Statistics for Spatial Data.

/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

1. Introduction and Overview. 2. Detecting Influential Observations and Outliers. 3. Detecting and Assessing Collinearity. 4. Applications and Remedies. 5. Research Issues and Directions for Extensions. Bibliography. Author Index. Subject Index.

Regression Diagnostics: Identifying Influential Data and Sources of Collinearity

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

Canonical Moments. Orthogonal Polynomials. Continued Fractions and the Stieltjes Transform. Special Sequences of Canonical Moments. Canonical Moments and Optimal DesignFirst Applications. Discrimination and Model Robust Designs. Applications in Approximation Theory. Canonical Moments and Random Walks. The Circle and Trigonometric Functions. Further Applications. Bibliography. Indexes.

The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis

We introduce a new class of `standardized' optimality criteria which depend on `standardized' covariances of the least squares estimators and provide an alternative to the commonly used criteria in design theory. Besides a nice statistical interpretation the new criteria satisfy an extremely useful invariance property which allows an easy calculation of optimal designs on many linearly transformed design spaces.

Designing Experiments with Respect to ‘Standardized’ Optimality Criteria

Linear rank statistics in nonparametric factorial designs are asymptotically normal and, in general, heteroscedastic. In a comprehensive simulation study, the asymptotic chi-squared law of the corresponding quadratic forms is shown to be a rather poor approximation of the finite-sample distribution. Motivated by this problem, we propose simple finite-sample size approximations for the distribution of quadratic forms in factorial designs under a normal heteroscedastic error structure. These approximations are based on an F distribution with estimated degrees of freedom that generalizes ideas of Patnaik and Box. Simulation studies show that the nominal level is maintained with high accuracy and in most cases the power is comparable to the asymptotic maximin Wald test. Data-driven guidelines are given to select the most appropriate test procedure. These ideas are finally transferred to nonparametric factorial designs where the same quadratic forms as in the parametric case are applied to the vector ...

Box-Type Approximations in Nonparametric Factorial Designs

The exact mean-squared error (MSE) of estimators of the variance in nonparametric regression based on quadratic forms is investigated. In particular, two classes of estimators are compared: Hall, Kay and Titterington's optimal difference-based estimators and a class of ordinary difference-based estimators which generalize methods proposed by Rice and Gasser, Sroka and Jennen-Steinmetz. For small sample sizes the MSE of the first estimator is essentially increased by the magnitude of the integrated first two squared derivatives of the regression function. It is shown that in many situations ordinary difference-based estimators are more appropriate for estimating the variance, because they control the bias much better and hence have a much better overall performance. It is also demonstrated that Rice's estimator does not always behave well. Data-driven guidelines are given to select the estimator with the smallest MSE.

Estimating the variance in nonparametric regression—what is a reasonable choice?

A new method for monotone estimation of a regression function is proposed, which is potentially attractive to users of conventional smoothing methods. The main idea of the new approach is to construct a density estimate from the estimated values m(i/N) (i = 1, ..., N) of the regression function and to use these 'data' for the calculation of an estimate of the inverse of the regression function. The final estimate is then obtained by a numerical inversion. Compared to the currently available techniques for monotone estimation the new method does not require constrained optimization. We prove asymptotic normality of the new estimate and compare the asymptotic properties with the unconstrained estimate. In particular, it is shown that for kernel estimates or local polynomials the bandwidths in the procedure can be chosen such that the monotone estimate is first order asymptotically equivalent to the unconstrained estimate. We also illustrate the performance of the new procedure by means of a simulation study.

/pdf/a-simple-nonparametric-estimator-of-a-strictly-monotone-2snti96cio.pdf

Holger Dette

Papers

The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis

Designing Experiments with Respect to ‘Standardized’ Optimality Criteria

Box-Type Approximations in Nonparametric Factorial Designs

Estimating the variance in nonparametric regression—what is a reasonable choice?

A simple nonparametric estimator of a strictly monotone regression function