Mixed random systematic sampling designs

In this paper, an alternative sampling procedure that is a mixture of simple random sampling and systematic sampling is proposed. It results in uniform inclusion probabilities for all individual units and positive inclusion probabilities for all pairs of units. As a result, the proposed sampling procedure enables us to estimate the population mean unbiasedly using the ordinary sample mean, and to provide an unbiased estimator of its sampling variance. It is also found that the suggested sampling procedure performs well especially when the size of simple random sample is small. Received August 2001     

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.     

An optimum sampling design to base HT-method of estimating a finite population total

The problem of constructing a sampling design with the value of the sum of second order inclusion probabilities attaining its lower bound for non-integral values of the expected effective size of a sample in the design has been considered in this paper. If the values of the characteristic of interest on all the units in the population are non-negative the design is admissible (in the sense of variance) with respect to Horvitz-Thompson estimator in the class of designs with the same set of values of the first order inclusion probabilities of the units. Again such a design is best to use Horvitz-Thompson estimator of population total in the sense of smallest average variance of the estimator under a special superpopulatio model.     

Small sample properties of the mixed regression estimator

In this paper, the small sample properties of the mixed regression estimator are examined when prior information may be biased and when the ration of the variance of the prior restriction errors to the variance of the sample errors is unknown. The mean square error of the mixed regression estimator is derived, and it is shown that the mixed regression estimator gets dominated by the ordinary least squares estimator in terms of the mean square error as the bias of prior information gets larger.     

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.     

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.     

Asymptotics and smoothing parameter selection for penalized spline regression with various loss functions

Penalized splines are used in various types of regression analyses, including non‐parametric quantile, robust and the usual mean regression. In this paper, we focus on the penalized spline estimator with general convex loss functions. By specifying the loss function, we can obtain the mean estimator, quantile estimator and robust estimator. We will first study the asymptotic properties of penalized splines. Specifically, we will show the asymptotic bias and variance as well as the asymptotic normality of the estimator. Next, we will discuss smoothing parameter selection for the minimization of the mean integrated squares error. The new smoothing parameter can be expressed uniquely using the asymptotic bias and variance of the penalized spline estimator. To validate the new smoothing parameter selection method, we will provide a simulation. The simulation results show that the consistency of the estimator with the proposed smoothing parameter selection method can be confirmed and that the proposed estimator has better behavior than the estimator with generalized approximate cross‐validation. A real data example is also addressed.     

Two estimators of the long-run variance: Beyond short memory

This paper deals with the estimation of the long-run variance of a stationary sequence. We extend the usual Bartlett-kernel heteroskedasticity and autocorrelation consistent (HAC) estimator to deal with long memory and antipersistence. We then derive asymptotic expansions for this estimator and the memory and autocorrelation consistent (MAC) estimator introduced by Robinson [Robinson, P. M., 2005. Robust covariance matrix estimation: HAC estimates with long memory/antipersistence correction. Econometric Theory 21, 171–180]. We offer a theoretical explanation for the sensitivity of HAC to the bandwidth choice, a feature which has been observed in the special case of short memory. Using these analytical results, we determine the MSE-optimal bandwidth rates for each estimator. We analyze by simulations the finite-sample performance of HAC and MAC estimators, and the coverage probabilities for the studentized sample mean, giving practical recommendations for the choice of bandwidths.     

On the exact variance of an unbiased product type estimator and its comparison with the sample mean

We consider the usual product estimator adjusted for its bias, considered by Robson (1957). The exact variance of this estimator has been obtained through a direct method. Its superiority over the sample mean and the product estimator is demonstrated. The research of these authors has been partially supported by their respective NSERC grants A3661 and A3989.     

A finite sample correction for the variance of linear efficient two-step GMM estimators

Monte Carlo studies have shown that estimated asymptotic standard errors of the efficient two-step generalized method of moments (GMM) estimator can be severely downward biased in small samples. The weight matrix used in the calculation of the efficient two-step GMM estimator is based on initial consistent parameter estimates. In this paper it is shown that the extra variation due to the presence of these estimated parameters in the weight matrix accounts for much of the difference between the finite sample and the usual asymptotic variance of the two-step GMM estimator, when the moment conditions used are linear in the parameters. This difference can be estimated, resulting in a finite sample corrected estimate of the variance. In a Monte Carlo study of a panel data model it is shown that the corrected variance estimate approximates the finite sample variance well, leading to more accurate inference.     

Effective nonparametric estimation in the case of severely discretized data

Often economic data are discretized or rounded to some extent. This paper proposes a regression and a density estimator that work especially well when discretization causes conventional kernel-based estimators to behave poorly. The estimator proposed here is a weighted average of neighboring frequency estimators, and the weights are composed of cubic B-splines. Interestingly, we show that this estimator can have both a smaller bias and variance than frequency estimators. As a means to obtain asymptotic normality and rates of convergence, we assume that the discreteness becomes finer as the sample size increases.     

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.     

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.     

One-Sided Coverage Intervals for a Proportion Estimated from a Stratified Simple Random Sample

Using an Edgeworth expansion to speed up the asymptotics, we develop one-sided coverage intervals for a proportion based on a stratified simple random sample. To this end, we assume the values of the population units are generated from independent random variables with a common mean within each stratum. These stratum means, in turn, may either be free to vary or are assumed to be equal. The more general assumption is equivalent to a model-free randomization-based framework when finite population correction is ignored. Unlike when an Edgeworth expansion is used to construct one-sided intervals under simple random sampling, it is necessary to estimate the variance of the estimator for the population proportion when the stratum means are allowed to differ. As a result, there may be accuracy gains from replacing the normal  z -score in the Edgeworth expansion with a  t -score.     

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.     

On the Construction of Imputation Classes in Surveys

This paper explores the problem of the construction of imputation classes using the score method, sometimes called predictive mean stratification or response propensity stratification, depending on the context. This method was studied in Thomsen (1973) , Little (1986) and Eltinge & Yansaneh (1997) . We use a different framework to evaluate the properties of the resulting imputed estimator of a population mean. In our framework, we condition on the realized sample. This enables us to considerably simplify our theoretical developments in the frequent situation where the boundaries and the number of classes are sample‐dependent. We find that the key factor for reducing the non‐response bias is to form classes homogeneous with respect to the response probabilities and/or the conditional expectation of the variable of interest. In the latter case, the non‐response/imputation variance is also reduced. Finally, we performed a simulation study to fully evaluate various versions of the score method and to compare them with a cross‐classification method, which is frequently used in practice. The results showed the superiority of the score method in general.     

Sequential estimation of normal mean under asymmetric loss function with a shrinkage stopping rule

The problem of estimating a normal mean with unknown variance is considered under an asymmetric loss function such that the associated risk is bounded from above by a known quantity. In the absence of a fixed sample size rule, a sequential stopping rule and two sequential estimators of the mean are proposed and second-order asymptotic expansions of their risk functions are derived. It is demonstrated that the sample mean becomes asymptotically inadmissible, being dominated by a shrinkage-type estimator. Also a shrinkage factor is incorporated in the stopping rule and similar inadmissibility results are established. Received September 1997     

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.     

The LIML estimator has finite moments!

The Limited Information Maximum Likelihood estimator of the vector of coefficients of a structural equation in a simultaneous equation model is the vector that defines the linear combination maximizing the effect variance relative to the error variance. If this “eigenvector” solution is normalized by setting a designated coefficient equal to 1, the second-order moment of the estimator may be unbounded. However, the second-order moment is finite if the normalization sets the sample error variance of the linear combination equal to 1.     

Asymptotics of SIMEX-based variance estimation

In this paper, we study the asymptotic properties of simulation extrapolation (SIMEX) based variance estimation that was proposed by Wang et al. (J R Stat Soc Series B 71:425–445, 2009). We first investigate the asymptotic normality of the parameter estimator in general parametric variance function and the local linear estimator for nonparametric variance function when permutation SIMEX (PSIMEX) is used. The asymptotic optimal bandwidth selection with respect to approximate mean integrated squared error (AMISE) for nonparametric estimator is also studied. We finally discuss constructing confidence intervals/bands of the parameter/function of interest. Other than applying the asymptotic results so that normal approximation can be used, we recommend a nonparametric Monte Carlo algorithm to avoid estimating the asymptotic variance of estimator. Simulation studies are carried out for illustration.