Weighting and estimation

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All (638)

All (638) (0 to 10 of 638 results)

1. Exact and approximation formulas for second-order inclusion probabilities in randomized systematic sampling with unequal probabilities and without replacement
Articles and reports: 12-001-X202600100003
Description: Probability-proportional-to-size sampling is widely used by national statistical offices. Here population units are selected with probabilities proportional to an auxiliary variable. Variance formulas in such designs require both first- and second-order inclusion probabilities. The computation of second-order inclusion probabilities is particularly challenging for large populations, and has been the subject of extensive research. This article presents some new exact and approximation formulas for second-order inclusion probabilities in randomized systematic sampling with unequal probabilities and without replacement.
Release date: 2026-06-29
2. Improving the coverage of confidence intervals with respect to degrees of freedom: Application to the Canadian Census
Articles and reports: 12-001-X202600100005
Description: Confidence intervals are very often constructed based on a probability distribution that uses a certain number of degrees of freedom as a parameter. This is the case with the Student and the modified Wilson confidence intervals, discussed in this article, which use quantiles from the Student distribution where the number of degrees of freedom is generally unknown. For the length of a confidence interval to be representative of the reliability of an estimate, the actual coverage rate must match the nominal rate. To that end, the number of degrees of freedom in the probability distribution used in practice to calculate the confidence interval must be estimated as precisely as possible. An approximate rule is often used, although it tends to overestimate the actual number of degrees of freedom. In this article, a more precise version of degrees of freedom, derived from the Satterthwaite approximation, is obtained in the context of the Canadian Census of Population. The sampling design is equivalent to a simple random design without replacement, cluster-stratified, and the variance estimation method is an adaptation of the balanced repeated replication method. An explicit expression of the degrees of freedom is obtained under these conditions, enabling the factors influencing them to be identified. For comparison, the degree of freedom formula is also established for the conventional variance estimator. A simulation study shows that using this version of degrees of freedom corrects the undercoverage problem observed with the approximate rule, showing the importance of accurately assessing this number.
Release date: 2026-06-29
3. The inverse-variance trap: A simple ratio fix for combining skewed data
Articles and reports: 12-001-X202600100009
Description: Combining estimates from independent surveys via inverse-variance weights can lead to negative bias when unknown variances are estimated and the target variable is non-negative and positively skewed. In such cases, strong positive correlations typically arise between the estimators and their corresponding variance estimators, causing standard linear combinations with inverse-variance weights to exhibit negative bias. We introduce a strikingly simple method to reduce bias: replace the standard weight with the ratio of the estimator to the variance estimator. Under a linear model linking the two, we show that the new ratio-weighted estimator is approximately unbiased, whereas the conventional inverse-variance combination exhibits downward bias. Through simulations, we demonstrate that the new method brings both the bias and the mean squared error closer to the optimum for a wide range of different target variables. As our method uses only standardly reported summary statistics, it can be immediately adopted to reduce this widespread bias and improve the reliability of scientific findings in various fields.
Release date: 2026-06-29
4. Variance of the generalized regression estimator under measurement error
Articles and reports: 12-001-X202600100010
Description: With the exception of two-phase sampling, the standard variance approximation of the generalized regression (GREG) estimator assumes that the population totals in the weighting scheme are observed without error. If the weighting model of the GREG estimator contains population totals that are observed with measurement error sources other than the sampling error of first-phase estimates, then this uncertainty will be ignored by the variance approximation of the GREG estimator. This paper proposes a variance approximation for the GREG estimator that accounts for additional uncertainty arising from measurement error in one or more of the population totals used in the weighting scheme. This approach has been developed for, and is being applied to, the Dutch Labour Force Survey (DLFS). The monthly publications of the DLFS are obtained with a time series model, which corrects for rotation group bias and discontinuities caused by major redesigns and the loss of face-to-face interviews during COVID-19. The GREG estimates for the quarterly figures are benchmarked to the average of the monthly publications to enforce numerical consistency between monthly and quarterly publication tables. The standard variance approximation of the GREG estimator assumes that these population totals are observed without error. This results in an underestimation of the variance of the GREG estimator. The variance approximation proposed in this paper results in more realistic standard errors for the quarterly GREG estimates.
Release date: 2026-06-29
5. Bayesian differential privacy for small counties and individual commodities
Articles and reports: 12-001-X202600100011
Description: We construct a hybrid Bayesian method, which includes a differentially private mechanism, to mask Census county totals for a U.S. state on acreage of a commodity. We use surrogates for data collected at the farm level from a past U.S. Census of Agriculture to illustrate our procedure. We use two Bayesian small area models (parametric and mixture) to accommodate the smaller counties with fewer farms and some counties with large acres. In these models, the Laplace distribution provides a differentially private mechanism. In pre-processing, we also incorporate the Census weights to form the observed total acreage, a scaling factor to the Laplace mechanism for each county, a square-root transformation of the observed total acreage to avoid negative masked estimates especially for small counties, and the p-percent rule and the 3+ rule to partition the counties into suppressed counties, non-sensitive counties and sensitive counties. Because of difficulties in specifying and tuning the privacy budget (an unknown parameter), to balance security and utility, we specify a prior for the privacy budget, where the values are not specified, and the Gibbs sampler is used to fit the hierarchical Bayesian models. In post-processing, we use Bayesian predictive inference to obtain masked county acreages, and this includes a benchmarking so that the masked state total matches the observed state total. As a measure of reliability of the Bayesian procedure, we use the posterior coefficients of variation for the masked posterior means of the counties. As a measure of utility, we use the absolute relative errors for the individual counties, together with other global measures. For the sensitive counties, there are some differences between the two small area models but both are much better than an individual area model; the mixture model being the best compromise for security and utility.
Release date: 2026-06-29
6. Small area estimation of general indicators in off-census years
Articles and reports: 12-001-X202600100012
Description: We propose small area estimators of general indicators in off-census years, which avoid the use of deprecated census microdata, but are nearly optimal in census years. The procedure is based on replacing the obsolete census file with a larger unit-level survey that adequately covers the areas of interest and contains the values of useful auxiliary variables. However, the minimal data requirement of the proposed method is a single survey with microdata on the target variable and suitable auxiliary variables for the period of interest. We also develop an estimator of the mean squared error (MSE) that accounts for the uncertainty introduced by the large survey used to replace the census of auxiliary information. Our empirical results indicate that the proposed predictors perform clearly better than the alternative predictors when census data are outdated, and are very close to optimal ones when census data are correct. They also illustrate that the proposed total MSE estimator corrects for the bias of purely model-based MSE estimators that do not account for the large survey uncertainty.
Release date: 2026-06-29
7. A Framework for Projecting Housing Use Associated with Immigrants and Non-Permanent Residents
Surveys and statistical programs – Documentation: 11-633-X2026002
Description: Recent changes in Canada’s immigration levels have heightened interest in understanding how immigration affects housing demand. This article develops a methodological framework for projecting housing use associated with permanent residents (PRs) and non-permanent residents (NPRs) under alternative immigration scenarios. The framework applies observed per capita housing use rates from the Census of Population to estimate incremental housing use by tenure over time.
Release date: 2026-04-24
8. Effects of model misspecification on small area estimators
Articles and reports: 12-001-X202500200001
Description: Nested error regression models are commonly used to incorporate unit specific auxiliary variables to improve small area estimates. When the mean structure of the model is misspecified, the design-based mean squared prediction error (MSPE) of Empirical Best Linear Unbiased Predictors (EBLUP) generally increases. The Observed Best Prediction (OBP) method has been proposed with the intent to improve on the design-based MSPE over EBLUP. In this paper, we conduct a Monte Carlo simulation experiments to understand the effect of misspsecification of mean structures on different small area estimators. Our findings suggest that the OBP using unit-level auxiliary variables does not outperform the EBLUP in terms of design-based MSPE, unless the number of small areas m is extremely large. Conversely, the performance of OBP significantly improves when area-level auxiliary variables are employed. This paper includes both analytical and numerical evidence to demonstrate these observations, providing practical insights for addressing model misspecification in small area estimation (SAE).
Release date: 2025-12-23
9. Estimating monthly indicators for Consumer Confidence using Structural Time Series Models
Articles and reports: 12-001-X202500200003
Description: In this paper a model-based inference procedure based on a multivariate structural time series model is developed for the production of monthly figures about consumer confidence. The input for the model are five series of direct estimates for the indices that measure consumer confidence, which are derived from the Dutch Consumer Survey. The model improves the accuracy of the direct estimates, since it provides a better separation of measurement errors and sampling errors from estimated target parameters. The standard errors for the month-to-month changes are clearly smaller under the time series model. A second problem addressed in this paper is related to the transition to a new survey process in 2017. Structural time series models in combination with a parallel run are applied to estimate discontinuities induced by the redesign. An algorithm designed for the consumer confidence variables is developed to construct uninterrupted input series for the aforementioned structural time series model. This inference method facilitated a smooth transition to a new survey design and resulted in uninterrupted series about consumer confidence that date back to 1986. The method is implemented for the production of official monthly figures on consumer confidence in the Netherlands.
Release date: 2025-12-23
10. Quantile balancing inverse probability weighting for non-probability samples
Articles and reports: 12-001-X202500200005
Description: The use of non-probability data sources for statistical purposes and for official statistics has become increasingly popular in recent years. However, statistical inference based on non-probability samples is made more difficult by nature of their biasedness and lack of representativity. In this paper we propose quantile balancing inverse probability weighting estimator (QBIPW) for non-probability samples. We apply the idea of Harms and Duchesne (2006) allowing the use of quantile information in the estimation process to reproduce known totals and the distribution of auxiliary variables. We discuss the estimation of the QBIPW probabilities and its variance. Our simulation study has demonstrated that the proposed estimators are robust against model mis-specification and, as a result, help to reduce bias and mean squared error. Finally, we applied the proposed methods to estimate the share of job vacancies aimed at Ukrainian workers in Poland using an integrated set of administrative and survey data about job vacancies.
Release date: 2025-12-23

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Analysis (610)

Analysis (610) (600 to 610 of 610 results)

601. Some estimators of population totals from simple random samples containing large units Archived
Articles and reports: 12-001-X197700100006
Description: The problem considered is the estimation of population total of some characteristic from a simple random sample containing a few large or extreme observations. The effect of these large units in the sample is to distort the estimate of the population total. It is therefore important to correct the weights for such units or deflate their values at the estimation stage once they have been sampled and identified as unusually large units. In this paper, three estimators which alter the usual sampling weights have been considered. The efficiencies of these estimators have been worked out in terms of the ratio of the variance of the usual estimator of the population total to the mean square error of these estimators. An empirical study of these estimators is also discussed.
Release date: 1977-06-20
602. Stratification index: Methodology and analysis Archived
Articles and reports: 12-001-X197600200002
Description: To obtain estimates of means or totals for a universe, a sample of units is often drawn to represent the universe and these units are then surveyed. One of the most important procedures used in the selection of the units is that of stratification, whereby the universe is split up into strata and independent samples of units are drawn from each stratum. A stratification index is developed to indicate the approximate fractional reduction in the sampling variance from that which would result if no stratification were undertaken. Also the methodology is extended to examine the effect of stratification on the sampling variance at different levels of stratification through the concept of a summary index. The stratification index is also extended to the case of ratio estimates using independent source data to re-weight the sample data. The index has been applied to the Canadian Labour Force Survey (LFS), a typical multi-stage stratified sample where ratio estimation, using projected age-sex population estimates is applied and empirical data are presented and analyzed.
Release date: 1976-12-13
603. The estimation of total variance in the 1976 Census Archived
Articles and reports: 12-001-X197600200004
Description: Published reports for the 1976 Census will include estimates of Total Variance as indicators of the reliability of the figures in these reports. In order to obtain these estimates of Total Variance, an Interpenetrating Design Experiment was incorporated into the collection methods for a sample of enumeration areas. In this paper we derive the formula for Total Variance in terms of variances due to sampling, correlated response and simple response. We then show how the Total Variance, and its components, can be estimated from the design and we give the estimators that will be used for the 1976 Census. The estimates of sampling and correlated response variance are unbiased but the simple response variance estimate is not.
Release date: 1976-12-13
604. Raking ratio estimators Archived
Articles and reports: 12-001-X197600100003
Description: This paper presents large sample results for the bias and variance of raking-ratio estimators for up to four iterations. Estimators of the bias and variance are also presented. An expression for the asymptotic covariance matrix of the maximum likelihood estimators of the cell proportions in a two-way table with known marginals is also given.
Release date: 1976-06-14
605. On the improvement of sample survey estimates Archived
Articles and reports: 12-001-X197500254830
Description: This paper focuses on the improvement of sample survey estimates in the particular situation where the survey sample, or part of it, is included in a larger sample from which auxiliary information is available. The properties of a method of estimation - sometimes applied in specific circumstances - are investigated and the limitations of its application are found. The application of the method to rotation designs in continuing surveys is more closely studied in the context of composite estimation.
Release date: 1975-12-15
606. On a ratio estimate with post-stratified weighting Archived
Articles and reports: 12-001-X197500254832
Description: A ratio estimate based on an auxiliary variable is considered for the case when the sample is post-stratified using information on another auxiliary variable. The variance of the ratio estimate is derived by the method of linearization [3,4]. An application to subprovincial estimation in the Canadian Labour Force Survey is discussed.
Release date: 1975-12-15
607. Measurement of response errors in Censuses and sample surveys Archived
Articles and reports: 12-001-X197500254824
Description:
Madow [1968] has proposed a two-phase sampling scheme under which response bias can be eliminated from sample surveys by obtaining “true” values for a subsample of the original sample. Often in cases of Censuses or ongoing surveys, the subsample data are not used to correct the main survey estimates but to assess their reliability. The main purpose of this paper is to present methods by which reliability estimates can be obtained when true values can be determined for a subsample of units.
Release date: 1975-12-15
608. The development of an automated estimation system Archived
Articles and reports: 12-001-X197500100001
Description: Although a survey is designed to satisfy a specific set of survey constraints, some steps involved in designing a survey, such as stratification, sample allocation and sample selection are common to all surveys. The steps involved in the creation of survey design systems are to identify, develop and implement common methods and procedures for such stages which, when taken together, constitute a survey design. The paper describes some methodological considerations in the development of an automated system for three methods of ratio estimation.
Release date: 1975-06-16
609. Some estimators for domain totals Archived
Articles and reports: 12-001-X197500100004
Description: A major concern in large scale surveys is the problem of sub-population estimation (domain estimation). This paper presents a study of four estimators for estimating domain totals. The domain considered in the study is an area type of domain, that is, a domain consisting of a combination of a certain number of area units belonging to different strata. This paper uses some actual data and some fictitious data to compare variances and mean square errors of the four estimators.
Release date: 1975-06-16
610. Components of variance model in multi-stage stratified samples Archived
Articles and reports: 12-001-X197500100007
Description: There are several multi-stage sample designs in various countries, such as the Current Population Survey in U.S.A., Labour Survey in Sweden, and the General Household Survey in United Kingdom. From each survey, estimated totals of Employed, Unemployed, and other characteristics may be obtained.
The Canadian Labour Force Survey is a monthly household survey in which the dwelling is the ultimate unit of sampling requiring two to four stages of selection. Each province is split up into strata and sampling units at various stages so that the sampling variance contains up to four components of variance whose actual formulae and estimation formulae are derived, utilizing those formerly derived by Yates and Grundy [12]. Ratio estimation is employed and the formulas are modified accordingly. To analyze the components of variance, it is necessary to express them in terms of components of sampling ratios and the sizes of sampling units at the various stages at provincial and national levels and approximate variance functions are thus derived.
Release date: 1975-06-16

Reference (28)

Reference (28) (10 to 20 of 28 results)

11. The Effects of the Revised Estimation Methodology on Estimates from Household Expenditure Surveys Archived
Surveys and statistical programs – Documentation: 62F0026M2005002
Description:
This document will provide an overview of the differences between the old and the new weighting methodologies and the effect of the new weighting system on estimations.
Release date: 2005-06-30
12. Variance estimation with plausible value achievement data: Two STATA programs for use with the YITS/PISA data Archived
Surveys and statistical programs – Documentation: 12-002-X20040016891
Description:
These two programs are designed to estimate variability due to measurement error beyond the sampling variance introduced by the survey design in the Youth in Transition Survey / Programme of International Student Assessment (YITS/PISA). Program code is included in an appendix.
Release date: 2004-04-15
13. Chain Fisher Volume Index Methodology Archived
Surveys and statistical programs – Documentation: 13-604-M2003042
Description:
On May 31, 2001, the quarterly income and expenditure accounts adopted the Chain Fisher Index formula, chained quarterly, as the official measure of real gross domestic product (GDP) in terms of expenditures. This formula was also adopted for the Provincial Accounts on October 31, 2002.
There were two reasons for adopting this formula: to provide users with a more accurate measure of real GDP growth between two consecutive periods and to make the Canadian measure comparable with the Income and Product Accounts of the United States, which has used the Chain Fisher Index formula since 1996 to measure real GDP.
Release date: 2003-11-06
14. Improvements in 2000 to the Labour Force Survey (LFS) Archived
Surveys and statistical programs – Documentation: 71F0031X2000001
Description:
This paper introduces and explains modifications made to the Labour Force Survey estimates in January 2000. Some of these modifications include the adjustment of all LFS estimates to reflect population counts based on the 1996 Census plus the implementation of a new estimation methodology called composite estimation. This new method results in more efficient estimates of month to month change, while improving the quality of monthly level estimates.
Release date: 2001-06-29
15. A donor imputation system to create a census database fully adjusted for underenumeration Archived
Surveys and statistical programs – Documentation: 11-522-X19990015668
Description:
Following the problems with estimating underenumeration in the 1991 Census of England and Wales the aim for the 2001 Census is to create a database that is fully adjusted to net underenumeration. To achieve this, the paper investigates weighted donor imputation methodology that utilises information from both the census and census coverage survey (CCS). The US Census Bureau has considered a similar approach for their 2000 Census (see Isaki et al 1998). The proposed procedure distinguishes between individuals who are not counted by the census because their household is missed and those who are missed in counted households. Census data is linked to data from the CCS. Multinomial logistic regression is used to estimate the probabilities that households are missed by the census and the probabilities that individuals are missed in counted households. Household and individual coverage weights are constructed from the estimated probabilities and these feed into the donor imputation procedure.
Release date: 2000-03-02
16. Fusion of data and estimation by entropy maximization Archived
Surveys and statistical programs – Documentation: 11-522-X19990015672
Description:
Data fusion as discussed here means to create a set of data on not jointly observed variables from two different sources. Suppose for instance that observations are available for (X,Z) on a set of individuals and for (Y,Z) on a different set of individuals. Each of X, Y and Z may be a vector variable. The main purpose is to gain insight into the joint distribution of (X,Y) using Z as a so-called matching variable. At first however, it is attempted to recover as much information as possible on the joint distribution of (X,Y,Z) from the distinct sets of data. Such fusions can only be done at the cost of implementing some distributional properties for the fused data. These are conditional independencies given the matching variables. Fused data are typically discussed from the point of view of how appropriate this underlying assumption is. Here we give a different perspective. We formulate the problem as follows: how can distributions be estimated in situations when only observations from certain marginal distributions are available. It can be solved by applying the maximum entropy criterium. We show in particular that data created by fusing different sources can be interpreted as a special case of this situation. Thus, we derive the needed assumption of conditional independence as a consequence of the type of data available.
Release date: 2000-03-02
17. Spatial statistics and environmental epidemiology using routine data Archived
Surveys and statistical programs – Documentation: 11-522-X19990015674
Description:
The effect of the environment on health is of increasing concern, in particular the effects of the release of industrial pollutants into the air, the ground and into water. An assessment of the risks to public health of any particular pollution source is often made using the routine health, demographic and environmental data collected by government agencies. These datasets have important differences in sampling geography and in sampling epochs which affect the epidemiological analyses which draw them together. In the UK, health events are recorded for individuals, giving cause codes, a data of diagnosis or death, and using the unit postcode as a geographical reference. In contrast, small area demographic data are recorded only at the decennial census, and released as area level data in areas distinct from postcode geography. Environmental exposure data may be available at yet another resolution, depending on the type of exposure and the source of the measurements.
Release date: 2000-03-02
18. Estimation using the generalized weight share method: The case of record linkage Archived
Surveys and statistical programs – Documentation: 11-522-X19990015680
Description:
To augment the amount of available information, data from different sources are increasingly being combined. These databases are often combined using record linkage methods. When there is no unique identifier, a probabilistic linkage is used. In that case, a record on a first file is associated with a probability that is linked to a record on a second file, and then a decision is taken on whether a possible link is a true link or not. This usually requires a non-negligible amount of manual resolution. It might then be legitimate to evaluate if manual resolution can be reduced or even eliminated. This issue is addressed in this paper where one tries to produce an estimate of a total (or a mean) of one population, when using a sample selected from another population linked somehow to the first population. In other words, having two populations linked through probabilistic record linkage, we try to avoid any decision concerning the validity of links and still be able to produce an unbiased estimate for a total of the one of two populations. To achieve this goal, we suggest the use of the Generalised Weight Share Method (GWSM) described by Lavallée (1995).
Release date: 2000-03-02
19. Simultaneous calibration of several surveys Archived
Surveys and statistical programs – Documentation: 11-522-X19990015684
Description:
Often, the same information is gathered almost simultaneously for several different surveys. In France, this practice is institutionalized for household surveys that have a common set of demographic variables, i.e., employment, residence and income. These variables are important co-factors for the variables of interest in each survey, and if used carefully, can reinforce the estimates derived from each survey. Techniques for calibrating uncertain data can apply naturally in this context. This involves finding the best unbiased estimator in common variables and calibrating each survey based on that estimator. The estimator thus obtained in each survey is always a linear estimator, the weightings of which can be easily explained and the variance can be obtained with no new problems, as can the variance estimate. To supplement the list of regression estimators, this technique can also be seen as a ridge-regression estimator, or as a Bayesian-regression estimator.
Release date: 2000-03-02
20. A method of generating a sample of artificial data from several existing tables: Application based on the residential electric power market Archived
Surveys and statistical programs – Documentation: 11-522-X19990015690
Description:
The artificial sample was generated in two steps. The first step, based on a master panel, was a Multiple Correspondence Analysis (MCA) carried out on basic variables. Then, "dummy" individuals were generated randomly using the distribution of each "significant" factor in the analysis. Finally, for each individual, a value was generated for each basic variable most closely linked to one of the previous factors. This method ensured that sets of variables were drawn independently. The second step consisted in grafting some other data bases, based on certain property requirements. A variable was generated to be added on the basis of its estimated distribution, using a generalized linear model for common variables and those already added. The same procedure was then used to graft the other samples. This method was applied to the generation of an artificial sample taken from two surveys. The artificial sample that was generated was validated using sample comparison testing. The results were positive, demonstrating the feasibility of this method.
Release date: 2000-03-02

Date modified:: 2026-09-12