How the instability of ranks under long memory affects large-sample inference

Under long memory, the limit theorems for normalized sums of random variables typically involve a positive integer called "Hermite rank". There is a different limit for each Hermite rank. From a statistical point of view, however, we argue that a rank other than one is unstable, whereas, a rank equal to one is stable. We provide empirical evidence supporting this argument. This has important consequences. Assuming a higher-order rank when it is not really there usually results in underestimating the order of the fluctuations of the statistic of interest. We illustrate this through various examples involving the sample variance, the empirical processes and the Whittle estimator.Accepted manuscrip

Variance Estimation and Kriging Prediction for a Class of Non-Stationary Spatial Models

This paper discusses the estimation and plug-in kriging prediction non-stationary spatial process assuming a smoothly varying variance an additive independent measurement error. A difference-based kernel estimator of the variance function and a modified likelihood estimator of the mea surement error variance are used for parameter estimation. Asymptotic properties of these estimators and the plug-in kriging predictor are established. A simula tion study is presented to test our estimation-prediction procedure. Our kriging predictor is shown to perform better than the spatial adaptive local polynomial regression estimator proposed by Fan and Gijbels (1995) when the measurement error is small

How the instability of ranks under long memory affects large-sample inference

Under long memory, the limit theorems for normalized sums of random variables typically involve a positive integer called "Hermite rank". There is a different limit for each Hermite rank. From a statistical point of view, however, we argue that a rank other than one is unstable, whereas, a rank equal to one is stable. We provide empirical evidence supporting this argument. This has important consequences. Assuming a higher-order rank when it is not really there usually results in underestimating the order of the fluctuations of the statistic of interest. We illustrate this through various examples involving the sample variance, the empirical processes and the Whittle estimator.Accepted manuscrip

Empirical Bayes Estimation in Cross-Classified Gaussian Models With Unbalanced Design

The James-Stein estimator and its Bayesian interpretation demonstrated the usefulness of empirical Bayes methods in facilitating competitive shrinkage estimators for multivariate problems consisting of nonrandom parameters. When transitioning from homoscedastic to heteroscedastic Gaussian data, empirical ``linear Bayes estimators typically lose attractive properties such as minimaxity, and are usually justified mainly from Bayesian viewpoints. Nevertheless, by appealing to frequentist considerations, traditional empirical linear Bayes estimators can be modified to better accommodate the asymmetry in unequal variance cases. This work develops empirical Bayes estimators for cross-classified (factorial) data with unbalanced design that are asymptotically optimal within classes of shrinkage estimators, and in particular asymptotically dominate traditional parametric empirical Bayes estimators as well the usual (unbiased) estimator

Reassessing the Paradigms of Statistical Model-Building

Statistical model-building is the science of constructing models from data and from information about the data-generation process, with the aim of analysing those data and drawing inference from that analysis. Many statistical tasks are undertaken during this analysis; they include classification, forecasting, prediction and testing. Model-building has assumed substantial importance, as new technologies enable data on highly complex phenomena to be gathered in very large quantities. This creates a demand for more complex models, and requires the model-building process itself to be adaptive. The word “paradigm” refers to philosophies, frameworks and methodologies for developing and interpreting statistical models, in the context of data, and applying them for inference. In order to solve contemporary statistical problems it is often necessary to combine techniques from previously separate paradigms. The workshop addressed model-building paradigms that are at the frontiers of modern statistical research. It tried to create synergies, by delineating the connections and collisions among different paradigms. It also endeavoured to shape the future evolution of paradigms

Autocovariance estimation in regression with a discontinuous signal and mm-dependent errors: A difference-based approach

We discuss a class of difference-based estimators for the autocovariance in nonparametric regression when the signal is discontinuous (change-point regression), possibly highly fluctuating, and the errors form a stationary $m$-dependent process. These estimators circumvent the explicit pre-estimation of the unknown regression function, a task which is particularly challenging for such signals. We provide explicit expressions for their mean squared errors when the signal function is piecewise constant (segment regression) and the errors are Gaussian. Based on this we derive biased-optimized estimates which do not depend on the particular (unknown) autocovariance structure. Notably, for positively correlated errors, that part of the variance of our estimators which depends on the signal is minimal as well. Further, we provide sufficient conditions for $\sqrt{n}$-consistency; this result is extended to piecewise Holder regression with non-Gaussian errors. We combine our biased-optimized autocovariance estimates with a projection-based approach and derive covariance matrix estimates, a method which is of independent interest. Several simulation studies as well as an application to biophysical measurements complement this paper.Comment: 41 pages, 3 figures, 3 table

Wavelet estimations of the derivatives of variance function in heteroscedastic model

This paper studies nonparametric estimations of the derivatives $r^{(m)}(x)$ of the variance function in a heteroscedastic model. Using a wavelet method, a linear estimator and an adaptive nonlinear estimator are constructed. The convergence rates under L^{\tilde{p}} (1\leq \tilde{p} < \infty) risk of those two wavelet estimators are considered with some mild assumptions. A simulation study is presented to validate the performances of the wavelet estimators