Minimax strategies in survey sampling

The risk of a sampling strategy is a function on the parameter space, which is the set of all vectors composed of possible values of the variable of interest. It seems natural to ask for a minimax strategy, minimizing the maximal risk. So far answers have been provided for completely symmetric parameter spaces. Results available for more general spaces refer to sample size 1 or to large sample sizes allowing for asymptotic approximation. In the present paper we consider arbitrary sample sizes, derive a lower bound for the maximal risk under very weak conditions and obtain minimax strategies for a large class of parameter spaces. Our results do not apply to parameter spaces with strong deviations from symmetry. For such spaces a minimax strategy will prescribe to consider only a small number of samples and takes a non-random and purposive character, which is in accordance with the common practice of completely sampling a stratum of large units.     

Variance estimation in model assisted survey sampling

Two versions of Yates-Grundy type variance estimators are usually employed for large samples when estimating a survey population total by a generalized regression (Greg, in brief) predictor motivated by consideration of a linear regression model. Their two alternative modifications are developed so that the limiting values of the design expectations of the model expectations of variance estimators 'match' respectively the (I) model expectations of the Taylor approximation of the design variance of the Greg predictor and the (II) limiting value of the design expectation of the model expectation of the squared difference between the Greg predictor and the population total. The exercise is extended to yield modifications needed when randomized response (RR) is only available rather than direct response (DR) when one encounters sensitive issues demanding protection of privacy. A comparative study based on simulation is presented for illustration..     

Optional randomized response technique in two-phase sampling

Two-phase optional randomized response (TORR) technique has been proposed. The proposed randomized response (RR) performs much better in general than the existing RR techniques. Theoretical and numerical illustrations identified conditions of superiority of the proposed TORR over the existing comparable optional randomized response (ORR) techniques.     

Generalized franklin's model for randomized response sampling

In this paper we introduce two estimators of a population proportion when randomized response sampling with a normal randomizing distribution is used* The estimators have been obtained by using the method of moments. Both of the proposed estimators are shown to be more efficient than the corresponding estimators of Eranklin (1989 b).     

Exact confidence intervals for randomized response strategies

For surveys with sensitive questions, randomized response sampling strategies are often used to increase the response rate and encourage participants to provide the truth of the question while participants' privacy and confidentiality are protected. The proportion of responding ‘yes’ to the sensitive question is the parameter of interest. Asymptotic confidence intervals for this proportion are calculated from the limiting distribution of the test statistic, and are traditionally used in practice for statistical inference. It is well known that these intervals do not guarantee the coverage probability. For this reason, we apply the exact approach, adjusting the critical value as in [10 J. Frey and A. Pérez, Exact binomial confidence intervals for randomized response, Amer. Statist.66 (2012), pp. 8–15. Available at http://dx.doi.org/10.1080/00031305.2012.663680.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], to construct the exact confidence interval of the proportion based on the likelihood ratio test and three Wilson-type tests. Two randomized response sampling strategies are studied: the Warner model and the unrelated model. The exact interval based on the likelihood ratio test has shorter average length than others when the probability of the sensitive question is low. Exact Wilson intervals have good performance in other cases. A real example from a survey study is utilized to illustrate the application of these exact intervals.     

Penalized calibration in survey sampling: Design-based estimation assisted by mixed models

Calibration techniques in survey sampling, such as generalized regression estimation (GREG), were formalized in the 1990s to produce efficient estimators of linear combinations of study variables, such as totals or means. They implicitly lie on the assumption of a linear regression model between the variable of interest and some auxiliary variables in order to yield estimates with lower variance if the model is true and remaining approximately design-unbiased even if the model does not hold. We propose a new class of model-assisted estimators obtained by releasing a few calibration constraints and replacing them with a penalty term. This penalization is added to the distance criterion to minimize. By introducing the concept of penalized calibration, combining usual calibration and this ‘relaxed’ calibration, we are able to adjust the weight given to the available auxiliary information. We obtain a more flexible estimation procedure giving better estimates particularly when the auxiliary information is overly abundant or not fully appropriate to be completely used. Such an approach can also be seen as a design-based alternative to the estimation procedures based on the more general class of mixed models, presenting new prospects in some scopes of application such as inference on small domains.     

Pseudo-Bayes and pseudo-empirical Bayes estimators in randomized response sampling

In this article, new pseudo-Bayes and pseudo-empirical Bayes estimators for estimating the proportion of a potentially sensitive attribute in a survey sampling have been introduced. The proposed estimators are compared with the recent estimator proposed by Odumade and Singh [Efficient use of two decks of cards in randomized response sampling, Comm. Statist. Theory Methods 38 (2009), pp. 439–446] and Warner [Randomized response: A survey technique for eliminating evasive answer bias, J. Amer. Statist. Assoc. 60 (1965), pp. 63–69].     

A bayesian comparison of randomized and voluntary response sampling models

This paper is concerned with making a Bayesian comparison between the two types of survey sampling procedures where the individuals are asked sensitive questions. The randomized response procedure is designed to obtain information in such a way that the respondent is protected against revealing his/her status about the sensitive issue. In voluntary response model each individual chooses either to respond or not to respond. It is assumed that the respondents do not falsify their answers. The problem of interest is to estimate the proportion of the sampled population who belong to the sensitive group. A beta prior distribution is used for the proportion of interest. The loss function is taken to be the quadratic loss function.     

A new quasi empirical bayes estimate in randomized response sampling

In this article, a new notion of “quasi-empirical” Bayes estimation is developed for estimating the proportion of a sensitive attribute in a population by making use of both a prior distribution of prevalence of the sensitive attribute in addition to the known prior distribution of an unrelated characteristic. The proposed quasi-empirical Bayes estimate is compared with those of the unrelated question model due to Greenberg et al. by means of a simulation study.     

An alternative estimator for randomized response sampling with continuous distributions from a dichotomous population

In this paper, we introduce an alternative estimator of a population proportion from a dichotomous population when using randomized response sampling with continuous randomizing distributions. We also propose the alternative use of exponential randomizing densities. The estimator is obtained by method of moments and is compared with Franklin's (1989) estimator using normal and exponential distributions. The proposed estimator is more efficient than Franklin's (1989) estimator under suitable conditions for the two distributions.     

On conditional sampling strategies

Estimation of the population average by means of a conditional strategy has been considered e.g. in [2–6,9] and [10]. Let us assume that the sampling design depends on a function of an auxiliary variable called an auxiliary statistic like: the sample mean or the sample variance. Under the conditional versions of these designs several estimators of the population mean are considered: the Horvitz-Thompson statistic, ratio and regression type estimators. The unbiasedness and accuracy of the strategies are considered. The empirical example of the accuracy comparisons of strategies is developed. The conditional strategies could improve the accuracy of estimation and protect it against outliers observations.     

Utilization of higher order moments of scrambling variables in randomized response sampling

In this paper, a new estimator for estimating the proportion of a potentially sensitive attribute in survey sampling has been introduced. The proposed estimator makes use of higher order moments of the scrambling variable at the estimation stage. The proposed estimator has been found to be more efficient than the estimator due to Kuk [1990. Asking sensitive questions indirectly. Biomerika 77(2), 436–438] and Franklin [1989. A comparison of estimators for randomized response sampling with continuous distributions from a dichotomous population. Comm. Statist. Theory Methods 18, 489–505] type estimators in randomized response sampling. Recently, Guerriero and Sandri [2007. A note on the comparison of some randomized response procedures. J. Statist. Plann. Inference 137, 2184–2190] have shown that the family of randomized response models proposed by Kuk [1990. Asking sensitive questions indirectly. Biomerika 77(2), 436–438] is better than the Simmons’ family in terms of efficiency and protection.     

Admissible and optimal sampling strategy for estimating finite population mean in randomized response surveys with multiple responses

ABSTRACT

We consider the problem of estimation of a finite population mean (or proportion) related to a sensitive character under a randomized response model when independent responses are obtained from each sampled individual as many times as he/she is selected in the sample and prove the admissibility of a sampling strategy in a class of comparable linear unbiased strategies. We prove that the admissible strategy is also optimal in this class under a super-population model.     

Admissibility of a variance estimator in finite population sampling under Warner's randomized response plan with multiple responses

ABSTRACT

Let P be the proportion of individuals in a finite population possessing a sensitive attribute. We consider the problem of unbiased estimation of (i) the variance of a linear unbiased estimator of P and (ii) the population variance P (1—P) for a given probability sampling design under Warner's (1965 Warner, S.L. (1965). Randomized response - A survey technique for eliminating evasive answer bias. J. Amer. Statist. Assoc. 60:63–69.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) randomized response (RR) plan when independent responses are obtained from each sampled individual as many times as he/she is selected in the sample and prove the admissibility of a quadratic unbiased estimator for each.     

Variance estimation with randomized response

Presenting a general procedure of eliciting a randomized response (RR) from selected persons in order to estimate the total of a sensitive variable related to a finite survey population, we consider two estimators along with variance estimators treating the case of sampling with probabilities proportional to (known) size measures (PPS) with replacement (WR), drawing analogies with multi-stage sampling and note their relative efficacies.     

Model selection in randomized response techniques for binary responses

Randomized response techniques (RRTs) have been proposed in survey sampling literature as a solution to the problem of social desirability bias (SDB) while dealing with sensitive questions. All RRTs reduce the SDB by introducing privacy protection for the respondents, but the variances of the estimates become larger compared with the ones obtained from direct questioning technique (DQT). The success of RRTs heavily depends on the assumption that the variable of interest is in fact sensitive for the population under study. There might be situations, however, where a presumably sensitive question is not considered to be sensitive in some populations, in which case using an RRT instead of the DQT would inflate the variance of the estimates unreasonably. In this study, we propose a two-stage sampling procedure for a binary response, which enables one to accurately estimate both the prevalence of the sensitive characteristic and the probability of cheating in a population. The proposed model allows one to choose between an RRT and the DQT. We support our theoretical results with numerous simulations.     

A comparison of two randomized response survey methods with consideration for the level of respondent protection

In recent years randomized response methods have been introduced in an attempt to improve the accuracy and honesty in personalized response surveys of very sensitive questions. Two randomized response methods are compared, taking into account the protection afforded the respondent. In addition, we point out that the estimators, which previous authors have claimed to be the maximum likelihood estimators of the population proportion with the sensitive characteristic, are in fact not the maximum likelihood estimators.     

Robust sampling strategies for regression estimation

For probability linear regression estimation, conditions are derived where sampling will be robust against violations of the commonly assumed heterogeneous variance model. Stratified pps (spps) and stratified random sampling (spscx) are shown to satisfy these conditions approximately and are more efficient generally than restricted simple random sampling (RSRS) for some real populations and for artificial populations with weights of k = 0, 0.5, 1.0, 1.5 and 2.0. The criteria needs some additional refinement to better predict relative efficiency of spps and spscx.