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

Convergence of Probability Measures

Preface * Introduction * The Bayes Error * Inequalities and alternatedistance measures * Linear discrimination * Nearest neighbor rules *Consistency * Slow rates of convergence Error estimation * The regularhistogram rule * Kernel rules Consistency of the k-nearest neighborrule * Vapnik-Chervonenkis theory * Combinatorial aspects of Vapnik-Chervonenkis theory * Lower bounds for empirical classifier selection* The maximum likelihood principle * Parametric classification *Generalized linear discrimination * Complexity regularization *Condensed and edited nearest neighbor rules * Tree classifiers * Data-dependent partitioning * Splitting the data * The resubstitutionestimate * Deleted estimates of the error probability * Automatickernel rules * Automatic nearest neighbor rules * Hypercubes anddiscrete spaces * Epsilon entropy and totally bounded sets * Uniformlaws of large numbers * Neural networks * Other error estimates *Feature extraction * Appendix * Notation * References * Index

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A Probabilistic Theory of Pattern Recognition

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Why is Nonparametric Regression Important? * How to Construct Nonparametric Regression Estimates * Lower Bounds * Partitioning Estimates * Kernel Estimates * k-NN Estimates * Splitting the Sample * Cross Validation * Uniform Laws of Large Numbers * Least Squares Estimates I: Consistency * Least Squares Estimates II: Rate of Convergence * Least Squares Estimates III: Complexity Regularization * Consistency of Data-Dependent Partitioning Estimates * Univariate Least Squares Spline Estimates * Multivariate Least Squares Spline Estimates * Neural Networks Estimates * Radial Basis Function Networks * Orthogonal Series Estimates * Advanced Techniques from Empirical Process Theory * Penalized Least Squares Estimates I: Consistency * Penalized Least Squares Estimates II: Rate of Convergence * Dimension Reduction Techniques * Strong Consistency of Local Averaging Estimates * Semi-Recursive Estimates * Recursive Estimates * Censored Observations * Dependent Observations

http://www.gbv.de/dms/hebis-darmstadt/toc/106925350.pdf

A Distribution-Free Theory of Nonparametric Regression

This paper presents an introduction to inference for copula models, based on rank methods. By working out in detail a small, fictitious numerical example, the writers exhibit the various steps involved in investigating the dependence between two random variables and in modeling it using copulas. Simple graphical tools and numerical techniques are presented for selecting an appropriate model, estimating its parameters, and checking its goodness-of-fit. A larger, realistic application of the methodology to hydrological data is then presented.

http://quantlabs.net/academy/download/free_quant_instituitional_books_/%5BJournal%20of%20Hydrologic%20Engineering,%20Genest%5D%20Everything%20You%20Always%20Wanted%20to%20Know%20about%20Copula%20Modeling%20but%20Were%20Afraid%20to%20Ask.pdf

Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask

This paper deals with estimation of the density of a copula function as well as with that of the Radon-Nikodym derivative of a bivariate distribution function with respect to the product of its marginal distribution functions. Strong uniform consistency and asymptotic normality of kernel-type estimators are proved under various conditions on the bandwidth and on the smoothness of the kernel. As an application, the estimation of Neyman-Pearson curves in the testing of independence problem is discussed.

Estimating the density of a copula function

Estimation of Hurst exponent revisited

In this paper we provide a detailed characterization of the asymptotic behavior of kernel density estimators for one-sided linear processes The conjecture that asymptotic normality for the kernel density estimator holds under short-range dependence is proved under minimal assumptions on bandwidths We also depict the dichotomous and trichotomous phenomena for various choices of bandwidths when the process is long-range dependent

https://projecteuclid.org/download/pdf_1/euclid.aos/1035844982

Kernel density estimation for linear processes

We consider the fixed-design regression model with long-range dependent normal errors and show that the finite-dimensional distributions of the properly normalized Gasser-Miiller and Priestley-Chao estimators of the regression function converge to those of a white noise process. Furthermore, the distributions of the suitably renormalized maximal deviations over an increasingly finer grid converge to the Gumbel distribution. These results contrast with our previous findings for the corresponding problem of estimating the marginal density of long-range dependent stationary sequences.

/pdf/nonparametric-regression-under-long-range-dependent-normal-ojz9vnw22r.pdf

Nonparametric Regression Under Long-Range Dependent Normal Errors

The kernel estimator is a widely used tool for the estimation of a density function. In this paper its adaptation to censored data using the Kaplan-Meier estimator is considered. Asymptotic properties of four estimators, arising naturally as a result of considering various types of bandwidths, are investigated. In particular we show that (i) both proposed estimators stemming from the nearest neighbor estimator have censoring-free variances and (ii) one of them is pointwise mean consistent.

/pdf/some-asymptotic-properties-of-kernel-estimators-of-a-density-36eeu08ptg.pdf

Jan Mielniczuk

Papers

Estimating the density of a copula function

Estimation of Hurst exponent revisited

Kernel density estimation for linear processes

Nonparametric Regression Under Long-Range Dependent Normal Errors

Some Asymptotic Properties of Kernel Estimators of a Density Function in Case of Censored Data