Jan Picek

Rank tests and regression rank score tests in measurement error models

The rank and regression rank score tests of linear hypothesis in the linear regression model are modified for measurement error models. The modified tests are still distribution free. Some tests of linear subhypotheses are invariant to the nuisance parameter, others are based on the aligned ranks using the R-estimators. The asymptotic relative efficiencies of tests with respect to tests in models without measurement errors are evaluated. The simulation study illustrates the powers of the tests.

A Class of Tests on the Tail Index

AbstractIn the family of distribution functions with nondegenerate right tail, we test the hypothesis

M-tests for Detection of Structural Changes in Regression

H_{m_0 }

Estimates of the Tail Index Based on Nonparametric Tests

with a hypothetical m0 > 0 and with some x0 ≥ 0. The proposed test is fully nonparametric and is based on splitting the set of observations into N subsamples of size n and on the empirical distribution function of the extremes of the subsamples; the asymptotics is for N → ∞ and fixed n (eventually small), and the asymptotic null distribution of the test criterion is normal. The test is consistent against exponentially tailed alternatives, as well as against heavy tailed alternatives with index m > m0, and is asymptotically unbiased for the broad family of distributions represented by

Bayesian Analysis for Likelihood-Based Nonparametric Regression

H_{m_0 }

Behavior of R-estimators under measurement errors

and its alternative. It may be used as a supplement to the usual tests of the Gumbel hypothesis m = ∞ against m < ∞, namely in the situation that the latter tests reject the hypothesis of exponentiality, and we need to know how heavy-tailed F really can be. This knowledge may be very important in the applications. The performance of the proposed test is illustrated on simulated data; we see that it distinguishes well the tails even for moderate samples. For an illustration, the proposed (nonparametric) test is numerically compared with the (parametric) likelihood ratio test for the class of generalized Pareto distributions. As it can be expected, the parametric test behaves well, provided F is exactly generalized Pareto, while the nonparametric test performs better for all other considered distribution shapes.