Non-standard rank tests
The monograph develops a methodology to construct rank tests and to study their asymptotic properties for models where the standard regularity conditions \((L^ 1\)-differentiability of the underlying densities or \(L^ 2\)-differentiability of the square root of the densities) do not hold. Chapter I is devoted to finite sample results for rank tests while the next chapters are mainly concerned with asymptotic properties of rank tests for lifetime models where the underlying random variables are of the form \(X_ i=\gamma (Z_ i+\theta_ 0)+\theta\), \(Z_ i\) are i.i.d. with densities \(f(x)=x^ ar(x)1_{(0,\infty)}(x),\) \(a\geq 0\), with a as the shape parameter and a smooth function r(x) [which \(=(1+a)\exp (- x^{1+a})\) in the Weibull case]. Both parametric models (for \(\theta\)) and semi-parametric models \((\gamma,\theta_ 0,a\) and the function r(x) as nuisance parameter) are considered. The following three null hypotheses are treated: \(H_ 0\) independent and identical distributions, \(H_ 1\) symmetric distributions, \(H_ 2\) independence. Table of Contents: I. Locally most powerful rank-tests - finite sample results (1. Preliminaries; 2. Locally most powerfull rank test for \(H_ 0\); 3. Locally most powerfull rank test for \(H_ 1\); 4. Locally most powerfull rank test for \(H_ 2\) against dependence); II. Asymptotic results for locally most powerful tests (5. Approximate rank tests; 6. Asymptotic results for locally most powerful rank tests with respect to \(H_ 0\); 7. Asymptotic results for \(H_ 1\); 8. Asymptotic results for \(H_ 2\); 9. Asymptotic results for locally most powerful tests in the case \(a=0);\) III. Asymptotic results for rank tests under alternatives (10. Limit distributions under alternatives; 11. Rank tests under almost regular models); IV. Tests based on minimum ranks (12. The minimum rank test (finite sample results); 13. The minimum rank tests (asymptotic results)); V. Parametric results for almost regular models (14. Local asymptotic normality under almost regular assumptions); VI Semiparametric models and Monte Carlo results (15. Adaptive tests for semiparametric regression alternatives; 16. A comparison of rank tests and parametric tests: the Monte Carlo approach); Appendix: Statistical experiments with non-regular densities.
Statistical experiments with non-regular densities
scientific article; zbMATH DE number 48815