CONVERGENCE OF INVARIANT MEASURES FOR MULTIVALUED STOCHASTIC DIFFERENTIAL EQUATIONS

This article is concerned with the weak convergence of invariant measures asso- ciated with multivalued stochastic differential equations in the finite dimensional space.

Testing conditional independence with data missing at random

It is known that conditional independence is a quite basic assumption in many fields of statistics. How to test its validity is of great importance and has been extensively studied by the literature. Nevertheless, all of the existing methods focus on the case that data are fully observed, but none of them seems having taken into account of the scenario when missing data are present. Motivated by this, this paper develops two testing statistics to handle such a situation relying on the idea of inverse probability weighted and augmented inverse probability weighted techniques. The asymptotic distributions of the proposed statistics are also derived under the null hypothesis. The simulation studies indicate that both testing statistics perform well in terms of size and power.

STABILITY OF THE STOCHASTIC θ-METHOD FOR SUPER-LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY

This paper deals with numerical stability properties of super-linear stochastic differential equations with unbounded delay. Sufficient conditions for mean square and almost sure decay stability of the above system and its stochastic θ-method approximation are investigated in this paper. The author establishes numerical stability under a monotone-type condition in unbounded delay setting. An example is presented to illustrate the result.

Finite sample breakdown point of Tukey＇s halfspace median

Tukey's halfspace median(HM), servicing as the multivariate counterpart of the univariate median,has been introduced and extensively studied in the literature. It is supposed and expected to preserve robustness property(the most outstanding property) of the univariate median. One of prevalent quantitative assessments of robustness is finite sample breakdown point(FSBP). Indeed, the FSBP of many multivariate medians have been identified, except for the most prevailing one—the Tukey's halfspace median. This paper presents a precise result on FSBP for Tukey's halfspace median. The result here depicts the complete prospect of the global robustness of HM in the finite sample practical scenario, revealing the dimension effect on the breakdown point robustness and complimenting the existing asymptotic breakdown point result.

Strong convergence rate of truncated Euler-Maruyama method for stochastic differential delay equations with Poisson jumps

We study a class of super-linear stochastic differential delay equations with Poisson jumps (SDDEwPJs). The convergence and rate of the convergence of the truncated Euler-Maruyama numerical solutions to SDDEwPJs are investigated under the generalized Khasminskii-type condition.

Distribution Approximation of Shrinkage Estimate in Censored Regression Model via Randomly Weighting Method

Censored regression("Tobit") model is a special case of limited dependent variable regression model, and plays an important role in econometrics. Based on this model, all kinds of methods for variable or group variable selection have been developed and the corresponding shrinkage parameter estimates are widely studied. However, asymptotic distributions of the shrinkage estimates involve unknown nuisance parameters,such as density function of error term. To avoid estimating nuisance parameters, this paper presents a randomly weighting method to approximate to the asymptotic distribution of the shrinkage estimate. A computation procedure of random approximation is provided and asymptotic properties of the randomly weighting estimates are also obtained. The proposed methods are evaluated with extensively numerical studies and a women labor supply example.

Computing Halfspace Depth Contours Based on the Idea of a Circular Sequence

This paper presents a new effcient algorithm for exactly computing the halfspace depth contours based on the idea of a circular sequence. Unlike the existing methods, the proposed algorithm segments the unit sphere directly relying on the permutations that correspond to the projections of observations onto some unit directions, without having to use the technique of parametric programming.Some data examples are also provided to illustrate the performance of the proposed algorithm.

Estimation of Parameters of Partially Linear Errors-in-variables Models with Replicated Net Points of Observation

A kind of partially linear errors-in-variables models with replicated net points of observation are studied in this paper. Estimators of unknown parameters are given. Under certain regular conditions, it is shown that the estimators of the unknown parameters are strongly consistent and their a.s. convergence rates are achieved.

The Limit of Finite Sample Breakdown Point of Tukey’s Halfspace Median for General Data

Under special conditions on data set and underlying distribution, the limit of finite sample breakdown point of Tukey＇s halfspace median （1） has been obtained in the literature. In this paper, we establish the result under weaker assumptions imposed on underlying distribution （weak smoothness） and on data set （not necessary in general position）. The refined representation of Tukey＇s sample depth regions for data set not necessary in general position is also obtained, as a by-product of our derivation.

Model-free feature screening for high-dimensional survival data

With the rapid-growth-in-size scientific data in various disciplines, feature screening plays an important role to reduce the high-dimensionality to a moderate scale in many scientific fields. In this paper, we introduce a unified and robust model-free feature screening approach for high-dimensional survival data with censoring, which has several advantages: it is a model-free approach under a general model framework, and hence avoids the complication to specify an actual model form with huge number of candidate variables; under mild conditions without requiring the existence of any moment of the response, it enjoys the ranking consistency and sure screening properties in ultra-high dimension. In particular, we impose a conditional independence assumption of the response and the censoring variable given each covariate, instead of assuming the censoring variable is independent of the response and the covariates. Moreover, we also propose a more robust variant to the new procedure, which possesses desirable theoretical properties without any finite moment condition of the predictors and the response. The computation of the newly proposed methods does not require any complicated numerical optimization and it is fast and easy to implement. Extensive numerical studies demonstrate that the proposed methods perform competitively for various configurations. Application is illustrated with an analysis of a genetic data set.

Pentavalent vertex-transitive tiameter two graphs

We classify the family of pentavalent vertex-transitive graphs F with diameter 2. Suppose that the automorphism group of F is transitive on the set of ordered distance 2 vertex pairs. Then we show that either F is distancetransitive or F is one of C8-, K5 K2, C5[K2], 2C4, or K3 K4.

Vertex-transitive Diameter Two Graphs

We investigate the family of vertex-transitive graphs with diameter 2.LetΓbe such a graph.Suppose that its automorphism group is transitive on the set of ordered non-adjacent vertex pairs.Then eitherΓis distance-transitive orΓhas girth at most 4.Moreover,ifΓhas valency 2,thenΓ≌C4 or C5;and for any integer n≥3,there exist such graphsΓof valency n such that its automorphism group is not transitive on the set of arcs.Also,we determine this family of graphs of valency less than 5.Finally,the family of diameter 2 circulants is characterized.

Empirical Likelihood-based Inferences in Varying Coefficient Models with Missing Data

In this paper, we consider the empirical likelihood-based inferences for varying coefficient models Y = X^τα（U） ＋ ε when X are subject to missing at random. Based on the inverse probability-weighted idea, a class of empirical log-likelihood ratios, as well as two maximum empirical likelihood estimators, are developed for α（u）. The resulting statistics are shown to have standard chi-squared or normal distributions asymptotically.Simulation studies are also constructed to illustrate the finite sample properties of the proposed statistics.