模糊线性回归的稳健方法

考虑两类模糊回归模型：一类是设计点为实数,参数为模糊数;另一类是观察值为模糊数、参数为实数。就这两类回归模型,从稳健统计的角度提出相应的稳健方法,并通过例子与现有的方法进行比较,说明所提方法的稳健性。     

Robust Simulation-Based Estimation of ARMA Models

This article proposes a new approach to the robust estimation of a mixed autoregressive and moving average (ARMA) model. It is based on the indirect inference method that originally was proposed for models with an intractable likelihood function. The estimation algorithm proposed is based on an auxiliary autoregressive representation whose parameters are first estimated on the observed time series and then on data simulated from the ARMA model. To simulate data the parameters of the ARMA model have to be set. By varying these we can minimize a distance between the simulation-based and the observation-based auxiliary estimate. The argument of the minimum yields then an estimator for the parameterization of the ARMA model. This simulation-based estimation procedure inherits the properties of the auxiliary model estimator. For instance, robustness is achieved with GM estimators. An essential feature of the introduced estimator, compared to existing robust estimators for ARMA models, is its theoretical tractability that allows us to show consistency and asymptotic normality. Moreover, it is possible to characterize the influence function and the breakdown point of the estimator. In a small sample Monte Carlo study it is found that the new estimator performs fairly well when compared with existing procedures. Furthermore, with two real examples, we also compare the proposed inferential method with two different approaches based on outliers detection.     

Robust canonical correlations: A comparative study

Summary  Several approaches for robust canonical correlation analysis will be presented and discussed. A first method is based on the definition of canonical correlation analysis as looking for linear combinations of two sets of variables having maximal (robust) correlation. A second method is based on alternating robust regressions. These methods are discussed in detail and compared with the more traditional approach to robust canonical correlation via covariance matrix estimates. A simulation study compares the performance of the different estimators under several kinds of sampling schemes. Robustness is studied as well by breakdown plots.     

The Robust Design with the Pole Assignment

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Robust nonuniform dichotomies and parameter dependence

We establish the robustness of linear cocycles with an exponential dichotomy, under sufficiently small Lipschitz perturbations, in the sense that the existence of an exponential dichotomy for a given cocycle persists under these perturbations. We consider cocycles in Banach spaces, as well as the general case of nonuniform exponential dichotomies, and also the general case of an exponential behavior ecρ(n), given by an arbitrary sequence ρ(n) including the usual exponential behavior ρ(n)=n as a very special case. Moreover, we show that the projections of the exponential dichotomies obtained from the perturbation vary continuously with the parameter, and in fact that they are locally Lipschitz on finite-dimensional parameters.     

Robust optimal strategies in Markov decision problems

An optimal strategy in a Markov decision problem is robust if it is optimal in every decision problem (not necessarily stationary) that is close to the original problem. We prove that when the state and action spaces are finite, an optimal strategy is robust if and only if it is the unique optimal strategy.     

空间自相关地理加权回归模型的估计

地理加权回归作为一类能有效处理回归分析中空间非平稳性现象的建模技术,在多类问题的研究得到了广泛的应用.主要讨论这类空间计量经济学模型在空间自相关情形下的估计问题.首先,对于因变量含有空间滞后项的地理加权回归模型,分别给出了局部似然估计和两步估计两种方法.其次,考虑了误差空间自相关下地理加权回归模型的估计问题.     

Robust simulation with a base environmental scenario

In the design of a system, the comparison of possible solutions using simulation is generally performed with fixed environmental conditions. In practice, however, unexpected changes can occur for example in the part mix of a manufacturing facility or in the customer demand. Such changes, which are considered as modifications in environmental factors, can impact the system response. As a consequence, a solution A that is better than B for a given environment, can yield poorer performance than B for another environment. Therefore, we are interested in robust simulation studies, which aim at taking into account several possible environments. In methods based on Taguchi’s principles, no distinction is made between these environments in the robustness computation. In the suggested heuristic approach, we focus on problems where a particular environment is expected when the system will be in operation (the others being unexpected environments). This particular environment will be considered in the study as a “base environmental scenario”. The robustness of a solution of the design problem is computed as an approximate measure of what will be saved or lost if the environment becomes the unexpected. Reference curves are suggested to allow these solutions to be empirically compared in accordance with the decision-maker’s requirements. A simplified example is provided. The results are different from those obtained using a signal to noise ratio, which is typically used in Taguchian approaches.     

Robust Depth-Weighted Wavelet for Nonparametric Regression Models

In the nonparametric regression models, the original regression estimators including kernel estimator, Fourier series estimator and wavelet estimator are always constructed by the weighted sum of data, and the weights depend only on the distance between the design points and estimation points. As a result these estimators are not robust to the perturbations in data. In order to avoid this problem, a new nonparametric regression model, called the depth-weighted regression model, is introduced and then the depth-weighted wavelet estimation is defined. The new estimation is robust to the perturbations in data, which attains very high breakdown value close to 1/2. On the other hand, some asymptotic behaviours such as asymptotic normality are obtained. Some simulations illustrate that the proposed wavelet estimator is more robust than the original wavelet estimator and, as a price to pay for the robustness, the new method is slightly less efficient than the original method.     

A Deterministic Algorithm for Robust Location and Scatter

Most algorithms for highly robust estimators of multivariate location and scatter start by drawing a large number of random subsets. For instance, the FASTMCD algorithm of Rousseeuw and Van Driessen starts in this way, and then takes so-called concentration steps to obtain a more accurate approximation to the MCD. The FASTMCD algorithm is affine equivariant but not permutation invariant. In this article, we present a deterministic algorithm, denoted as DetMCD, which does not use random subsets and is even faster. It computes a small number of deterministic initial estimators, followed by concentration steps. DetMCD is permutation invariant and very close to affine equivariant. We compare it to FASTMCD and to the OGK estimator of Maronna and Zamar. We also illustrate it on real and simulated datasets, with applications involving principal component analysis, classification, and time series analysis. Supplemental material (Matlab code of the DetMCD algorithm and the datasets) is available online.     

Robust fuzzy rough classifiers

Fuzzy rough sets, generalized from Pawlak's rough sets, were introduced for dealing with continuous or fuzzy data. This model has been widely discussed and applied these years. It is shown that the model of fuzzy rough sets is sensitive to noisy samples, especially sensitive to mislabeled samples. As data are usually contaminated with noise in practice, a robust model is desirable. We introduce a new model of fuzzy rough set model, called soft fuzzy rough sets, and design a robust classification algorithm based on the model. Experimental results show the effectiveness of the proposed algorithm.     

Huber's Minimax Approach in Distribution Classes with Bounded Variances and Subranges with Applications to Robust Detection of Signals

A brief survey of former and recent results on Huber's minimax approach in robust statistics is given. The least mformative distributions minimizing Fisher information for location over several distribution classes with upper-bounded variances and subranges are written down. These least informative distributions are qualitatively different from classical Huber's solution and have the following common structure: (i) with relatively small variances they are short-tailed, in particular normal; (ii) with relatively large variances they are heavytailed, in particular the Laplace; (iii) they are compromise with relatively moderate variances. These results allow to raise the efficiency of minimax robust procedures retaining high stability as compared to classical Huber's procedure for contaminated normal populations. In application to signal detection problems, the proposed minimax detection rule has proved to be robust and close to Huber's for heavy-tailed distributions and more efficient than Huber's for short-tailed ones both in asymptotics and on finite samples.     

Robust portfolios that do not tilt factor exposure

Robust portfolios reduce the uncertainty in portfolio performance. In particular, the worst-case optimization approach is based on the Markowitz model and form portfolios that are more robust compared to mean–variance portfolios. However, since the robust formulation finds a different portfolio from the optimal mean–variance portfolio, the two portfolios may have dissimilar levels of factor exposure. In most cases, investors need a portfolio that is not only robust but also has a desired level of dependency on factor movement for managing the total portfolio risk. Therefore, we introduce new robust formulations that allow investors to control the factor exposure of portfolios. Empirical analysis shows that the robust portfolios from the proposed formulations are more robust than the classical mean–variance approach with comparable levels of exposure on fundamental factors.     

Huber＇s Minimax Approach in Distribution Classes with Bounded Variances and Subranges with Applications to Robust Detection of Signals

A brief survey of former and recent results on Huber‘s minimax approach in robust statistics is given. The least informative distributions minimizing Fisher information for location over several distribution classes with upper-bounded variances and subranges are written down. These least informative distributions are qualitatively different from classical Huber‘s solution and have the following common structure: (i) with relatively small variances they are short-tailed, in particular normal；(ii) with relatively large variances they are heavytailed, in particular the Laplace; (iii) they are compromise with relatively moderate variances. These results allow to raise the efficiency of minimax robust procedures retaining high stability as compared to classical Huber‘s procedure for contaminated normal populations. In application to signal detection problems, the proposed minimax detection rule has proved to be robust and close to Huber‘s for heavy-tailed distributions and more efficient than Huber‘s for short-tailed ones both in asymptotics and on finite samples。     

Robust Frequency Estimation Using Elemental Sets

Abstract

The extraction of sinusoidal signals from time-series data is a classic problem of ongoing interest in the statistics and signal processing literatures. Obtaining least squares estimates is difficult because the sum of squares has local minima O(1/n) apart in the frequencies. In practice the frequencies are often estimated using ad hoc and inefficient methods. Problems of data quality have received little attention. An elemental set is a subset of the data containing the minimum number of points such that the unknown parameters in the model can be identified. This article shows that, using a variant of the classical method of Prony, parameter estimates for a sum of sinusoids can be obtained algebraically from an elemental set. Elemental set methods are used to construct finite algorithm estimators that approximately minimize the least squares, least trimmed sum of squares, or least median of squares criteria. The elemental set estimators prove able in simulations to resolve the frequencies to the correct local minima of the objective functions. When used as the first stage of an MM estimator, the constructed estimators based on the trimmed sum of squares and least median of squares criteria produce final estimators which have high breakdown properties and which are simultaneously efficient when no outliers are present. The approach can also be applied to sums of exponentials, and sums of damped sinusoids. The article includes simulations with one and two sinusoids and two data examples.     

Robust Estimation for Generalized Additive Models

This article studies M-type estimators for fitting robust generalized additive models in the presence of anomalous data. A new theoretical construct is developed to connect the costly M-type estimation with least-squares type calculations. Its asymptotic properties are studied and used to motivate a computational algorithm. The main idea is to decompose the overall M-type estimation problem into a sequence of well-studied conventional additive model fittings. The resulting algorithm is fast and stable, can be paired with different nonparametric smoothers, and can also be applied to cases with multiple covariates. As another contribution of this article, automatic methods for smoothing parameter selection are proposed. These methods are designed to be resistant to outliers. The empirical performance of the proposed methodology is illustrated via both simulation experiments and real data analysis. Supplementary materials are available online.     

Robust improvement schemes for road networks under demand uncertainty

This paper is concerned with development of improvement schemes for road networks under future travel demand uncertainty. Three optimization models, sensitivity-based, scenario-based and min–max, are proposed for determining robust optimal improvement schemes that make system performance insensitive to realizations of uncertain demands or allow the system to perform better against the worst-case demand scenario. Numerical examples and simulation tests are presented to demonstrate and validate the proposed models.     

Robust and asymptotically unbiased estimation of extreme quantiles for heavy tailed distributions

A robust and asymptotically unbiased extreme quantile estimator is derived from a second order Pareto-type model and its asymptotic properties are studied under suitable regularity conditions. The finite sample properties of the proposed estimator are investigated with a small simulation experiment.     

Robust Parameter Estimation for Stochastic Differential Equations

We consider the estimation of parameters in stochastic differential equations (SDEs). The problem is treated in the setting of nonlinear filtering theory with a degenerate diffusion matrix. A robust stochastic Feynman–Kac representation for solutions of SDEs of Zakai-type is derived. It is verified that these solutions are conditional densities for the conditional measures defined by degenerate filtering problems. We show that the corresponding estimator for the parameters is robust in the following sense: It depends continuously on both the measurement path and on the intensity of the measurement noise. An algorithm based on a Monte-Carlo approach is given for the practical application of the estimator, and numerical results are reported. Mathematics Subject Classifications (2000) Primary: 62M05, 62M20; secondary: 62F15.     

Robust Estimation of the Generalized Pareto Distribution

One approach used for analyzing extremes is to fit the excesses over a high threshold by a generalized Pareto distribution. For the estimation of the shape and scale parameters in the generalized Pareto distribution, under some restrictions on the value of the scale parameter, maximum likelihood, method of moments and probability weighted moments' estimators are available. However, these are not robust estimators. In this paper we implement a robust estimation procedure known as the method of medians (He and Fung, 1999) to estimate the parameters in the generalized Pareto distribution. The asymptotic distribution of our estimator is normal for any value of the shape parameter except –1.