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The well-known statistic PageRank was created in 1998 by co-founders of Google, Sergey Brin and Larry Page, to optimize the ranking of websites for their search engine outcomes. It is computed using an iterative algorithm, based on the idea that nodes with a larger number of incoming edges are more important. Google's PageRank involves some information from ``aliens''; the 15% of information is regarded as the connections from the outside of the network system under consideration. In this paper, seeking a stable statistic which is ``close'' to an ``intrinsic'' version of PageRank, we will introduce a new statistic called MarkovRank. A special attention will be paid to the comparison of rank statistics among standard-PageRank,``intrinsic-PageRank'' and MarkovRank. It is concluded that the rank statistic of MarkovRank, which is always well-defined, is identical to that of ``intrinsic-PageRank'', as far as the latter is well-defined., Comment: 16 pages, 4 figures

A change point detection procedure using the method of moment estimators is proposed. The test statistics is based on a suitable $Z$-process. The asymptotic behavior of this process is established under both the null and the alternative hypothesis and the consistency of the test is also proved. An estimator for the change point is proposed and its consistency is derived. Some examples of this method applied to a parametric family of random variables are presented.

As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a stochastic maximal inequality derived by using the Taylor expansion, is presented. The inequality dealing with finite-dimensional discrete-time martingales is pulled up to infinite-dimensional ones by using the monotone convergence arguments. The main results are some weak convergence theorems for sequences of separable random fields of discrete-time martingales under the uniform topology with the help also of entropy methods. As special cases, some new results for i.i.d. random sequences, including a new Donsker theorem and a moment bound for suprema of empirical processes indexed by classes of sets or functions, are obtained., Comment: This paper has been withdrawn by the author due to a crucial error

In this paper, weak convergences of marked empirical processes in $L^2(\mathbb{R},\nu)$ and their applications to statistical goodness-of-fit tests are provided, where $L^2(\mathbb{R},\nu)$ is the set of equivalence classes of the square integrable functions on $\mathbb{R}$ with respect to a finite Borel measure $\nu$. The results obtained in our framework of weak convergences are, in the topological sense, weaker than those in the Skorokhod topology on a space of c\'adl\'ag functions or the uniform topology on a space of bounded functions, which have been well studied in previous works. However, our results have the following merits: (1) avoiding conditions which do not suit for our purpose; (2) treating a weight function which makes us possible to propose an Anderson--Darling type test statistics for goodness-of-fit tests. Indeed, the applications presented in this paper are novel., Comment: 22 pages

As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a {\em stochastic maximal inequality} derived by using the formula for integration by parts and on a new concept named {\em strict countability}, is presented. The main results are some weakconvergence theorems for sequences of separable random fields of discrete-time martingales under the uniform topology with the help also of entropy methods. As special cases, some new results for i.i.d.\ random sequences, including a new Donsker theorem and a moment bound for suprema of empirical processes indexed by classes of sets or functions, are obtained., Comment: This paper has been withdrawn by the author due to an crucial error

The Dantzig selector for a special parametric model of diffusion processes is studied in this paper. In our model, the diffusion coefficient is given as the exponential of the linear combination of other processes which are regarded as covariates. We propose an estimation procedure which is an adaptation of the Dantzig selector for linear regression models and prove the $l_q$ consistency of the estimator for all $q \in [1,\infty]$., Comment: 15 pages

The Dantzig selector for the proportional hazards model proposed by D.R. Cox is studied in a high-dimensional and sparse setting. We prove the $l_q$ consistency for all $q \geq 1$ of some estimators based on the compatibility factor, the weak cone invertibility factor, and the restricted eigenvalue for certain deterministic matrix which approximates the Hessian matrix of log partial likelihood. Our matrix conditions for these three factors are weaker than those of previous researches., Comment: 13 pages

As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a stochastic maximal inequality derived by using It\^o's formula and on a new concept named strict countability, is presented. The main results are some weak convergence theorems for sequences of separable random fields of locally square-integrable martingales under the uniform topology with the help also of entropy methods. As special cases, some new results for i.i.d. random sequences, including a new Donsker theorem and a moment bound for suprema of empirical processes indexed by classes of sets or functions, are obtained. An application to statistical estimation in semiparametric models is presented with an illustration to construct adaptive estimators in Cox's regression model., Comment: This paper has been withdrawn by the author due to a crucial error in the proof of Lemma 16 (i)

The problem to establish not only the asymptotic distribution results for statistical estimators but also the moment convergence of the estimators has been recognized as an important issue in advanced theories of statistics. One of the main goals of this paper is to present a metod to derive the moment convergence of $Z$-estimators as it has been done for $M$-estimators. Another goal of this paper is to develop a general, unified approach, based on some partial estimation functions which we call "$Z$-process", to the change point problems for ergodic models as well as some models where the Fisher information matrix is random and inhomogeneous in time. Applications to some diffusion process models and Cox's regression model are also discussed., Comment: 28 pages

This paper generalizes a part of the theory of $Z$-estimation which has been developed mainly in the context of modern empirical processes to the case of stochastic processes, typically, semimartingales. We present a general theorem to derive the asymptotic behavior of the solution to an estimating equation $\theta\leadsto \Psi_n(\theta,\widehat{h}_n)=0$ with an abstract nuisance parameter $h$ when the compensator of $\Psi_n$ is random. As its application, we consider the estimation problem in an ergodic diffusion process model where the drift coefficient contains an unknown, finite-dimensional parameter $\theta$ and the diffusion coefficient is indexed by a nuisance parameter $h$ from an infinite-dimensional space. An example for the nuisance parameter space is a class of smooth functions. We establish the asymptotic normality and efficiency of a $Z$-estimator for the drift coefficient. As another application, we present a similar result also in an ergodic time series model., Comment: Published in at http://dx.doi.org/10.1214/09-AOS693 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider a nonparametric goodness of fit test problem for the drift coefficient of one-dimensional small diffusions. Our test is based on discrete observation of the processes, and the diffusion coefficient is a nuisance function which is estimated in our testing procedure. We prove that the limit distribution of our test is the supremum of the standard Brownian motion, and thus our test is asymptotically distribution free. We also show that our test is consistent under any fixed alternatives.

This note extends some results of Nishiyama [Ann. Probab. 28 (2000) 685--712]. A maximal inequality for stochastic integrals with respect to integer-valued random measures which may have infinitely many jumps on compact time intervals is given. By using it, a tightness criterion is obtained; if the so-called quadratic modulus is bounded in probability and if a certain entropy condition on the parameter space is satisfied, then the tightness follows. Our approach is based on the entropy techniques developed in the modern theory of empirical processes., Comment: Published at http://dx.doi.org/10.1214/009117906000000755 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

A goodness of fit test for the drift coefficient of an ergodic diffusion process is presented. The test is based on the score marked empirical process. The weak convergence of the proposed test statistic is studied under the null hypotheses and it is proved that the limit process is a continuous Gaussian process. The structure of its covariance function allows to calculate the limit distribution and it turns out that it is a function of a standard Brownian motion and so exact reject regions can be constructed. The proposed test is asymptotically distribution free and it is consistent under any simple fixed alternative.