An optional scrambled randomized response technique for practical surveys

Eichhorn and Hayre (J Stat Plan Inference 7:307–316, 1983) introduced the scrambled response technique to gather information on sensitive quantitative variables. Singh and Joarder (Metron 15:151–157, 1997), Gupta et al. (J Stat Plan Inference 100:239–247, 2002) and Bar-Lev et al. (Metrika 60:255–260, 2004) permitted the respondents either to report their true values on the sensitive quantitative variable or the scrambled response and developed the optional randomized response (ORR) technique based on simple random sample with replacement (SRSWR). While developing the ORR procedure, these authors made the assumption that the probability of disclosing the true response or the randomized response (RR) is the same for all the individuals in a population. This is not a very realistic assumption as in practical survey situations the probability of reporting the true value or the RR generally varies from unit to unit. Moreover, if one generalizes the ORR method as developed by these authors relaxing the ‘constant probability’ assumption, the variance of an unbiased estimator for the population total or mean can not be estimated as this involves the unknown parameter, ‘the probability of revealing the true response’. Here we propose a modified ORR procedure for stratified unequal probability sampling after relaxing the assumption of ‘constant probability’ of providing the true response. It is also demonstrated with a numerical exercise that our procedure produces better estimator for a population total than that provided by the method suggested by the earlier authors.     

Mean square error estimation in multi-stage sampling

Summary: Suppose for a homogeneous linear unbiased function of the sampled first stage unit (fsu)-values taken as an estimator of a survey population total, the sampling variance is expressed as a homogeneous quadratic function of the fsu-values. When the fsu-values are not ascertainable but unbiased estimators for them are separately available through sampling in later stages and substituted into the estimator, Raj (1968) gave a simple variance estimator formula for this multi-stage estimator of the population total. He requires that the variances of the estimated fsu-values in sampling at later stages and their unbiased estimators are available in certain `simple forms'. For the same set-up Rao (1975) derived an alternative variance estimator when the later stage sampling variances have more ‘complex forms’. Here we pursue with Raj's (1968) simple forms to derive a few alternative variance and mean square error estimators when the condition of homogeneity or unbiasedness in the original estimator of the total is relaxed and the variance of the original estimator is not expressed as a quadratic form.  We illustrate a particular three-stage sampling strategy and present a simulation-based numerical exercise showing the relative efficacies of two alternative variance estimators. Received: 19 February 1999     

Estimation from two-stage unequal probability sampling with missing units

A formula is presented for an unbiased estimator for the variance of an unbiased estimator of a survey population total as well as for an unbiased estimator of its variance based on sampling in two-stages following Rao et al. J Roy Stat Soc B 24: 482–491 (1962) scheme in both stages when the originally selected units in both stages cannot be fully covered in the survey but are to be randomly sub-sampled. The development is helpful to tackle non-responses if assumed to have occurred at random in either or both the stages     

Estimation of bias and standard error of an improved estimator of normal mean

This paper considers an improved estimator of normal mean which is obtained by considering a feasible version of minimum mean squared error estimator. The exact expression for the bias and the mean squared error are fairly complicated and do not provide any guidelines as how to estimate the standard error of improved estimator. As is well known that any estimator without a formula for standard error has little practical utility. We therefore derive unbiased estimators for the bias and mean squared error of the improved estimator. Incidently, they turn out to be minimum variance unbiased estimators. Further, this exercise yields a simple formula for estimating the standard error. Based on the criterion of estimated standard error, the efficiency of the improved estimator with respect to the traditional unbiased estimator (i.e., sample mean) is examined numerically. The relationship with asymptotic standard error is also studied.     

Almost unbiased estimation in multiplicative models

In the lognormal linear models the estimation of constant term presents problems. In this paper we use weighted jackknife procedure (suggested by Hinkley 1977) for reducing the bias of the maximum likelihood estimator. The resulting estimator is unbiased upto order (1/T),T being the number of observations, and has the same MSE as that of the MLE to the same order of approximation; moreover, being the jackknife estimator it enjoys all the desirable large sample properties like any other jackknife estimator. The research of this author is partially supported through a research grant from NSERC of Canada.     

A comparison between PPSWR and chaudhuri’s IPPS procedures

Chaudhuri (1975) suggested a simple procedure for extending any IPPS procedure involving two draws to an IPPS procedure for a general sample size, provided the size measures satisfy a certain condition. It is proved that the variance of the HTE based on Chaudhuri’s procedure is smaller than the variance of the usual unbiased estimator based on a PPSWR sample in the same number of draws, if the same is true for the procedure to start with.     

Christofides’ randomized response technique in complex sample surveys

Christofides (2003) has given an improved modification of Warners (1965) pioneering randomized response (RR) technique in estimating an unknown proportion of people bearing a sensitive characteristic in a given community. As both these RR devices are shown to yield unbiased estimators based only on simple random sampling (SRS) with replacement (WR) but in practice samples are mostly taken with unequal selection probabilities without replacement (WOR), here we present methods of estimation when Christofides RR data are available from unequal probability samples. Warners (1965) RR device was earlier shown by Chaudhuri (2001) to be applicable in complex surveys. For completeness we present estimators for the variance of our estimator and also describe what to do if some people opt to divulge truths.This research is partially supported by CSIR grant No. 21(0539)/02/EMR-II     

Admissible unbiased variance estimation in finite population sampling under randomized response

LetP be the proportion of units in a finite population possessing a sensitive attribute. We prove the admissibility of (i) an unbiased estimator of the variance of a general homogeneous linear unbiased estimator ofP and (ii) an unbiased estimator of the population varianceP(1−P), based on an arbitrary but fixed sampling design, under the randomized response plans due to Warner (1965) and Eriksson (1973). Admissibility of an unbiased strategy for estimating the population variance is also established.     

The power of four tests of autocorrelation in the linear regression model

The power of each of four tests of first-order autocorrelation in the linear regression model is determined for a simple and multiple regression model whose parameters are presumed to be known. The tests are: Durbin-Watson bounds test, a test based on Theil's best linear unbiased scalar estimator, a test devised by Abrahamse, Koerts and Louter, and an exact test devised by Durbin.For positive values of the coefficient of autocorrelation the Durbin-Watson bounds test is generally better than the tests based on the estimator proposed by Abrahamse, Koerts and Louter, the best linear unbiased scalar estimator, and the Durbin exact test. For negative values of the coefficient of autocorrelation, the pattern of results is mixed for all four test procedures. A byproduct of these experiments is the demonstrated feasibility of enumerating the distribution of the Durbin-Watson test statistic for any regression matrix and thus eliminating the region of indeterminacy from the Durbin-Watson test procedure.     

Spatial heteroskedasticity and autocorrelation consistent estimation of covariance matrix

This paper considers spatial heteroskedasticity and autocorrelation consistent (spatial HAC) estimation of covariance matrices of parameter estimators. We generalize the spatial HAC estimator introduced by Kelejian and Prucha (2007) to apply to linear and nonlinear spatial models with moment conditions. We establish its consistency, rate of convergence and asymptotic truncated mean squared error (MSE). Based on the asymptotic truncated MSE criterion, we derive the optimal bandwidth parameter and suggest its data dependent estimation procedure using a parametric plug-in method. The finite sample performances of the spatial HAC estimator are evaluated via Monte Carlo simulation.     

Mean squared error of the empirical best linear unbiased predictor in an orthogonal finite discrete spectrum linear regression model

The mean squared error (MSE) of the empirical best linear unbiased predictor in an orthogonal finite discrete spectrum linear regression model is derived and a comparison with the MSE of the best linear unbiased predictor in this model is made. It is shown that under weak conditions these two mean square errors are asymptotically the same.     

Mean squared error of ridge estimators in logistic regression

It is well known that the maximum likelihood estimator (MLE) is inadmissible when estimating the multidimensional Gaussian location parameter. We show that the verdict is much more subtle for the binary location parameter. We consider this problem in a regression framework by considering a ridge logistic regression (RR) with three alternative ways of shrinking the estimates of the event probabilities. While it is shown that all three variants reduce the mean squared error (MSE) of the MLE, there is at the same time, for every amount of shrinkage, a true value of the location parameter for which we are overshrinking, thus implying the minimaxity of the MLE in this family of estimators. Little shrinkage also always reduces the MSE of individual predictions for all three RR estimators; however, only the naive estimator that shrinks toward 1/2 retains this property for any generalized MSE (GMSE). In contrast, for the two RR estimators that shrink toward the common mean probability, there is always a GMSE for which even a minute amount of shrinkage increases the error. These theoretical results are illustrated on a numerical example. The estimators are also applied to a real data set, and practical implications of our results are discussed.     

FORECASTING REALIZED VOLATILITY WITH LINEAR AND NONLINEAR UNIVARIATE MODELS

Abstract In this paper, we consider a nonlinear model based on neural networks as well as linear models to forecast the daily volatility of the S&P 500 and FTSE 100 futures. As a proxy for daily volatility, we consider a consistent and unbiased estimator of the integrated volatility that is computed from high‐frequency intraday returns. We also consider a simple algorithm based on bagging (bootstrap aggregation) in order to specify the models analysed in this paper.     

Distributional and Inferential Properties of the Estimated Precision Cp Based on Multiple Samples

Process precision index C_p has been widely used in the manufacturing industry to provide numerical measures on process potential. Pearn et al. (1998) considered an unbiased estimator of C_p for one single sample. They showed that the unbiased estimator is the UMVUE. They also proposed an efficient test for C_p based on one single sample, and showed that the test is the UMP test. In this paper, we consider an unbiased estimator of C_p for multiple samples. We show that the unbiased estimator is the UMVUE of C_p, which is asymptotically efficient. We also consider an efficient test for C_p, and show that the test is the UMPtest for multiple samples. The practitioners can use the proposed test on theirin-plant applications to obtain reliable decisions.     

Variance estimation using auxiliary information: An almost unbiased multivariate ratio estimator

The goal of this paper is to investigate the repeated substitution method (seeSrivastava, 1967) estimating population variance in finite population sample surveys. We propose an almost unbiased multivariate ratio estimator that has a smaller mean squared error than the conventional biased multivariate ratio estimator (established byIsaki (1983)) and with the same precision as the multivariate regression estimator. Furthermore, it is a computationally much more interesting estimator since to compute it we only need to have knowledge of correlation among available variables, which it is common to have in several practical situations. A comparison of the multivariate ratio estimator proposed and the multivariate regression estimator is given.     

Design and estimation problems when estimating a regression coefficient from survey data

Summary The values of a variablex are assumed known for all elements in a finite population. Between this variable and another variableY, whose values are registered in a sample survey, there is the usual linear regression relationship. This paper considers problems of design and of estimation of the regression coefficienta and the interceptb. The followingGodambe type theorem is proved: There exists no minimum variance unbiased linear estimator ofa andb. We also derive that the usual estimators ofa andb have minimum variance if attention is restricted to the class of linear estimators unbiased in any given sample.     

Estimation of correlation for a finite universe

Summary The formula for thePearsonion correlation coefficient, based on a simple random sample, is a consistent estimator of the parent correlation between two given measurable characteristics of the elements of a finite universe. However, when the universe is stratified, and the elements in each stratum are drawn without replacement and with equal probabilities at each draw, the formula for a consistent estimator is much more complex. Generally speaking, the formula for a consistent estimator of the parent correlation varies with the sampling design. The results of this paper are relevant to the analysis of sociological data obtained through sample surveys. In the literature of the theory of statistical sampling the problem of estimating the correlation between pairs of variate values of the identifiable elements constituting a universe has so far not been considered. Needless to say the solution of this problem has an important bearing on sociological studies based on sample surveys.     

On the performance of the tick test

In financial research, the sign of a trade (or identity of trade aggressor) is not always available in the transaction dataset and it can be estimated using a simple set of rules called the tick test. In this paper we investigate the accuracy of the tick test from an analytical perspective by providing a closed formula for the performance of the prediction algorithm. By analyzing the derived equation, we provide formal arguments for the use of the tick test by proving that it is bounded to perform better than chance (50/50) and that the set of rules from the tick test provides an unbiased estimator of the trade signs. On the empirical side of the research, we compare the values from the analytical formula against the empirical performance of the tick test for fifteen heavily traded stocks in the Brazilian equity market. The results show that the formula is quite realistic in assessing the accuracy of the prediction algorithm in a real data situation.     

Repeated substitution method: The ratio estimator for the population variance

This paper proposes a new unbiased estimator for the population variance in finite population sample surveys using auxiliary information. This estimator has a smaller mean squared error than the conventional unbiased estimator, the ratio estimator established by Isaki (1983) and it has the same precision than the regression estimator. Furthermore, it is a much more interesting estimator from the computation viewpoint.     

On the existence of locally mvu estimators

It is proved that there exists an unbiased estimator for some real parameter of a class of distributions, which has minimal variance for some fixed distribution among all corresponding unbiased estimators, if and. only if the corresponding minimal variances for all related unbiased estimation problems concerning finite subsets of the underlying family of distributions are bounded. As an application it is shown that there does not exist some unbiased estimator for θ^k+c(ε≥0) with minimal variance for θ =0 among all corresponding unbiased estimators on the base of k i.i.d. random variables with a Cauchy-distribution, where θ denotes some location parameter.