Pons Odile. Statistical Tests for Nonparametric Hypotheses. Asymptotic Theory

World Scientific, 2014. — 304 p. — ISBN: 978-9814531740.In tests of hypotheses concerning the whole distribution function of samples, the optimal test statistics rely on the empirical processes or on their functionals. The approach is similar for tests about the form of densities and regression curves, Kolmogorov-Smirnov and Cramer-von Mises type statistics can be defined for them and for other functions. The main subjects of the book are tests of hypotheses in regular nonparametric models, they include tests based on empirical processes and smooth estimators of density functions, regression functions and regular functions defining the distribution of point processes and Gaussian diffusion processes. The asymptotic behavior of the statistics and the asymptotic properties of the tests are detailed.The last part generalizes tests built for samples to sequential tests, especially tests for processes observed continuously in time. There is no unified theory of sequential tests and several approaches are considered.
Many tests have been studied since several decades and special conferences have early been devoted to the theory of the tests, they give an interesting account of the main advances. This book is not exhaustive, its emphasis on the asymptotically optimal nonparametric test originated in the empirical processes theory and in a nonparametric version of Lecam’s asymptotic theory of testing hypotheses which are presented and generalized in this book.Definitions
Rank tests and empirical distribution functions
Hypotheses of the tests
Weak convergence of the test statistics
Tests for densities and curves
Asymptotic levels of tests
Permutation and bootstrap tests
Relative efficiency of tests
Asymptotic theory
Parametric tests
Parametric likelihood ratio tests
Likelihood ratio tests against local alternatives
Nonparametric likelihood ratio tests
Nonparametric tests for empirical functionals
Tests of homogeneity
Mixtures of exponential distributions
Nonparametric bootstrap tests
Exercises
Nonparametric tests for one sampleKolmogorov-Smirnov tests for a distribution function
Tests for symmetry of a density
Tests about the form of a density
Goodness of fit test in biased length models
Goodness of fit tests for a regression function
Tests about the form of a regression function
Tests based on observations by intervals
Exercises
Two-sample testsTests of independence
Test of homogeneity
Goodness of fit tests in R2
Tests of symmetry for a bivariate density
Tests about the form of densities
Comparison of two regression curves
Tests based on observations by intervals
Exercises
Multi-dimensional testsTests of independence
Test of homogeneity of k sub-samples
Test of homogeneity of k rescaled distributions
Test of homogeneity of several variables of Rk
Test of equality of marginal distributions
Test of exchangeable components for a random variable
Tests in single-index models
Comparison of k curves
Tests in proportional odds models
Tests for observations by intervals
Competing risks
Tests for Markov renewal processes
Tests in Rkn as kn tends to infinity
Exercises
Nonparametric tests for processesGoodness of fit tests for an ergodic process
Poisson process
Poisson processes with scarce jumps
Point processes in R+
Marked point processes
Spatial Poisson processes
Tests of stationarity for point processes
Diffusion processes
Comparison of diffusion processes
Exercises
Nonparametric tests under censoring or truncationComparison of right-censored distributions
Likelihood ratio test of homogeneity
Tests of homogeneity against general local alternatives
Goodness of fit for the hazard functions ratio
Tests of comparison of k samples
Goodness of fit tests for k samples
Tests of independence of two censored variables
Comparison of two bivariate distributions
Tests for left-censored samples
Tests for the mean residual life and excess life
Tests for right or left-truncated samples
Comparison of censored or truncated regression curves
Observation in random intervals
Exercises
Sequential testsDefinitions and properties
Sequential likelihood ratio test
Sequential algorithms for test statistics
Properties of the record variables
Sequential tests for point processes
Sequential tests for hazard functions
Sequential tests for regressions and diffusion processes
Appendix Functional estimation and probability