procedures have been developed to analyze such data while taking into account the resulting dependence of observations. Most of these developments require a multivariate normality distribution assumption. It is important to know whether normal theory-based inference can still be valid when applied to nonnormal hierarchical data sets. Using an analytical approach for balanced data and numerical illustrations for unbalanced data, this paper shows that the likelihood ratio statistic based on the normality assumption is asymptotically robust for many nonnormal distributions. The result extends the scope of asymptotic robustness theory that has been established in different contexts.     

Asymptotic normality for discretely observed Markov jump processes with an absorbing state

For a continuous-time Markov process, occasionally, only discrete-time observations are available. For a simple sample of homogeneous Markov jump processes with an absorbing state, observed each on a stochastic grid of time points, we establish asymptotic normality of the maximum likelihood estimator and close the gap in Kremer and Weißbach (2013). By showing that the solution of the Kolmogorov backward equation system is continuous differentiable, we can apply results for M-estimators.