Pier Francesco Perri

New scrambled response models for estimating the mean of a sensitive quantitative character

Moving from the scrambling mechanism recently suggested by Saha [25], three scrambled randomized response (SRR) models are introduced with the intent to realize a right trade-off between efficiency and privacy protection. The models perturb the true response on the sensitive variable by resorting to the multiplicative and additive approaches in different ways. Some analytical and numerical comparisons of efficiency are performed to set up the conditions under which improvements upon Sahas model can be obtained and to quantify the efficiency gain. The use of auxiliary information is also discussed in a class of estimators for the sensitive mean under a generic randomization scheme. The class includes also the three proposed SRR models. Finally, some graphical comparisons are carried out from the double perspective of the accuracy in the estimates and respondents’ privacy protection.

An improved class of estimators for the population mean

Starting from the Rao (Commun Stat Theory Methods 20:3325–3340, 1991) regression estimator, we propose a class of estimators for the unknown mean of a survey variable when auxiliary information is available. The bias and the mean square error of the estimators belonging to the class are obtained and the expressions for the optimum parameters minimizing the asymptotic mean square error are given in closed form. A simple condition allowing us to improve the classical regression estimator is worked out. Finally, in order to compare the performance of some estimators with the regression one, a simulation study is carried out when some population parameters are supposed to be unknown.

Improved Estimators of the Population Mean for Missing Data

Motivated by a recent work by Kadilar and Cingi (2008), we proposed three regression-type estimators to overcome the problem of missing data for a study variable. The estimators make optimal use of the available auxiliary information. We show that, given the same amount of information, these estimators are simpler and more efficient than those proposed by Kadilar and Cingi. A numerical illustration, performed on three different populations, highlights the efficiency gain from using our proposal. Finally, a suggestion is made regarding the optimal use of auxiliary information in sampling practice.

Some developments on the log-Dagum distribution

Skewed and fat-tailed distributions frequently occur in many applications. Models proposed to deal with skewness and kurtosis may be difficult to treat because the density function cannot usually be written in a closed form and the moments might not exist. The log-Dagum distribution is a flexible and simple model obtained by a logarithmic transformation of the Dagum random variable. In this paper, some characteristics of the model are illustrated and the estimation of the parameters is considered. An application is given with the purpose of modeling kurtosis and skewness that mark the financial return distribution.

A calibration-based approach to sensitive data: a simulation study

In this paper, we discuss the use of auxiliary information to estimate the population mean of a sensitive variable when data are perturbed by means of three scrambled response devices, namely the additive, the multiplicative and the mixed model. Emphasis is given to the calibration approach, and the behavior of different estimators is investigated through simulated and real data. It is shown that the use of auxiliary information can considerably improve the efficiency of the estimates without jeopardizing respondent privacy.

Statistical Modeling of Temporal Dependence in Financial Data via a Copula Function

In financial analysis it is useful to study the dependence between two or more time series as well as the temporal dependence in a univariate time series. This article is concerned with the statistical modeling of the dependence structure in a univariate financial time series using the concept of copula. We treat the series of financial returns as a first order Markov process. The Archimedean two-parameter BB7 copula is adopted to describe the underlying dependence structure between two consecutive returns, while the log-Dagum distribution is employed to model the margins marked by skewness and kurtosis. A simulation study is carried out to evaluate the performance of the maximum likelihood estimates. Furthermore, we apply the model to the daily returns of four stocks and, finally, we illustrate how its fitting to data can be improved when the dependence between consecutive returns is described through a copula function.

A Property of the CHAID Partitioning Method for Dichotomous Randomized Response Data and Categorical Predictors

In this paper, we present empirical and theoretical results on classification trees for randomized response data. We considered a dichotomous sensitive response variable with the true status intentionally misclassified by the respondents using rules prescribed by a randomized response method. We assumed that classification trees are grown using the Pearson chi-square test as a splitting criterion, and that the randomized response data are analyzed using classification trees as if they were not perturbed. We proved that classification trees analyzing observed randomized response data and estimated true data have a one-to-one correspondence in terms of ranking the splitting variables. This is illustrated using two real data sets.

Scrambled Response Models Based on Auxiliary Variables

We discuss the problem of obtaining reliable data on a sensitive quantitative variable without jeopardizing respondent privacy. The information is obtained by asking respondents to perturb the response through a scrambling mechanism. A general device allowing for the use of multi-auxiliary variables is illustrated as well as a class of estimators for the unknown mean of a sensitive variable. A number of scrambled response models are shown and others discussed in terms of the efficiency of the estimates and the privacy guaranteed to respondents.

Randomized response surveys: A note on some privacy protection measures

We discuss a number of privacy protection measures in situations where people are asked highly confidential questions concerning a quantitative sensitive variable. Most of the discussion will be devoted to the measures proposed by [8,11,12,30,31] with particular reference to the trade-off between the level of privacy disclosure and the efficiency of the estimates. Determination of the optimal sample size which would allow researchers to attain a predetermined level of efficiency and privacy is also considered.

Horvitz-Thompson estimation with randomized response and nonresponse

We consider the nonstandard situation where the randomized response theory is applied in a nonresponse set-up. The customary Horvitz-Thompson estimator for the population total of a sensitive variable is adjusted to take into account missing data, true response and randomized response, and the expression of its variance and the corresponding estimator provided. Finally, a numerical study is carried out to investigate how the estimation process can be affected by scrambled responses and how possible improvements in respondent cooperation can compensate for certain levels of nonresponse.