从x出发的漂移Brownian Motion的极值分布

该文研究了从x出发的正漂移Brownian Motion的极值问题，给出了关于这种随机过程的两种极大值的定义，并主要利用Brownian Motion的一些重要性质，比如正交不变性、时空齐次性及在有限停时上的强Markov性等，获得了两种极大值的分布函数的精确表达式。     

非时齐马尔科夫链及其轨道的概率构造

给出了一种由局部刻画构造非时齐马尔科夫链的概率方法.以任意时刻之后第一次跳的时间为条件,证明了过程的马尔科夫性,同时得到了转移概率的递归表达式—这种做法使其概率意义得以明确呈现,并且进一步证明了强马尔科夫性等一系列性质.这为数值模拟,即以Monte Carlo生成非时齐Q-过程的随机轨道提供了严格的理论基础.     

Practical powerful wavelet packet tests for second-order stationarity

Methods designed for second-order stationary time series can be misleading when applied to nonstationary series, often resulting in inaccurate models and poor forecasts. Hence, testing time series stationarity is important especially with the advent of the ‘data revolution’ and the recent explosion in the number of nonstationary time series analysis tools. Most existing stationarity tests rely on a single basis. We propose new tests that use nondecimated basis libraries which permit discovery of a wider range of nonstationary behaviours, with greater power whilst preserving acceptable statistical size. Our tests work with a wide range of time series including those whose marginal distributions possess heavy tails. We provide freeware R software that implements our tests and a range of graphical tools to identify the location and duration of nonstationarities. Theoretical and simulated power calculations show the superiority of our wavelet packet approach in a number of important situations and, hence, we suggest that the new tests are useful additions to the analyst's toolbox.     

证券组合的Markov链模型

本文将证券价格时间序列分解成趋势变动序列和 Markov链 ,建立了证券组合的 Markov链模型 ,应用 Markov链理论对此模型进行了分析 ,给出了充分大的一个时间内的收益率 ,风险和切点组合的计算公式     

Subsampling tests of parameter hypotheses and overidentifying restrictions with possible failure of identification

We introduce a general testing procedure in models with possible identification failure that has exact asymptotic rejection probability under the null hypothesis. The procedure is widely applicable and in this paper we apply it to tests of arbitrary linear parameter hypotheses as well as to tests of overidentification in time series models given by unconditional moment conditions. The main idea is to subsample classical tests, like for example the Wald or the J test. More precisely, instead of using critical values based on asymptotic theory, we compute data-dependent critical values based on the subsampling technique.We show that under full identification the resulting tests are consistent against fixed alternatives and that they have exact asymptotic rejection probabilities under the null hypothesis independent of identification failure. Furthermore, the subsampling tests of parameter hypotheses are shown to have the same local power as the original tests under full identification.An algorithm is provided that automates the block size choice needed to implement the subsampling testing procedure. A Monte Carlo study shows that the tests have reasonable size properties and often outperform other robust tests in terms of power.     

基于改进的GM(1,1)-Markov链组合模型广东省单位GDP能耗预测

本文以灰色系统理论的GM(1,1)模型和随机过程理论的Markov链模型为基础构建了一个动态GM(1,1)-Markov链组合预测模型。该模型同时利用了GM(1,1)模型对序列趋势因素良好的拟合能力和Markov链模型对残差序列信息的提取能力。为进一步提高该模型的预测精度,用泰勒(Taylor)近似方法和新信息优先的思想对该模型进行了改进。最后,以1991-2014年广东省单位GDP能耗数据实证了该模型的预测效果。     

Assessing Markov property in multistate transition models with applications to credit risk modeling

Multistate transition models are increasingly used in credit risk applications as they allow us to quantify the evolution of the process among different states. If the process is Markov, analysis and prediction are substantially simpler, so analysts would like to use these models if they are applicable. In this paper, we develop a procedure for assessing the Markov hypothesis and discuss different ways of implementing the test procedure. One issue when sample size is large is that the statistical test procedures will detect even small deviations from the Markov model when these differences are not of practical interest. To address this problem, we propose an approach to formulate and test the null hypothesis of “weak non‐Markov.” The situation where the transition probabilities are heterogeneous is also examined, and approaches to accommodate this case are indicated. Simulation studies are used extensively to study the properties of the procedures, and two applications are to illustrate the results.     

IS THERE UNIT ROOT IN THE NITROGEN OXIDES EMISSIONS: A MONTE CARLO INVESTIGATION?

Abstract Use of the time‐series econometric techniques to investigate issues about environmental regulation requires knowing whether air pollution emissions are trend stationary or difference stationary. It has been shown that results regarding trend stationarity of the pollution data are sensitive to the methods used. I conduct a Monte Carlo experiment to study the size and power of two unit root tests that allow for a structural change in the trend at a known time using the data‐generating process calibrated to the actual pollution series. I find that finite sample properties of the Perron test are better than the Park and Sung Phillips‐Perron (PP) type test. Severe size distortions in the Park and Sung PP type test can explain the rejection of a unit root in air pollution emissions reported in some environmental regulation analyses.     

Optimizing Markovian modeling of chaotic systems with recurrent neural networks

In this paper, we propose a methodology for optimizing the modeling of an one-dimensional chaotic time series with a Markov Chain. The model is extracted from a recurrent neural network trained for the attractor reconstructed from the data set. Each state of the obtained Markov Chain is a region of the reconstructed state space where the dynamics is approximated by a specific piecewise linear map, obtained from the network. The Markov Chain represents the dynamics of the time series in its statistical essence. An application to a time series resulted from Lorenz system is included.     

A Score Type Test for General Autoregressive Models in Time Series

This paper is devoted to the goodness-of-fit test for the general autoregressive models in time series. By averaging for the weighted residuals, we construct a score type test which is asymptotically standard chi-squared under the null and has some desirable power properties under the alternatives. Specifically, the test is sensitive to alternatives and can detect the alternatives approaching, along a direction, the null at a rate that is arbitrarily close to n-1/2. Furthermore, when the alternatives are not directional, we construct asymptotically distribution-free maximin tests for a large class of alternatives. The performance of the tests is evaluated through simulation studies.     

Runs tests for assessing volatility forecastability in financial time series

In this work we refine a nonparametric methodology firstly applied in Christoffersen and Diebold [Review of Economics and Statistics 82 (2000) 12] for assessing volatility forecastability in financial time series based on discretization and on the use of runs tests. Empirical results are provided for SP500 and MIB30 indexes that lead naturally to a discretized one-period Markov chain. The results are confirmed with other persistence measures and their robustness is studied via numerical simulation.     

A Tabu Search heuristic procedure in Markov chain bootstrapping

Markov chain theory is proving to be a powerful approach to bootstrap finite states processes, especially where time dependence is non linear. In this work we extend such approach to bootstrap discrete time continuous-valued processes. To this purpose we solve a minimization problem to partition the state space of a continuous-valued process into a finite number of intervals or unions of intervals (i.e. its states) and identify the time lags which provide “memory” to the process. A distance is used as objective function to stimulate the clustering of the states having similar transition probabilities. The problem of the exploding number of alternative partitions in the solution space (which grows with the number of states and the order of the Markov chain) is addressed through a Tabu Search algorithm. The method is applied to bootstrap the series of the German and Spanish electricity prices. The analysis of the results confirms the good consistency properties of the method we propose.     

Continuous Markovian Model for Unexpected Shift in SPC

Various process models for discrete manufacturing systems (parts industry) can be treated as bounded discrete-space Markov chains, completely characterized by the original in-control state and a transition matrix for shifts to an out-of-control state. The present work extends these models by using a continuous-state Markov chain, incorporating non-random corrective actions. These actions are to be realized according to the statistical process control (SPC) technique and should substantially affect the model. The developed stochastic model yields Laplace distribution of a process mean. Real-data tests confirm its applicability for the parts industry and show that the distribution parameter is mainly controlled by the SPC sample size.     

A markov chain sampler for contingency table exact inference

Summary  In the inference of contingency table, when the cell counts are not large enough for asymptotic approximation, conditioning exact method is used and often computationally impractical for large tables. Instead, various sampling methods can be used. Based on permutation, the Monte Carlo sampling may become again impractical for large tables. For this, existing the Markov chain method is to sample a few elements of the table at each iteration and is inefficient. Here we consider a Markov chain, in which a sub-table of user specified size is updated at each iteration, and it achieves high sampling efficiency. Some theoretical properties of the chain and its applications to some commonly used tables are discussed. As an illustration, this method is applied to the exact test of the Hardy-Weinberg equilibrium in the population genetics context.     

Approximations for interactive Markov chains in discrete and continuous time

A class of models called interactive Markov chains is studied in both discrete and continuous time. These models were introduced by Conlisk and serve as a rich class for sociological modeling, because they allow for interactions among individuals. In discrete time, it is proved that the Markovian processes converge to a deterministic process almost surely as the population size becomes infinite. More importantly, the normalized process is shown to be asymptotically normal with specified mean vector and covariance matrix. In continuous time, the chain is shown to converge weakly to a diffusion process with specified drift and scale terms. The distributional results will allow for the construction of a likelihood function from interactive Markov chain data, so these results will be important for questions of statistical inference. An example from manpower planning is given which indicates the use of this theory in constructing and evaluating control policies for certain social systems.     

Comparative analysis of the power of goodness-of-fit tests for near competing hypotheses. I. The verification of simple hypotheses

Using statistical modeling methods, we analyze the power of a series of goodness-of-fit tests for simple and complex hypotheses. The estimates we give for the power of the tests for simple hypotheses versus some near competing hypotheses enable us to rank the goodness-of-fit tests.     

TESTING THE ADEQUACY OF GARCH-TYPE MODELS IN TIME SERIES

In this article a new approach for checking the adequacy of GARCH-type models in time series was proposed. The resulted tests involve weight functions, which provide them with the flexibility in choosing scores to enhance power performance. The choice of weight functions and the power properties of the tests are studied. For a large number of alternatives, asymptotically distribution-free maximin test is constructed. The tests are asymptotically chi-squared under the null hypothesis and easy to implement. Simulation results indicate that the tests perform well.     

Uniqueness of quantum Markov chains associated with an XY-model on a cayley tree of order 2

We propose the construction of a quantum Markov chain that corresponds to a “forward” quantum Markov chain. In the given construction, the quantum Markov chain is defined as the limit of finite-dimensional states depending on the boundary conditions. A similar construction is widely used in the definition of Gibbs states in classical statistical mechanics. Using this construction, we study the quantum Markov chain associated with an XY-model on a Cayley tree. For this model, within the framework of the given construction, we prove the uniqueness of the quantum Markov chain, i.e., we show that the state is independent of the boundary conditions.     

The analysis and comparison of start-up demonstration tests

Two new start-up demonstration tests are introduced as alternatives to a previously studied one. The first test is based on total successes and total failures (TSTF) while the second is based on consecutive successes and consecutive failures (CSCF). The probabilistic analysis of these tests in the independent and identically distributed case uses a Markov chain approach to avoid the complexities of the probability generating function approach. We discuss advantages of these new tests in certain situations. Practical guidance on choosing tests, estimation, and comparisons of the various criteria are studied. The Markov chain approach can be easily extended and generalized to study the probabilistic analysis of the non-i.i.d. case.     

The tail empirical process of regularly varying functions of geometrically ergodic Markov chains

We consider a stationary regularly varying time series which can be expressed as a function of a geometrically ergodic Markov chain. We obtain practical conditions for the weak convergence of the tail array sums and feasible estimators of cluster statistics. These conditions include the so-called geometric drift or Foster–Lyapunov condition and can be easily checked for most usual time series models with a Markovian structure. We illustrate these conditions on several models and statistical applications. A counterexample is given to show a different limiting behavior when the geometric drift condition is not fulfilled.