Asymptotic equivalence and adaptive estimation for robust nonparametric regression

Asymptotic equivalence theory developed in the literature so far are only for bounded loss functions. This limits the potential applications of the theory because many commonly used loss functions in statistical inference are unbounded. In this paper we develop asymptotic equivalence results for robust nonparametric regression with unbounded loss functions. The results imply that all the Gaussian nonparametric regression procedures can be robustified in a unified way. A key step in our equivalence argument is to bin the data and then take the median of each bin. The asymptotic equivalence results have significant practical implications. To illustrate the general principles of the equivalence argument we consider two important nonparametric inference problems: robust estimation of the regression function and the estimation of a quadratic functional. In both cases easily implementable procedures are constructed and are shown to enjoy simultaneously a high degree of robustness and adaptivity. Other problems such as construction of confidence sets and nonparametric hypothesis testing can be handled in a similar fashion.Comment: Published in at http://dx.doi.org/10.1214/08-AOS681 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Forecasting with measurement errors in dynamic models

This paper explores the effects of measurement error on dynamic forecasting models. The paper sets out to illustrate a trade off that confronts forecasters and policymakers when they use data that are measured with error. On the one hand, observations on recent data give valuable clues as to the shocks that are hitting the system and will be propagated into the variables to be forecast (and which ultimately will inform monetary policy). But on the other, those recent observations are likely to be those least well measured. Two broad classes of results are illustrated. The first relates to cases where it is imagined that the forecaster takes the coefficients in the data generating process as a given, and has to choose how much of the historical time series of data to use to form a forecast. It is shown that if recent data is sufficiently badly measured, relative to older data, that it can be optimal in this case not to use old data at all. The second class of results is more general. Here, it is shown that for a general class of linear autoregressive forecasting models, the optimal weight to place on a data observation of some age, relative to the weight in the true data generating process, will depend on the measurement error in that data. The gains to be had in forecasting are illustrated using a model of UK business investment growth.measurement error, forecasting, signal-extraction

Using time-varying VARs to diagnose the source of ‘Great Moderations’: a Monte Carlo analysis

In this paper, we assess the ability of time-varying VAR models to correctly diagnose the source of ‘Great Moderations’ generated in simulations of a learning model. We find that, in general, they can. For example, in data sets with Great Moderations generated by good policy, the VAR correctly identifies a downward shift in the policy disturbance. And it shows that if the policy behaviour associated with the latter part of the sample (during which policy is conducted well) are applied to the earlier part of the sample, the implied variances of output, inflation and interest rates would have been much lower. An important caveat to our results is that they appear to be sensitive to the method used to identification of monetary policy shocks. When we identify monetary policy shocks using a Cholesky decomposition, the VAR provides quite clear evidence in favour of the correct explanation for our simulated Great Moderations When sign restrictions are used to identify the monetary policy shocks, conclusions from the counterfactual experiments are less precise. The contrast between our results and previous work based on Monte Carlo evidence using RE models suggests that the ability of VARs to correctly diagnose the source of the Great Moderation may be dependent on the nature of the expectations-formation process in the private sector.

Law of Log Determinant of Sample Covariance Matrix and Optimal Estimation of Differential Entropy for High-Dimensional Gaussian Distributions

Differential entropy and log determinant of the covariance matrix of a multivariate Gaussian distribution have many applications in coding, communications, signal processing and statistical inference. In this paper we consider in the high dimensional setting optimal estimation of the differential entropy and the log-determinant of the covariance matrix. We first establish a central limit theorem for the log determinant of the sample covariance matrix in the high dimensional setting where the dimension $p(n)$ can grow with the sample size $n$. An estimator of the differential entropy and the log determinant is then considered. Optimal rate of convergence is obtained. It is shown that in the case $p(n)/n \rightarrow 0$ the estimator is asymptotically sharp minimax. The ultra-high dimensional setting where $p(n) > n$ is also discussed.Comment: 19 page

Optimal rates of convergence for covariance matrix estimation

Covariance matrix plays a central role in multivariate statistical analysis. Significant advances have been made recently on developing both theory and methodology for estimating large covariance matrices. However, a minimax theory has yet been developed. In this paper we establish the optimal rates of convergence for estimating the covariance matrix under both the operator norm and Frobenius norm. It is shown that optimal procedures under the two norms are different and consequently matrix estimation under the operator norm is fundamentally different from vector estimation. The minimax upper bound is obtained by constructing a special class of tapering estimators and by studying their risk properties. A key step in obtaining the optimal rate of convergence is the derivation of the minimax lower bound. The technical analysis requires new ideas that are quite different from those used in the more conventional function/sequence estimation problems.Comment: Published in at http://dx.doi.org/10.1214/09-AOS752 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Robust nonparametric estimation via wavelet median regression

In this paper we develop a nonparametric regression method that is simultaneously adaptive over a wide range of function classes for the regression function and robust over a large collection of error distributions, including those that are heavy-tailed, and may not even possess variances or means. Our approach is to first use local medians to turn the problem of nonparametric regression with unknown noise distribution into a standard Gaussian regression problem and then apply a wavelet block thresholding procedure to construct an estimator of the regression function. It is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the Besov classes, without prior knowledge of the smoothness of the underlying functions or prior knowledge of the error distribution. The estimator also automatically adapts to the local smoothness of the underlying function, and attains the local adaptive minimax rate for estimating functions at a point. A key technical result in our development is a quantile coupling theorem which gives a tight bound for the quantile coupling between the sample medians and a normal variable. This median coupling inequality may be of independent interest.Comment: Published in at http://dx.doi.org/10.1214/07-AOS513 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonparametric regression in exponential families

Most results in nonparametric regression theory are developed only for the case of additive noise. In such a setting many smoothing techniques including wavelet thresholding methods have been developed and shown to be highly adaptive. In this paper we consider nonparametric regression in exponential families with the main focus on the natural exponential families with a quadratic variance function, which include, for example, Poisson regression, binomial regression and gamma regression. We propose a unified approach of using a mean-matching variance stabilizing transformation to turn the relatively complicated problem of nonparametric regression in exponential families into a standard homoscedastic Gaussian regression problem. Then in principle any good nonparametric Gaussian regression procedure can be applied to the transformed data. To illustrate our general methodology, in this paper we use wavelet block thresholding to construct the final estimators of the regression function. The procedures are easily implementable. Both theoretical and numerical properties of the estimators are investigated. The estimators are shown to enjoy a high degree of adaptivity and spatial adaptivity with near-optimal asymptotic performance over a wide range of Besov spaces. The estimators also perform well numerically.Comment: Published in at http://dx.doi.org/10.1214/09-AOS762 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Understanding political ideas and movements: a guide for A2 politics students

Written specifically to cover the A2 component of the GCE Government and Politics A-level, this book is a comprehensive introduction to the political ideas and movements that have shaped the modern world. Underpinned by the work of major thinkers such as Hobbes, Locke, Marx, Mill, Weber and others, the first half of the book looks at political concepts including the state and sovereignty, the nation, democracy, representation and legitimacy, freedom, equality and rights, obligation and citizenship. There is also a specific chapter which addresses the role of ideology in the shaping of politics and society. The second half of the book addresses traditional theoretical subjects such as socialism, Marxism and nationalism, before moving on to more contemporary movements such as environmentalism, ecologism and feminism. The subject is covered in a clear, accessible style, and includes a number of student-friendly features, such as chapter summaries, key points to consider, definitions and tips for further sources of information. There is a definite need for a text of this kind. It will be invaluable for students of government and politics at introductory level

Understanding political ideas and movements

Written specifically to cover the A2 component of the GCE Government and Politics A-level, this book is a comprehensive introduction to the political ideas and movements that have shaped the modern world. Underpinned by the work of major thinkers such as Hobbes, Locke, Marx, Mill, Weber and others, the first half of the book looks at political concepts including the state and sovereignty, the nation, democracy, representation and legitimacy, freedom, equality and rights, obligation and citizenship. There is also a specific chapter which addresses the role of ideology in the shaping of politics and society. The second half of the book addresses traditional theoretical subjects such as socialism, Marxism and nationalism, before moving on to more contemporary movements such as environmentalism, ecologism and feminism. The subject is covered in a clear, accessible style, and includes a number of student-friendly features, such as chapter summaries, key points to consider, definitions and tips for further sources of information. There is a definite need for a text of this kind. It will be invaluable for students of government and politics at introductory level