关于累积和(CUSUM)检验的改进

对连续检验问题，常用的检测方法有三大类其一是众所周知的Shewhartt控制图，它是最常用的对生产过程进行连续监控的控制方法，不过，如果过程均值有小的漂移(即μ-μo小)时，Shewhart控制图的检验效果不是很好，除了Shewhart控制图外，另有二类常用的控制图法，其一是累积和控制图(CUSUM)，由Page^[1]基于似然比导出，其二是指数加权移动平均控制图(EWMA)，由Roberts^[2]给出，它们已被证明在检测小的漂移时效果不错。许多人对CUSUM与EWMA进行了比较，总的来说。最好的CUSUM与最好的EWMA在检测小的漂移方面难分优劣，但CUSUM是由似然比导出的，且其平均运行长度的计算相对来说要简便些，因此，CUSUM在与EWMA的比较中更具优势，应用更广．我们分析了CUSUM的导出过程和公式。指出CUSUM有二个可以进一步改进的方面在此基础上，我们给出了二个新的累积和检验统计量及其判断难则，它们分别是PCUSUM检验统计量Pn和DCUSUM检验统计量Sn．在连续检验问题中判断一个检验方法好坏的最重要的标难是其平均运行长度比较标难是在要求具有相同的受控状态下平均运行长度ARL0的条件下，比较其失控状态下的平均运行长度ARL1，ARL1越小越好我们对PCUSUM检验和DCUSUM检验都建立了其平均运行长度ARL的计算公式．通过对CUSUM，PCUSUM，DCUSUM的平均运行长度的比较我们发现我们提出的新的累积和控制方法确比原来的CUSUM有较大改进。     

Estimation with Missing Data

This paper reviews estimation problems with missing, or hidden data. We formulate this problem in the context of Markov models and consider two interrelated issues, namely, the estimation of a state given measured data and model parameters, and the estimation of model parameters given the measured data alone. We also consider situations where the measured data is, itself, incomplete in some sense. We deal with various combinations of discrete and continuous states and observations.     

具有可变抽样区间的非正态累积和控制图

当过程存在小波动时,累积和控制图比传统的休哈特控制图监控效果灵敏。为了提高控制图的监控效率,本文针对非正态情形下的累积和控制图进行可变抽样区间设计。首先用Burr分布近似各种非正态分布,构造可变抽样区间的非正态累积和控制图;其次利用马尓可夫链方法计算其平均报警时间;最后研究结果表明, 所设计的可变抽样区间非正态累积和控制图较固定抽样区间的非正态累积和控制图能更好地监控过程的变化。     

Clustering Multivariate Longitudinal Observations: The Contaminated Gaussian Hidden Markov Model

The Gaussian hidden Markov model (HMM) is widely considered for the analysis of heterogenous continuous multivariate longitudinal data. To robustify this approach with respect to possible elliptical heavy-tailed departures from normality, due to the presence of outliers, spurious points, or noise (collectively referred to as bad points herein), the contaminated Gaussian HMM is here introduced. The contaminated Gaussian distribution represents an elliptical generalization of the Gaussian distribution and allows for automatic detection of bad points in the same natural way as observations are typically assigned to the latent states in the HMM context. Once the model is fitted, each observation has a posterior probability of belonging to a particular state and, inside each state, of being a bad point or not. In addition to the parameters of the classical Gaussian HMM, for each state we have two more parameters, both with a specific and useful interpretation: one controls the proportion of bad points and one specifies their degree of atypicality. A sufficient condition for the identifiability of the model is given, an expectation-conditional maximization algorithm is outlined for parameter estimation and various operational issues are discussed. Using a large-scale simulation study, but also an illustrative artificial dataset, we demonstrate the effectiveness of the proposed model in comparison with HMMs of different elliptical distributions, and we also evaluate the performance of some well-known information criteria in selecting the true number of latent states. The model is finally used to fit data on criminal activities in Italian provinces. Supplementary materials for this article are available online     

Robust estimation in joint mean–covariance regression model for longitudinal data

In this paper, we develop robust estimation for the mean and covariance jointly for the regression model of longitudinal data within the framework of generalized estimating equations (GEE). The proposed approach integrates the robust method and joint mean–covariance regression modeling. Robust generalized estimating equations using bounded scores and leverage-based weights are employed for the mean and covariance to achieve robustness against outliers. The resulting estimators are shown to be consistent and asymptotically normally distributed. Simulation studies are conducted to investigate the effectiveness of the proposed method. As expected, the robust method outperforms its non-robust version under contaminations. Finally, we illustrate by analyzing a hormone data set. By downweighing the potential outliers, the proposed method not only shifts the estimation in the mean model, but also shrinks the range of the innovation variance, leading to a more reliable estimation in the covariance matrix.     

CUSUM,GLR,GEWMA以及RFCuscore控制图监测稳定过程均值漂移的效果比较(英)

本文给出了累积和控制图(CUSUM)监测稳定过程均值漂移的平均运行长度(ARL)的区间估计,并采用数字模拟的方法对CUSUM,GLR,GEWMA以及RFCuscore四种控制图监测稳定过程均值漂移的效果进行比较,结果显示CUSUM效果最好.     

Behavioral portfolio selection with loss control

In this paper we formulate a continuous-time behavioral (à la cumulative prospect theory) portfolio selection model where the losses are constrained by a pre-specified upper bound. Economically the model is motivated by the previously proved fact that the losses occurring in a bad state of the world can be catastrophic for an unconstrained model. Mathematically solving the model boils down to solving a concave Choquet minimization problem with an additional upper bound. We derive the optimal solution explicitly for such a loss control model. The optimal terminal wealth profile is in general characterized by three pieces: the agent has gains in the good states of the world, gets a moderate, endogenously constant loss in the intermediate states, and suffers the maximal loss (which is the given bound for losses) in the bad states. Examples are given to illustrate the general results.     

Optimization designs and performance comparison of two CUSUM schemes for monitoring process shifts in mean and variance

In statistical process control (SPC), when dealing with a quality characteristic x that is a variable, it is usually necessary to monitor both the mean value and variability. This article proposes an optimization algorithm (called the holistic algorithm) to design the CUSUM charts for this purpose. It facilitates the determination of the charting parameters of the CUSUM charts and considerably or significantly increases their overall detection effectiveness. A single CUSUM chart (called the ABS CUSUM chart) has been developed by the holistic algorithm and fully investigated. This chart is able to detect two-sided mean shifts and increasing variance shifts by inspecting the absolute value of sample mean shift. The results of performance studies show that the overall performance of the ABS CUSUM chart is nearly as good as an optimal 3-CUSUM scheme (a scheme incorporating three individual CUSUM charts). However, since the ABS CUSUM chart is easier for implementation and design, it may be more suitable for many SPC applications in which both mean and variance of a variable have to be monitored.     

A Geometric Approach to Maximum Likelihood Estimation of the Functional Principal Components From Sparse Longitudinal Data

In this article, we consider the problem of estimating the eigenvalues and eigenfunctions of the covariance kernel (i.e., the functional principal components) from sparse and irregularly observed longitudinal data. We exploit the smoothness of the eigenfunctions to reduce dimensionality by restricting them to a lower dimensional space of smooth functions. We then approach this problem through a restricted maximum likelihood method. The estimation scheme is based on a Newton–Raphson procedure on the Stiefel manifold using the fact that the basis coefficient matrix for representing the eigenfunctions has orthonormal columns. We also address the selection of the number of basis functions, as well as that of the dimension of the covariance kernel by a second-order approximation to the leave-one-curve-out cross-validation score that is computationally very efficient. The effectiveness of our procedure is demonstrated by simulation studies and an application to a CD4+ counts dataset. In the simulation studies, our method performs well on both estimation and model selection. It also outperforms two existing approaches: one based on a local polynomial smoothing, and another using an EM algorithm. Supplementary materials including technical details, the R package fpca, and data analyzed by this article are available online.     

非参数理论综述

对非参数理论进行了系统地综述.非参数理论中一个比较重要的内容是估计方法,常见的非参数估计方法有核估计、局部多项式估计、近邻估计等.光滑参数的选取、"维数灾难"与边界点问题也是与非参数理论有关的重要内容,也对这些方面进行综述.最后,文章还综述了非参数技术在时间序列模型中的有关应用问题.     

A two-step method for estimating high-dimensional Gaussian graphical models

The problem of estimating high-dimensional Gaussian graphical models has gained much attention in recent years. Most existing methods can be considered as one-step approaches, being either regression-based or likelihood-based. In this paper, we propose a two-step method for estimating the high-dimensional Gaussian graphical model. Specifically, the first step serves as a screening step, in which many entries of the concentration matrix are identified as zeros and thus removed from further consideration. Then in the second step, we focus on the remaining entries of the concentration matrix and perform selection and estimation for nonzero entries of the concentration matrix. Since the dimension of the parameter space is effectively reduced by the screening step,the estimation accuracy of the estimated concentration matrix can be potentially improved. We show that the proposed method enjoys desirable asymptotic properties. Numerical comparisons of the proposed method with several existing methods indicate that the proposed method works well. We also apply the proposed method to a breast cancer microarray data set and obtain some biologically meaningful results.     

Data-driven modeling of the temporal evolution of breakers’ states in the French electrical transmission grid

In electrical transmission grids, it is common to observe the states of circuit breakers. While they are known at irregular times, system modeling and grid state estimation are of the highest importance to ensure secure operations. This paper proposes a richer method to estimate the grid state over its reference configurations based on the temporal evolution of its breakers’ states. The first contribution consists in developing a general multi-observation continuous-time finite-state Hidden Markov Model with filter-based parameter estimation to infer the hidden state (e.g., the grid reference configuration) handling multiple observed processes with irregular “jump” times (e.g., the breakers’ states). As a second contribution, we build a numerical scheme with no discretization error adapted to all state jumps generated by the observed processes. Finally, we apply our model to simulated and real data to illustrate the approach’s performance. The available data consists of historical records of breakers’ states during the electrical transmission grid operated normally. For this real-data-driven application, we also present a clustering approach to identify the set of grid reference configurations.     

非参数模型均值函数结构变点的Bootstrap检测

本文检测非参数回归模型均值函数结构变点,针对均值函数跃度的长期均值为零时,基于残量的CUSUM统计量对均值函数结构变点检验无效的问题,本文提出了一种基于均值函数的核估计的检验统计量,得到统计量在原假设和备择假设下的极限分布,并构造Bootstrap方法对非参数回归模型均值函数结构变点进行检验,证明了检验和估计的一致性;模拟结果表明本文方法明显优于已有方法。     

Mean estimation in the presence of change points

In this study we address the problem of the mean estimation of the IBEX-35 index stock quotes in the presence of change points. We rely on nonparametric regression methods for detecting and estimating changes points, and for estimating the discontinuous regression function. Model-assisted and model-based estimators and their jump-preserving counterparts are used for mean estimation and an empirical comparison between the methods is performed.     

Maximum a Posteriori Sequence Estimation Using Monte Carlo Particle Filters

We develop methods for performing maximum a posteriori (MAP) sequence estimation in non-linear non-Gaussian dynamic models. The methods rely on a particle cloud representation of the filtering distribution which evolves through time using importance sampling and resampling ideas. MAP sequence estimation is then performed using a classical dynamic programming technique applied to the discretised version of the state space. In contrast with standard approaches to the problem which essentially compare only the trajectories generated directly during the filtering stage, our method efficiently computes the optimal trajectory over all combinations of the filtered states. A particular strength of the method is that MAP sequence estimation is performed sequentially in one single forwards pass through the data without the requirement of an additional backward sweep. An application to estimation of a non-linear time series model and to spectral estimation for time-varying autoregressions is described.     

Efficient Data Augmentation for Fitting Stochastic Epidemic Models to Prevalence Data

Stochastic epidemic models describe the dynamics of an epidemic as a disease spreads through a population. Typically, only a fraction of cases are observed at a set of discrete times. The absence of complete information about the time evolution of an epidemic gives rise to a complicated latent variable problem in which the state space size of the epidemic grows large as the population size increases. This makes analytically integrating over the missing data infeasible for populations of even moderate size. We present a data augmentation Markov chain Monte Carlo (MCMC) framework for Bayesian estimation of stochastic epidemic model parameters, in which measurements are augmented with subject-level disease histories. In our MCMC algorithm, we propose each new subject-level path, conditional on the data, using a time-inhomogenous continuous-time Markov process with rates determined by the infection histories of other individuals. The method is general, and may be applied to a broad class of epidemic models with only minimal modifications to the model dynamics and/or emission distribution. We present our algorithm in the context of multiple stochastic epidemic models in which the data are binomially sampled prevalence counts, and apply our method to data from an outbreak of influenza in a British boarding school. Supplementary material for this article is available online.     

Fully Bayesian Analysis of Switching Gaussian State Space Models

In the present paper we study switching state space models from a Bayesian point of view. We discuss various MCMC methods for Bayesian estimation, among them unconstrained Gibbs sampling, constrained sampling and permutation sampling. We address in detail the problem of unidentifiability, and discuss potential information available from an unidentified model. Furthermore the paper discusses issues in model selection such as selecting the number of states or testing for the presence of Markov switching heterogeneity. The model likelihoods of all possible hypotheses are estimated by using the method of bridge sampling. We conclude the paper with applications to simulated data as well as to modelling the U.S./U.K. real exchange rate.     

Change-Point Detection in Angular Data

We suggest a modification of the CUSUM procedure to detect changes in angular data. We obtain limit theorems for the test statistics under the no change null hypothesis. We discuss the estimation of the times of changes and show that the binary segmentation provides the times of all changes. Our method is applied to a data set on the activity of a pulsar.     

Particle Approximations of the Score and Observed Information Matrix for Parameter Estimation in State–Space Models With Linear Computational Cost

Poyiadjis, Doucet, and Singh showed how particle methods can be used to estimate both the score and the observed information matrix for state–space models. These methods either suffer from a computational cost that is quadratic in the number of particles, or produce estimates whose variance increases quadratically with the amount of data. This article introduces an alternative approach for estimating these terms at a computational cost that is linear in the number of particles. The method is derived using a combination of kernel density estimation, to avoid the particle degeneracy that causes the quadratically increasing variance, and Rao–Blackwellization. Crucially, we show the method is robust to the choice of bandwidth within the kernel density estimation, as it has good asymptotic properties regardless of this choice. Our estimates of the score and observed information matrix can be used within both online and batch procedures for estimating parameters for state–space models. Empirical results show improved parameter estimates compared to existing methods at a significantly reduced computational cost. Supplementary materials including code are available.     

COMPARISON OF TWO MODELS AND TWO ESTIMATION METHODS FOR CATCH AND EFFORT DATA

We consider the problem of estimating the optimal steady effort level from a time series of catch and effort data, taking account of errors in the observation of the “effective effort” as well as randomness in the stock-production function. The “total least squares” method ignores the time series nature of the data, while the “approximate likelihood” method takes it into account. We compare estimation schemes based upon these two methods by applying them to artificial data for which the “correct” parameters are known. We use a similar procedure to compare the effectiveness of a “power model” for stock and production with the “Ricker model.” We apply these estimation methods to some sets of real data, and obtain an interval estimate of the optimal effort.