Tomáš Hobza

Likelihood Divergence Statistics for Testing Hypotheses in Familial Data

Abstract Rényi Statistics, for testing composite hypotheses in parametric models, are defined as Rényi divergences between unrestricted and restricted estimated joint probability density functions. This family of statistics is proposed to test the equality of intraclass correlation coefficients in multivariate normal familial data. When maximum likelihood estimators are used, asymptotic distributions of test statistics under null hypothesis are obtained. Rényi statistics are compared with the likelihood ratio test statistic in terms of sizes and powers.

Generalized piecewise linear histograms

We extend the concept of piecewise linear histogram introduced recently by Beirlant, Berlinet and Gyorfi. The disadvantage of that histogram is that in many models it takes on negative values with probability close to 1. We show that for a wide set of models, the extended class of estimates contains a bona fide density with probability tending to 1 as the sample size n increases to infinity. The mean integrated absolute error in the extended class of estimators decreases with the same rate n–2/5 as in the original narrower class.

Inner structure of vehicular ensembles and random matrix theory

Abstract We introduce a special class of random matrices (DUE) whose spectral statistics corresponds to statistics of microscopical quantities detected in vehicular flows. Comparing the level spacing distribution (for ordered eigenvalues in unfolded spectra of DUE matrices) with the time-clearance distribution extracted from various areas of the flux-density diagram (evaluated from original traffic data measured on Czech expressways with high occupancies) we demonstrate that the set of classical systems showing an universality associated with Random Matrix Ensembles can be extended by traffic systems.

A Fay–Herriot Model with Different Random Effect Variances

A modification of the Fay–Herriot model is introduced to treat situations where small areas are divided in two groups and domain random effects have different variances across the groups. The model is applicable to data having a large subset of domains where direct estimates of the variable of interest cannot be described in the same way as in its complementary subset of domains. This is generally the case when domains are constructed by crossing geographical characteristics with sex. Algorithms and formulas to fit the model, to calculate EBLUPs and to estimate mean squared errors are given. Monte Carlo simulation experiments are presented to illustrate the gain of precision obtained by using the proposed model and to get some practical conclusions. A motivating application to Spanish Labour Force Survey data is also given.

Multi-sample Rényi test statistics

This paper focuses on testing composite hypotheses about parameters of s independent samples of different sizes. With this purpose, it introduces test statistics based on the family of Renyi divergences between likelihoods. The asymptotic distributions of the proposed test statistics and of the likelihood ratio statistic are derived under standard regularity assumptions. An application to test the homogeneity of variances in data from families belonging to different populations is described and, under this setup, a simulation experiment compares the small sample performance of the likelihood ratio test and some members of the Renyi family of tests. The experiment indicates that some of the Renyi tests perform better under null hypothesis.

On convergence of fisher informations in continuous models with quantized observations

Continuous location models with real observations and well defined Fisher information are considered and reduction of the Fisher information due to quantizations of the observation space intom intervals is studied. In fact, generalized Fisher informations of orders α≥1 are considered where α=2 corresponds to the classical Fisher information. By an example it is argued that in some models the information of order α=2 is infinite while the informations of some orders α↮2 are finite. Among the studied problems is the existence of optimal quantizations which maximize the reduced information for fixedm and α≥1 and the construction of simple and practically applicable quantizations for which the reduction converges to zero whenm→∞, uniformly for all α≥1. The rate of this convergence is estimated for all α≥1 and directly evaluated for α=1 and α=2. For special models the reductions are directly evaluated form=1.2,… either analytically or numerically.

A Wald-type test statistic based on robust modified median estimator in logistic regression models

ABSTRACT In this paper a new robust estimator, modified median estimator, is introduced and studied for the logistic regression model. This estimator is based on the median estimator considered in Hobza et al. [Robust median estimator in logistic regression. J Stat Plan Inference. 2008;138:3822–3840]. Its asymptotic distribution is obtained. Using the modified median estimator, we also consider a Wald-type test statistic for testing linear hypotheses in the logistic regression model and we obtain its asymptotic distribution under the assumption of random regressors. An extensive simulation study is presented in order to analyse the efficiency as well as the robustness of the modified median estimator and Wald-type test based on it.

A modified nested-error regression model for small area estimation

A nested-error regression model having both fixed and random effects is introduced to estimate linear parameters of small areas. The model is applicable to data having a proportion of domains where the variable of interest cannot be described by a standard linear mixed model. Algorithms and formulas to fit the model, to calculate EBLUP and to estimate mean-squared errors are given. A Monte Carlo simulation experiment is presented to illustrate the gain of precision obtained by using the proposed model and to obtain some practical conclusions. A motivating application to Spanish Labour Force Survey data is also given.

Small area estimation under random regression coefficient models

Statistical agencies are interested to report precise estimates of linear parameters from small areas. This goal can be achieved by using model-based inference. In this sense, random regression coefficient models provide a flexible way of modelling the relationship between the target and the auxiliary variables. Because of this, empirical best linear unbiased predictor (EBLUP) estimates based on these models are introduced. A closed-formula procedure to estimate the mean-squared error of the EBLUP estimators is also given and empirically studied. Results of several simulation studies are reported as well as an application to the estimation of household normalized net annual incomes in the Spanish Living Conditions Survey.

Small Area Estimation of Poverty Proportions under Random Regression Coefficient Models

In this paper a random regression coefficient model is used to provide estimates of small area poverty proportions. As poverty variable is dichotomic at the individual level, the sample data from Spanish Living Conditions Survey is previously aggregated to the level of census sections. EBLUP estimates based on the proposed model are obtained. A closed-formula procedure to estimate the mean squared error of the EBLUP estimators is given and empirically studied. Results of several simulations studies are reported as well as an application to real data.