Estimation of variance in multi-stage sampling

In multi-stage sampling when selection is without replacement at the first stage, estimation of the variance of the estimate of the population total is often done assuming sampling with replacement. This estimate is biased and the degree of bias is not negligible. In this paper, a procedure which gives unbiased estimates of the variance making use of only estimated primary sampling unit totals is suggested for the case when sampling at the second and subsequent stages is simple random without replacement. This procedure is based on sub-samples drawn from the selected second and subsequent stage units.     

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     

Optimal allocation in J-stage sampling with stratification from finite population

Summary  The known results of optimal allocation in two-stage simple random sampling with stratification from finite population are generalized for J-stages. The optimalization is treated by minimizing the cost function for given variance as well as by minimizing the variance for given cost. It turns out that for both cases the optimal values differ only for the first-stage units and are the same for all the subsequent. The general formula for minimum variance of the unbiased estimator of the population mean per element is given.     

Optimal allocation in J-stage sampling with stratification from finite population

Summary The known results of optimal allocation in two-stage simple random sampling with stratification from finite population are generalized for J-stages. The optimalization is treated by minimizing the cost function for given variance as well as by minimizing the variance for given cost. It turns out that for both cases the optimal values differ only for the first-stage units and are the same for all the subsequent. The general formula for minimum variance of the unbiased estimator of the population mean per element is given.     

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     

On the efficiency of systematic sampling in ratio and product estimation

Summary In this paper the possibility of gain in efficiency in systematic sampling as compared to simple random sampling has been considered when a ratio or product estimator is used to improve upon the conventional unbiased estimator. The expression for the variance of the estimators are derived for multistage design where systematic selection is used at the ultimate-stage with any probability scheme at the previous stages. In particular the results for the uni-stage systematic sampling and for two-stage sampling with systematic selection at the second-stage have been obtained in section 3.     

On the generalization of Rao,Hartley and Cochran's scheme

Summary The problem considered in this paper is a generalization of the usual Rao, Hartley and Cochran (RHC) scheme. In the usual RHC scheme the population ofN units is randomly divided inton groups wheren is the size of the sample. In this paper we propose to divide the population under consideration into (n+k) random groups wherek is some positive integer. Then a sample ofn groups is selected by using simple random sampling without replacement (SRSWOR). The expressions for the unbiased estimator of population total, its variance and the unbiased estimate of variance have been obtained under the proposed sheme. The condition under which the proposed sheme is more efficient than the usual RHC scheme has also been investigated.     

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     

Two-stage rank estimation of quantile index models

This paper estimates a class of models which satisfy a monotonicity condition on the conditional quantile function of the response variable. This class includes as a special case the monotonic transformation model with the error term satisfying a conditional quantile restriction, thus allowing for very general forms of conditional heteroscedasticity. A two-stage approach is adopted to estimate the relevant parameters. In the first stage the conditional quantile function is estimated nonparametrically by the local polynomial estimator discussed in Chaudhuri (Journal of Multivariate Analysis 39 (1991a) 246–269; Annals of Statistics 19 (1991b) 760–777) and Cavanagh (1996, Preprint). In the second stage, the monotonicity of the quantile function is exploited to estimate the parameters of interest by maximizing a rank-based objective function. The proposed estimator is shown to have desirable asymptotic properties and can then also be used for dimensionality reduction or to estimate the unknown structural function in the context of a transformation model.     

带未知异方差广义空间模型的有效估计

空间单元大小以及其它的经济特征上的差异,常常会导致空间异方差问题。本文给出了广义空间模型异方差问题的三种不同估计方法。第一种方法是将异方差形式参数化,来克服自由度的不足,使用ML估计进行实现。而针对异方差形式未知时,分别采用了基于2SLS的迭代GMM估计和更加直接的MCMC抽样方法加以解决,特别是MCMC方法表现得更加优美。蒙特卡罗模拟表明,给定异方差形式条件下, ML估计通过异方差参数化的方法依然可以获得较好的估计效果。而异方差形式未知的情况下,另外两种方法随着样本数的增大时也可以与ML的估计结果趋于一致。     

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.     

The (mis)specification of discrete duration models with unobserved heterogeneity: A Monte Carlo study

Empirical researchers usually prefer statistical models that can be easily estimated with the help of commonly available software packages. Sequential binary models with or without normal random effects are an example of such models that can be adopted to estimate discrete duration models with unobserved heterogeneity. But an easy-to-implement estimation may incur a cost. In this paper we conduct a Monte Carlo simulation to evaluate the consequences of omitting or misspecifying the unobserved heterogeneity distribution in single-spell discrete duration models.     

Detecting shocks: Outliers and breaks in time series

A single outlier in a regression model can be detected by the effect of its deletion on the residual sum of squares. An equivalent procedure is the simple intervention in which an extra parameter is added for the mean of the observation in question. Similarly, for unobserved components or structural time-series models, the effect of elaborations of the model on inferences can be investigated by the use of interventions involving a single parameter, such as trend or level changes. Because such time-series models contain more than one variance, the effect of the intervention is measured by the change in individual variances.We examine the effect on the estimated parameters of moving various kinds of intervention along the series. The horrendous computational problems involved are overcome by the use of score statistics combined with recent developments in filtering and smoothing. Interpretation of the resulting time-series plots of diagnostics is aided by simulation envelopes.Our procedures, illustrated with four example, permit keen insights into the fragility of inferences to specific shocks, such as outliers and level breaks. Although the emphasis is mostly on parameter estimation, forecast are also considered. Possible extensions include seasonal adjustment and detrending of series.     

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.     

Interpolation and optimal linear prediction

This paper is concerned with the interpolation of spatially distributed observations of a quantitative phenomenon, sometimes referred to as kriging. This activity can be understood as a prediction procedure for values of random functions under stationarity assumptions in a polynomial linear regression context. After a heuristic and an exact derivation of the best linear unbiased prediction procedure (and the variance of prediction error) if the covariance function relating covariance between two possible observations to their mutual distance is known, follows the introduction of weaker assumptions admitting the definition of the variance only for increments of a certain order by a pseudoco–variance function. A particular related case is the so–called semivariogram for increments of order one. The prediction procedure turns out to be similar to that in the previous situation. The weaker assumptions allow an unbiased estimation of the unknown pseudocovahance function of polynomial form under restrictions imposed by Fourier transformation. Extension from point–wise observations or predictions to area or volume averages is touched upon.     

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.     

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.     

Optimum allocation in stratified sampling with replacement

In a stratified sample, when sampling is done with replacement in each stratum a better estimate of the population mean can be achieved by considering the distinct units only. An explicit expression for the variance for the mean, of a stratified sample based on the distinct units only, is obtained. Then the optimum allocation for the different stratum are obtained by minimizing this variance subject to (i) total sample size being fixed, or (ii) the expected number of distinct units being fixed. Neyman’s solutions are obtained as special cases. The solutions finally arrived at are algebraically complex, hence, numerical methods are applied. In all examples, the variance of the estimates obtained by this method are smaller than the variances obtained by Neyman’s allocation. A part of this work was supported by the Office of the Ordinance Research; U.S.A. Grant (DA-AROL(D)-31-124-G83) when the author was at University of California, Berkeley.     

The effects of sampling on measures of association between variables based on sample means

Many studies that involve people's perceptions or behaviors focus on aggregate rather than individual responses. For example, variables describing public perceptions for some set of events may be represented as mean scores for each event. Event mean scores then become the unit of analysis for each variable. The variance of these mean scores for a variable is not only a function of the variation among the events themselves, but is also due to the variation among respondents and their possible responses. This is also the case for the covariances between variables based on event mean scores. In many contexts the variance and covariance components attributable to the sampling of respondents and their responses may be large; these components can be described as measurement error. In this paper we show how to estimate variances and covariances of aggregate variables that are free of these sources of measurement error. We also present a measure of reliability for the event means and examine the effect of the number of respondents on these spurious components. To illustrate how these estimates are computed, forty-two respondents were asked to rate forty events on seven risk perception variables. Computing the variances and covariances for these variables based on event means resulted in relatively large components attributable to measurement error. A demonstration is given of how this error is removed and the resulting effect on our estimates.     

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.