Randomized response: Estimating mean square errors of linear estimators and finding optimal unbiased strategies

Summary General procedures are described to generate quantitative randomized response (RR) required to estimate the finite population total of a sensitive variable. Permitting sample selection with arbitrary probabilities a formula for the mean square error (MSE) of a linear estimator of total based on RR is noted indicating the simple modification over one that might be based on direct response (DR) if the latter were available. A general formula for an unbiased estimator of the MSE is presented. A simple approximation is proposed in case the RR ratio estimator is employed based on a simple random sample (SRS) taken without replacement (WOR). Among sampling strategies employing unbiased but not necessarily linear estimators based on RR, certain optimal ones are identified under two alternative models analogously to well-known counterparts based on DR, if available. Unlike Warner’s (1965) treatment of categorical RR we consider quantitative RR here.     

Analysis of Divergence in Loglinear Models When Expected Frequencies are Subject to Linear Constraints

Consider the loglinear model for categorical data under the assumption of multinomial sampling. We are interested in testing between various hypotheses on the parameter space when we have some hypotheses relating to the parameters of the models that can be written in terms of constraints on the frequencies. The usual likelihood ratio test, with maximum likelihood estimator for the unspecified parameters, is generalized to tests based on -divergence statistics, using minimum -divergence estimator. These tests yield the classical likelihood ratio test as a special case. Asymptotic distributions for the new -divergence test statistics are derived under the null hypothesis.     

Residual‐based IV estimation of dynamic panel data models with fixed effects

In dynamic panel regression, when the variance ratio of individual effects to disturbance is large, the system‐GMM estimator will have large asymptotic variance and poor finite sample performance. To deal with this variance ratio problem, we propose a residual‐based instrumental variables (RIV) estimator, which uses the residual from regressing Δy_i,t?1 on as the instrument for the level equation. The RIV estimator proposed is consistent and asymptotically normal under general assumptions. More importantly, its asymptotic variance is almost unaffected by the variance ratio of individual effects to disturbance. Monte Carlo simulations show that the RIV estimator has better finite sample performance compared to alternative estimators. The RIV estimator generates less finite sample bias than difference‐GMM, system‐GMM, collapsing‐GMM and Level‐IV estimators in most cases. Under RIV estimation, the variance ratio problem is well controlled, and the empirical distribution of its t‐statistic is similar to the standard normal distribution for moderate sample sizes.     

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.     

Estimating and testing linkage disequilibrium from sib data

We consider estimation and testing of linkage equilibrium from genotypic data on a random sample of sibs, such as monozygotic and dizygotic twins. We compute the maximum likelihood estimator with an EM‐algorithm and a likelihood ratio statistic that takes the family structure into account. As we are interested in applying this to twin data we also allow observations on single children, so that monozygotic twins can be included. We allow non‐zero recombination fraction between the loci of interest, so that linkage disequilibrium between both linked and unlinked loci can be tested. The EM‐algorithm for computing the maximum likelihood estimator of the haplotype frequencies and the likelihood ratio test‐statistic, are described in detail. It is shown that the usual estimators of haplotype frequencies based on ignoring that the sibs are related are inefficient, and the likelihood ratio test for testing that the loci are in linkage disequilibrium.     

Design-based or Prediction-based Inference? Stratified Random vs Stratified Balanced Sampling

Early survey statisticians faced a puzzling choice between randomized sampling and purposive selection but, by the early 1950s, Neyman's design-based or randomization approach had become generally accepted as standard. It remained virtually unchallenged until the early 1970s, when Royall and his co-authors produced an alternative approach based on statistical modelling. This revived the old idea of purposive selection, under the new name of “balanced sampling”. Suppose that the sampling strategy to be used for a particular survey is required to involve both a stratified sampling design and the classical ratio estimator, but that, within each stratum, a choice is allowed between simple random sampling and simple balanced sampling; then which should the survey statistician choose? The balanced sampling strategy appears preferable in terms of robustness and efficiency, but the randomized design has certain countervailing advantages. These include the simplicity of the selection process and an established public acceptance that randomization is “fair”. It transpires that nearly all the advantages of both schemes can be secured if simple random samples are selected within each stratum and a generalized regression estimator is used instead of the classical ratio estimator.     

Successive Sampling to Estimate Quantiles with P-Auxiliary Variables

The successive sampling is a known technique that can be used in longitudinal surveys to estimate population parameters and measurements of difference or change of a study variable. The paper discusses the estimation of quantiles for the current occasion based on sampling in two successive occasions and using p-auxiliary variables obtained of the previous occasion. A multivariate ratio estimator from the matched portion is used to provide the optimum estimate of a quantile by weighting the estimates inversely to derived optimum weights. Its properties are studied under large–sample approximation and the expressions of the variances are established. The behavior of these asymptotic variances is analyzed on the basis of data from natural populations. A simulation study is also used to measure the precision of the proposed estimator.     

Predictive estimation of finite population mean using product estimator

If the predictive approach advocated byBasu [1971] is adopted for estimating the mean of a finite population, it is observed that the use of mean per unit estimator, regression estimator and ratio estimator as a predictor for the mean of unobserved units in the population result in the corresponding customary estimators of the mean of the whole population. Whereas if the product estimator is used as a predictor for the mean of unobserved units in the population, the resulting estimator of the mean of the whole population is different from the customary product estimator. The new estimator so obtained is compared with the customary product estimator.     

Empiric bayes estimation of hypergeometric probability

A Bayes-empiric Bayes estimator of a parameter of the hypergeometric distribution, based on orthogonal polynomials on non-negative integers, is introduced. It is shown that this estimator is asymptotically optimal; and the resulting estimator of the prior probability function is mean square consistent.     

Estimators for particulate sampling derived from a multinomial distribution

Counting the number of units is not always practical during the sampling of particulate materials: it is often much easier to sample a fixed volume or fixed mass of particles. Hence, a class of sampling designs is proposed which leads to samples that have approximately a constant mass or a constant volume. For these sampling designs, estimators were derived which are a ratio of arbitrary sample totals. A Taylor expansion was used to obtain a first-order approximation for the expected value and variance in the limit of a large batch-to-sample size ratio. Furthermore, a π -estimator for a ratio of batch totals was found by deriving expressions for the first- and second-order inclusion probabilities. Practical application of the π -estimator is limited because it requires inaccessible batch information. However, when the denominator of the estimated batch ratio is the batch size, the π -estimator becomes equal to a sample total divided by the sample size in the limit of a large sample-to-particle size ratio. As a consequence, the obtained sample ratio becomes an unbiased estimator for the corresponding batch ratio. Retaining unbiasedness, the Horvitz–Thompson estimator for the variance, which also contains inaccessible batch information, is replaced by an estimator containing sample information only. Practical application of this estimator is illustrated for the sampling of slag, produced during the production of steel.     

Calibration methods for estimating quantiles

We propose a calibrated estimator of the quantiles of sample survey data and discuss the asymptotic theory behind it. This estimator is defined for any sampling design and uses the information available on J auxiliary variables. A simulation study based on a real population is used to compare the estimator with various methods proposed previously.     

A chain ratio-type estimator in finite population double sampling using two auxiliary variables

Summary In this paper we construct a chain ratio-type estimator using two auxiliary variables. The performance of the constructed estimator relative to the simple mean, ratio-type estimate based on double sampling andChand's ratio-type estimator is investigated. A numerical illustration is given.     

Influence diagnostic in survey sampling: Estimating the conditional bias

The conditional bias has been proposed by Moreno Rebollo et al. (1999) as an influence diagnostic in survey sampling, when the inference is based on the randomization distribution generated by a random sampling. The conditional bias is a population parameter. So, from an applied point of view, it must be estimated. In this paper, we propose an estimator of the conditional bias and we study conditions that guarantee its unbiasedness. The results are applied in a Simple Random Sampling and in a Proportional Probability Aggregated Size Sampling, when the ratio estimator is used. Received October 2000     

A semiparametric model for binary response and continuous outcomes under index heteroscedasticity

This paper formulates a likelihood‐based estimator for a double‐index, semiparametric binary response equation. A novel feature of this estimator is that it is based on density estimation under local smoothing. While the proofs differ from those based on alternative density estimators, the finite sample performance of the estimator is significantly improved. As binary responses often appear as endogenous regressors in continuous outcome equations, we also develop an optimal instrumental variables estimator in this context. For this purpose, we specialize the double‐index model for binary response to one with heteroscedasticity that depends on an index different from that underlying the ‘mean response’. We show that such (multiplicative) heteroscedasticity, whose form is not parametrically specified, effectively induces exclusion restrictions on the outcomes equation. The estimator developed exploits such identifying information. We provide simulation evidence on the favorable performance of the estimators and illustrate their use through an empirical application on the determinants, and affect, of attendance at a government‐financed school. Copyright © 2009 John Wiley & Sons, Ltd.     

On the variance of the ratio estimator

Summary In an earlier paper [Rao 1966] an exact expression for the variance of the ratio estimator under theMidzuno-Sen sampling scheme is obtained and here we study some of the interesting properties of the coefficients involved in this expression which depend on the auxiliary information. Use of these coefficients is made of in finding out an exact expression for the Bias and Mean Square Error of the ratio estimator under Simple Random Sampling With-Out Replacement (SRSWOR) scheme.     

Distribution free estimation of heteroskedastic binary response models using Probit/Logit criterion functions

In this paper estimators for distribution free heteroskedastic binary response models are proposed. The estimation procedures are based on relationships between distribution free models with a conditional median restriction and parametric models (such as Probit/Logit) exhibiting (multiplicative) heteroskedasticity. The first proposed estimator is based on the observational equivalence between the two models, and is a semiparametric sieve estimator (see, e.g. Gallant and Nychka (1987), Ai and Chen (2003) and Chen et al. (2005)) for the regression coefficients, based on maximizing standard Logit/Probit criterion functions, such as NLLS and MLE. This procedure has the advantage that choice probabilities and regression coefficients are estimated simultaneously. The second proposed procedure is based on the equivalence between existing semiparametric estimators for the conditional median model (,  and ) and the standard parametric (Probit/Logit) NLLS estimator. This estimator has the advantage of being implementable with standard software packages such as Stata. Distribution theory is developed for both estimators and a Monte Carlo study indicates they both perform well in finite samples.     

Inference in semiparametric binary response models with interval data

This paper studies the semiparametric binary response model with interval data investigated by Manski and Tamer (2002). In this partially identified model, we propose a new estimator based on MT’s modified maximum score (MMS) method by introducing density weights to the objective function, which allows us to develop asymptotic properties of the proposed set estimator for inference. We show that the density-weighted MMS estimator converges at a nearly cube-root-n rate. We propose an asymptotically valid inference procedure for the identified region based on subsampling. Monte Carlo experiments provide supports to our inference procedure.     

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.     

Combining kernel estimators in the uniform deconvolution problem

We construct a density estimator and an estimator of the distribution function in the uniform deconvolution model. The estimators are based on inversion formulas and kernel estimators of the density of the observations and its derivative. Initially the inversions yield two different estimators of the density and two estimators of the distribution function. We construct asymptotically optimal convex combinations of these two estimators. We also derive pointwise asymptotic normality of the resulting estimators, the pointwise asymptotic biases and an expansion of the mean integrated squared error of the density estimator. It turns out that the pointwise limit distribution of the density estimator is the same as the pointwise limit distribution of the density estimator introduced by Groeneboom and Jongbloed (Neerlandica, 57, 2003, 136), a kernel smoothed nonparametric maximum likelihood estimator of the distribution function.     

Control variate method for stationary processes

The sample mean is one of the most natural estimators of the population mean based on independent identically distributed sample. However, if some control variate is available, it is known that the control variate method reduces the variance of the sample mean. The control variate method often assumes that the variable of interest and the control variable are i.i.d. Here we assume that these variables are stationary processes with spectral density matrices, i.e. dependent. Then we propose an estimator of the mean of the stationary process of interest by using control variate method based on nonparametric spectral estimator. It is shown that this estimator improves the sample mean in the sense of mean square error. Also this analysis is extended to the case when the mean dynamics is of the form of regression. Then we propose a control variate estimator for the regression coefficients which improves the least squares estimator (LSE). Numerical studies will be given to see how our estimator improves the LSE.