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This study proposes the estimators for the mean and its variance of the number of respondents who possessed a rare sensitive attribute based on stratified sampling schemes (stratified sampling and stratified double sampling). This study deals with the extension of the estimation reported in Land et al. [Estimation of a rare sensitive attribute using Poisson distribution, Statistics (2011), in press. DOI: 10.1080/02331888.2010.524300] using a Poisson distribution and an unrelated question randomized response model reported in Greenberg et al. [The unrelated question randomized response model: Theoretical framework, J. Amer. Statist. Assoc. 64 (1969), 520–539]. In the stratified sampling, the estimators are proposed when the parameter of the rare unrelated attribute is known and unknown. The variances of estimators using a proportional and optimum allocation are also suggested. The proposed estimators are evaluated using a relative efficiency comparing variances of the estimators reported in Land et al. dep...

We propose new variants of Land et al.’s [Estimation of a rare sensitive attribute using Poisson distribution. Statistics. 2011. DOI: 10.1080/02331888.2010.524300] randomized response model when a population consists of some clusters and the population is stratified with some clusters in each stratum. The estimator for the mean number of persons who possess a rare sensitive attribute, its variance, and the variance estimator are suggested when the parameter of a rare unrelated attribute is assumed to be known and unknown. The clusters are selected with and without replacement. When they are selected with replacement, the selecting probabilities for each cluster are defined depending on the cluster sizes and with equal probability. In addition, the variance comparison between a probability proportional to size (PPS) and PPS for stratification are performed. When the parameters vary in clusters, the stratified PPS has better efficiency than the PPS.

To examine the asymmetry of financial data in detail, we have considered both the tail dependence with diverse copulas and Jung et al.'s (8) directional dependence by copula. From the empirical study in this paper, we have found that the tail dependence by Patton's (11) modified symmetrized Joe-Clayton copula function did not show the asymmetry property sufficiently because there is no tail dependence for the currency exchange rates of Republic of Korea and Japan against the US dollar after the 2008 financial crisis. However, by using the copula approach for directional dependence, we showed that there exists the asymmetry of the currency exchange rates of Republic of Korea and Japan against the US dollar before and after the 2008 financial crisis because the directional dependence only when existing an asymmetry of data can be found. We conclude that the copula approach for directional dependence can be supplemented to explain the asymmetry property in addition to the tail dependence approach.

We study a weighted least squares estimator for Aalen's additive risk model with right-censored survival data which allows for a very flexible handling of covariates. We divide the follow-up period into intervals and assume a constant hazard rate in each interval. The model is motivated as a piecewise approximation of a hazard function composed of three parts: arbitrary nonparametric functions for some covariate effects, smoothly varying functions for others, and known (or constant) functions for yet others. The proposed estimator is an extension of the grouped data version of the Huffer and McKeague (1991) estimator. For our model, since the number of parameters is finite (although large), conventional approaches (such as maximum likelihood) are easy to formulate and implement. The approach is illustrated by simulations, and is compared to the previous studies. The method is also applied to the Framingham study data.